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Author Message
Ed
PostPosted: Fri Jun 27, 2008 10:55 pm 
Offline
Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Old SSv3.2 scores: 
Killer rating table      
Rounded Score from SSv3.2  
pg# on this thread - PART B
(E) = Easy  (H) = Hard
======================================================================
|A ##       Rate  Score|A ##       Rate  Score|A ##       Rate  Score|
|----------------------+----------------------+----------------------|
|A.58v1.5  1.25    1.10|A.60RP-L E1.75    2.10|A.61              1.50|
|A.59      1.25    1.40|A.50v2   !3.0(t&E)2.95|A.61X     1.50    1.30|
|A.59v1.5          2.40|SKX4v2    4.0      DNF|A.62      0.75    0.85|
|A.60      1.50    1.60|A.61XV3            DNF|A.62v2    2.00    3.80|
|A.60RP    2.5(t&e)2.75|A.68v3    3 !(t&E)5.45|                      |
|====================================================================|
                        (+"Unsolvables" posts)
                              page #3
Old scores SSv3.3.0: 
Rounded Score from SSv3.3.0 
! = 0.10 change from previous Score
pg# on this thread - PART B
(E) = Easy  (H) = Hard
======================================================================
|A ##       Rate  Score|A ##       Rate  Score|A ##       Rate  Score|
|----------------------+----------------------+----------------------|
|A.58v1.5  1.25   !1.20|A.60RP-L E1.75   !2.55|A.61             !1.40|
|A.59      1.25    1.35|A.50v2   3.0(t&E)!5.00|A.61X     1.50   !1.50|
|A.59v1.5  1.75   !2.15|SKX4v2    4.0      DNF|A.62      0.75    0.85|
|A.60      1.50   !1.75|A.61XV3            DNF|A.62v2    2.00   !5.10|
|A.60RP   !2.5(t&e)5.50|A.68v3    3 !(t&E)7.30|                      |
|====================================================================|
                        (+"Unsolvables" posts)
                              page #3
Killer rating table
SudokuSolver Target range v3.6.3
Rating.....Score
0.50   =   0.85
0.75   =   0.90-0.95
1.00   =   1.00-1.20
1.25   =   1.25-1.45
1.50   =   1.50-1.70   (E) = Easy  (H) = Hard
===========================================================================================
|A ##     by     Rate    Score|A ##     by     Rate    Score|A ##     by     Rate    Score|
|-----------------------------+-----------------------------+-----------------------------|
|A58v1.5 mhp     1.25     1.30|A60RP-L mhp    E1.75     2.05|A.61    Ruud             1.40|
|A.59    Ruud    1.25     1.35|A.50v2  Ruud    3.0      5.20|A.61X   mhp     1.50     1.30|
|A59v1.5 JC      1.75     1.85|SKX4v2  Para    4.0       DNF|A.62    Ruud    0.75     1.05|
|A.60    Ruud    1.50     1.60|A.61XV3 mhp              9.95|A.62v2  Ruud    2.00     3.35|
|A.60RP  Ruud   H2.5      6.75|A.68v3  Ruud     3 (t&E) 5.10|                             |
|=========================================================================================|
                                         page #3


Assassin 58v1.5 by mhparker (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6912:6912:6912:2563:5380:5380:5894:5894:1800:6912:2314:2314:2563:5380:3854:3854:5894:1800:6912:4883:3092:2563:3606:3606:3606:5894:5894:4883:4883:3092:3092:2335:2335:2335:4130:3107:2084:3877:3877:3879:3879:3879:4130:4130:3107:2084:3877:4655:4655:4655:3634:3634:4148:4148:8246:8246:2616:2616:2616:4923:3634:4148:5950:1343:8246:3905:3905:2371:4923:3141:3141:5950:1343:8246:8246:2371:2371:4923:5950:5950:5950:
Solution: 
+-------+-------+-------+
| 9 7 3 | 5 8 6 | 4 1 2 |
| 2 1 8 | 4 7 9 | 6 3 5 |
| 6 5 4 | 1 3 2 | 9 8 7 |
+-------+-------+-------+
| 8 6 1 | 7 4 3 | 2 5 9 |
| 5 2 9 | 8 6 1 | 7 4 3 |
| 3 4 7 | 2 9 5 | 8 6 1 |
+-------+-------+-------+
| 7 8 2 | 3 5 4 | 1 9 6 |
| 1 3 6 | 9 2 8 | 5 7 4 |
| 4 9 5 | 6 1 7 | 3 2 8 |
+-------+-------+-------+
Quote:
mhparker, lead-in: It should be slightly harder than the original was intended to be, but still very do-able
mhparker, in rating thread: Agree with downgrading to 1.25
Afmob: After going through SudokuSolver's log I noticed that the ... in step 2f (of Afmob's Walkthrough) was unnecessary since the elimination could also be achieved by looking at ....So all in all this Assassin would be of rating (Hard) 1.0
Afmob: Rating: 1.25 Walkthrough
Andrew in 2011: a fun variant! I wish I'd finished it when it first appeared.... I...found....interesting step...I'll rate my walkthrough...at Hard 1.25
Walkthrough by Afmob:
Since this variant didn't have a wt in the archive, I decided to solve it and write a walkthrough. After going through SudokuSolver's log I noticed that the Killer triple in step 2f was unnecessary since the elimination could also be achieved by looking at the Outies of R1234 = 14(3).

So all in all this Assassin would be of rating (Hard) 1.0, I think.

A58 V1.5 Walkthrough:

1. R6789
a) Innies N7 = 8(2) = [17/62]
b) 15(2): R8C4 <> 6,7
c) Innies N9 = 10(2) <> 5
d) Innies R6789 = 7(2) <> 7,8,9; R6C2 <> 3
e) Innies+Outies R6789: 8 = R5C23 - R6C1 -> R5C23 <> 8 (IOU @ N4)
f) 8(2): R5C1 <> 1

2. R1234
a) Innies N1 = 9(2) <> 9, R3C3 <> 8
b) Innies N3 = 15(2) = {69/78}
c) Innies R1234 = 14(2) = [59/68/95]
d) 12(2): R5C9 = (347)
e) Innies+Outies R1234: 2 = R5C78 - R4C9 -> R5C78 <> 2 (IOU @ N6)
f) ! 16(3) <> 8 because R4C8 <> 1,7,8 and (358) is a Killer triple of 12(2)

3. R456+N19
a) 8 locked in 15(3) @ R5C4 @ R5 = 8{16/25/34} <> 7,9 -> 8 locked for N5
b) Innies+Outies N14: R4C4 = R6C3 <> 8
c) Innies+Outies N69: 3 = R6C6 - R4C7 -> R6C6 <> 1,2,3
d) 8 locked in R4C123 @ N4 for R4
e) Innies R1234 = 14(2) = {59} locked for R4+N6
f) 9 locked in 15(3) @ R5 @ R5C2 = 9{15/24} -> 9 locked for N4
g) 12(2) @ N6 <> 4
h) 19(3) = 8{47/56}
i) Innies R6789 = 7(2): R6C1 <> 1
j) 8(2) <> 7
k) 9(3) = 2{16/34} -> 2 locked for R4

4. R789+N4
a) 5(2) = {14} locked for C1+N7 because (23) is a Killer pair of 8(2)
b) Innies N7 = 8(2) = {26} -> R7C3 = 2, R8C3 = 6
c) Cage sum: R8C4 = 9
d) 19(3) @ N8 = 8{47/56} -> 8 locked for C6
e) 10(2) = 2{17/35}
f) Innies N9 = 10(2) = {19/46} because (37) is a Killer pair of 10(3)
g) 12(2) <> 3

5. R456+C6
a) Hidden triple (123) @ R345C6 @ C6 -> R345C6 = {123}
b) Hidden Single: R5C4 = 8 @ C4 -> R5C56 = 7(2): R5C5 <> 1,2,3
c) 7 locked in R5C789 @ R5 for N6
d) 18(3) <> 1
e) Innies+Outies N14: R6C3 = R4C4 = (347)
f) 18(3) can only have one of (347) and R6C3 = (347) -> R6C45 <> 3,4,7

6. C789 !
a) Innies N69 = 11(3) = {128/146/236} <> 9
b) ! Killer pair (68) locked in Innies N3 + Innies N69 for C7
c) 16(3) @ R4C8: R5C8 <> 1 because R5C7 <> 6,9
d) 16(3) @ R6: R7C8 <> 1,4 because 9 only possible there
e) 16(3) @ R4C8 <> 1 because R7C8 = (69) blocks {69} @ R45C8
f) 16(3) @ R4C8 = 4{39/57} because R4C8 = (59) -> 4 locked for R5+N6
g) Innies N9 = [19/46]
h) Innies N69 = 11(3) = 1{28/46} because R7C7 = (14) -> 1 locked for C7
i) Innies+Outies N69: 3 = R6C6 - R4C7 -> R6C6 <> 6,7

7. N45
a) Hidden Single: R6C3 = 7 @ R6, R4C4 = 7 @ N5
b) 12(3) = {147} -> 1,4 locked for C3
c) 3 locked in 8(2) @ N4 = {35} locked for C1+N4
d) 15(3) = {249} -> R5C3 = 9, R5C2 = 2, R6C2 = 4
e) 19(3) = {568} -> R3C2 = 5, {68} locked for R4
f) 15(3) @ N5 = {168} -> R5C5 = 6, R5C6 = 1
g) 18(3) = {279} -> R6C4 = 2, R6C5 = 9
h) 14(3) = {158} -> R6C6 = 5, R7C7 = 1, R6C7 = 8
i) R4C3 = 1, R3C3 = 4, R4C6 = 3

8. N236
a) R3C6 = 2 -> R3C57 = 12(2) = [39] -> R3C5 = 3, R3C7 = 9
b) 10(3) = {145} -> R3C4 = 1, {45} locked for C4+N2
c) Innies N3 = 15(2) = {69} -> R2C7 = 6
d) 7(2) <> 1
e) 9(2) = {18} -> R2C3 = 8, R2C2 = 1
f) Killer pair (35) locked in 7(2) + 12(2) for C9

9. N9
a) Hidden Single: R9C3 = 5 @ N7
b) 5 locked in 12(2) @ N9 = {57} locked for R8+N9

10. Rest is singles.

Rating: 1.25
2011 Walkthrough by Andrew:
Many thanks Mike for a fun variant! I wish I'd finished it when it first appeared and posted my walkthrough then; however it was when we were moving from Calgary to Lethbridge so I was limiting myself to doing basic Assassins for at least several weeks.

Afmob suggested that if he'd spotted an alternative to his step 2f it would have reduced his rating. I agree that this would have given a slightly simpler solving path but not that it would have reduced the rating; IMHO there's are other steps in Afmob's walkthrough which are as hard as step 2f.

As I've commented after my step 16, if I hadn't missed something simple then I wouldn't have found the interesting step 26 and probably wouldn't have posted this walkthrough.

Here is my walkthrough for A58 V1.5

Prelims

a). R12C9 = {16/25/34}, no 7,8,9
b). R2C23 = {18/27/36/45}, no 9
c). R2C67 = {69/78}
d). R45C9 = {39/48/57}, no 1,2,6
e). R56C1 = {17/26/35}, no 4,8,9
f). R89C1 = {14/23}
g). R8C34 = {69/78}
h). R8C67 = {39/48/57}, no 1,2,6
i). 10(3) in N2 = {127/136/145/235}, no 8,9
j). 21(3) cage in N2 = {489/579/678}, no 1,2,3
k). 19(3) cage at R3C2 = {289/379/469/478/568}, no 1
l). 9(3) cage at R4C5 = {126/135/234}, no 7,8,9
m). 10 cage at R7C3 = {127/136/145/235}, no 8,9
n). 19(3) cage in N8 = {289/379/469/478/568}, no 1
o). 9(3) cage in N8 = {126/135/234}, no 7,8,9
p). 32(5) cage in N7 = {26789/35789/45689}, no 1

[I originally started with 32(5) cage, 8,9 locked for N7, as a step resulting from the Prelims, but this 45 is more effective.]
1. 45 rule on N7 2 innies R78C3 = 8 = [17/26], clean-up: no 6,7 in R8C4
1a. Killer pair 1,2 in R89C1 and R7C3, locked for N7

2. 45 rule on N1 2 innies R3C23 = 9 = {27/36/45}/[81], no 9, no 8 in R3C3

3. 45 rule on N3 2 innies R23C7 = 15 = {69/78}
3a. Min R3C7 = 6 -> max R3C56 = 8, no 8,9 in R3C56

4. 45 rule on N9 2 innies R7C78 = 10 = {19/28/37/46}, no 5

5. 45 rule on R1234 2 innies R4C89 = 14 = {59}/[68], clean-up: no 5,8,9 in R5C9

6. 45 rule on R6789 2 innies R6C12 = 7 = {16/25}/[34], no 7,8,9, no 3 in R6C2, clean-up: no 1 in R5C1

7. 45 rule on N69 1 outie R6C6 = 1 innie R4C7 + 3 -> no 1,2,3 in R6C6

8. 19(3) cage in N8 = {379/469/478/568} (cannot be {289} which clashes with R8C4), no 2

9. 45 rule on C1 5 innies R12347C1 = 32 = {26789/35789/45689}, no 1

10. 45 rule on N14 3 innies R346C3 = 12 = {138/147/246/345} (cannot be {129} which clashes with R7C3, cannot be {156/237} which clash with R78C3), no 9
10a. 45 rule on N14 1 outie R4C4 = 1 innie R6C3, no 9 in R4C4

11. 10(3) cage at R7C3 = {127/145/235} (cannot be {136} which clashes with 9(3) cage in N8), no 6

12. 16(3) cage at R4C8 = {169/349/367/457} (cannot be {178} because R4C8 only contains 5,6,9, cannot be {358} which clashes with R45C9, cannot be {259/268} which clashes with R4C89, CCC), no 2,8
12a. 5 of {457} must be in R4C8 -> no 5 in R5C78
[With hindsight the combination eliminations can be made more simply by
45 rule on R1234 2 outies R5C67 = 1 innie R4C9 + 2, IOU no 2 in R5C67
16(3) cage at R4C8 = {169/349/367/457} (cannot be {178} because R4C8 only contains 5,6,9, cannot be {358} which clashes with R45C9), no 8
Better still, after reading Afmob’s comment, use the IOU to eliminate 2 and then
45 rule on R1234 3 outies R5C789 = 14 cannot contain 8 because R5C9 = {347}]

13. 19(3) cage at R3C2 = {289/379/469/478/568}
13a. 2 of {289} must be in R4C12 (R4C12 cannot be {89} which clashes with R4C89) -> no 2 in R3C2, clean-up: no 7 in R3C3 (step 2)
13b. 5 of {568} must be in R3C2 (R4C12 cannot be {56/58} which clash with R4C89) -> no 5 in R4C12

14. 45 rule on N69 3 innies R467C7 = 11 = {128/137/146/236/245}, no 9, clean-up: no 1 in R7C8 (step 4)

15. 45 rule on R1234 3 outies R5C789 = 14 = {149/167/347}
15a. 45 rule on R6789 3 outies R5C123 = 16 = {169/259/349/367} (cannot be {178/268/358} which clash with R56C1, CCC, cannot be {457} which clashes with R5C789), no 8
[Again, with hindsight, step 15a can be done more simply by
45 rule on R6789 2 outies R5C23 = 1 innie R6C1 + 8, IOU no 8 in R5C23
45 rule on R6789 3 outies R5C123 = 16 = {169/259/349/367} (cannot be {457} which clashes with R5C789)]

16. 8 in R5 only in 15(3) cage at R5C4, locked for N5, clean-up: no 5 in R4C7 (step 7), no 8 in R6C3 (step 10a)
16a. 15(3) cage at R5C4 = {168/258/348}, no 7,9
16b. 9 in N5 only in R6C456, locked for R6

[At this stage I missed 8 in N4 only in R4C123, locked for R4 -> R4C89 = {59} ... which would have made my solving path shorter and I wouldn’t have seen the interesting step 26, plus I probably wouldn’t have posted this walkthrough.]

17. 18(3) cage at R6C3 = {279/369/459/567}, no 1, clean-up: no 1 in R4C4 (step 10a)

18. R5C123 (step 15a) = {169/259/349/367}
18a. 7 of {367} must be in R5C23 (R5C23 cannot be {36} because 15(3) cage in N4 cannot be {36}6) -> no 7 in R5C1, clean-up: no 1 in R6C1, no 6 in R6C2 (step 6)
[For an alternative way to achieve this result see my post Overlapping Cage and Hidden Cage in the Killer Techniques Forum.]

19. 1 in C1 only in R89C1 = {14}, locked for C1 and N7 -> R7C3 = 2, R8C3 = 6 (step 1), R8C4 = 9, clean-up: no 3,7 in R2C2, no 3,7 in R3C2 (step 2), no 2,6 in R4C4 (step 10a), no 4 in 10(3) cage at R7C3 (step 11), no 8 in R7C78 (step 4), no 3 in R8C78
19a. R7C78 (step 4) = [19/46/64] (cannot be {37} which clashes with 10(3) cage at R7C3), no 3,7

20. 19(3) cage in N8 (step 8) = {478/568}, no 3, 8 locked for C6, clean-up: no 7 in R2C7, no 8 in R3C7 (step 3)

21. 2 in N8 only in 9(3) cage = {126/234}, no 5

22. 8 in N2 only in 21(3) cage = {489/678}, no 5, 8 locked for C5
22a. R5C4 = 8 (hidden single in N5)

23. 19(3) cage at R3C2 = {289/469/478/568} (cannot be {379} because no 3,7,9 in R3C2), no 3

24. 14(3) cage at R6C6 = {149/158/167/347/356} (cannot be {239/257} because R7C7 only contains 1,4,6, cannot be {248} because 2,8 only in R6C7), no 2
24a. 3,8 of {158/356} must be in R6C7 -> no 5 in R6C7

25. R467C7 (step 14) = {128/137/146/236}
25a. 1 of {128/137} must be in R7C7, 1 of {146} must be in R67C7 (R67C7 cannot be {46} because 14(4) cage at R6C6 cannot be 4{46}) -> no 1 in R4C7, clean-up: no 4 in R6C6 (step 7)

[The next step has been available for a long time but I’ve only just spotted it.]
26. 7 in R4 only in R4C1234, R4C4 = R6C3 (step 10a) -> R4C123 + R6C3 must contain 7, locked for N4
[An alternative way to see this step is
45 rule on N1 4 outies R4C1234 = 22
45 rule on N4 4 innies R4C123 + R6C3 = 22
R4C1234 and R4C123 + R6C3 share three cells so must contain the same combination
7 in R4 only in R4C1234 -> R4C123 + R6C3 must contain 7, locked for N4]

27. 7 in R5 only in R5C789, locked for N6
27a. R5C789 (step 15) = {167/347}, no 9

28. 9 in N6 only in R4C89 (step 5) = {59}, locked for R4 and N6, clean-up: no 4 in R5C9, clean-up: no 5 in R6C3 (step 10a)

29. 19(3) cage at R3C2 (step 23) = {478/568}, no 2
29a. 5 of {568} must be in R3C2 -> no 6 in R3C2, clean-up: no 3 in R3C3 (step 2)

30. 9 in N4 only in 15(3) cage = {159/249}, no 3,6

31. 16(3) cage at R6C8 = {169/268/349}
31a. 9 of {349} must be in R7C8 -> no 4 in R7C8, clean-up: no 6 in R7C7 (step 4)

32. R467C7 (step 25) = {128/146} (cannot be {236} because R7C7 only contains 1,4), no 3, 1 locked for C7, clean-up: no 6 in R6C6 (step 7)
32a. Killer pair 6,8 in R23C7 and R46C7, locked for C7, clean-up: no 4 in R8C8

33. 16(3) cage at R4C8 (step 12) = {349/457} (cannot be {169} because 1,6 only in R5C8, cannot be {367} which clashes with R5C9), no 1,6
33a. Naked triple {347} in R5C789, locked for R5 and N6, clean-up: no 5 in R6C1, no 2 in R6C2 (step 6), no 7 in R6C6 (step 7)

34. 14(3) cage at R6C6 (step 24) = {149/158}, no 6

35. 7 in R6 only 18(3) cage at R6C3 (step 17) = {279} (only remaining combination, cannot be {567} = 7{56} which clashes with 15(3) cage in N5) -> R6C3 = 7, R6C45 = [29], R6C6 = 5, clean-up: no 2 in R2C2
35a. Naked pair {16} in R5C56, locked for R5 and N5
35b. Naked pair {34} in R4C56, locked for R4 -> R4C4 = 7, clean-up: no 1 in R7C5 (step 11)

36. Naked pair {68} in R4C12, locked for R4, N4 and 19(3) cage at R3C2 -> R4C3 = 1, R4C7 = 2, R6C1 = 3, R5C1 = 5, R5C23 = [29], R6C2 = 4, R3C2 = 5, R3C3 = 4, clean-up: no 8 in R2C2

37. 14(3) cage at R6C6 (step 34) = {158} (only remaining combination) -> R67C7 = [81], R7C8 = 9 (step 4), R4C8 = 5, R4C9 = 9, R5C9 = 3, clean-up: no 4 in R12C9, no 7 in R2C6, no 7 in R3C7 (step 3), no 7 in R8C7
37a. Naked pair {69} in R2C67, locked for R2, clean-up: no 1 in R1C9, no 3 in R2C3
37b. Naked pair {69} in R23C7, locked for C7 and N3, clean-up: no 1 in R2C9

38. R2C23 = [18], R1C3 = 3, R9C3 = 5
38a. Naked pair {25} in R12C9, locked for C9 and N3

39. Naked pair {47} in R15C7, locked for C7 -> R9C7 = 3, R8C7 = 5, R8C8 = 7, R5C78 = [74], R1C7 = 4, R2C8 = 3

40. 21(3) cage in N2 (step 22) = {489/678} -> R1C5 = 8
40a. 6,9 only in R1C6 -> R1C6 = {69}
40b. Naked pair {69} in R12C6, locked for C6 and N2 -> R56C1 = [61]
40c. Naked triple {478} in 19(3) cage in N8, locked for C6 and N8 -> R4C56 = [43], R2C5 = 7, R1C6 = 6 (step 40)

and the rest is naked singles.

Rating Comment. I'll rate my walkthrough for A58 V1.5 at Hard 1.25 because of step 26, plus I did a fair amount of analysis.


Assassin 59 by Ruud (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:2560:4610:4610:3332:2565:2565:3079:3079:5129:5129:5129:4610:3332:2565:3343:3343:3343:3090:3090:3348:4610:5398:2565:2584:4377:4377:4635:3090:3348:5398:5398:5398:2584:4377:7459:4635:4635:4635:4647:4647:4647:7459:7459:7459:4635:3374:3119:4144:4144:4144:2611:3636:7459:3374:3374:3119:5177:4144:6459:2611:3636:3636:3903:3903:3903:5177:2115:6459:3909:3909:3909:3400:3400:5177:5177:2115:6459:6459:2639:2639:
Solution: 
+-------+-------+-------+
| 2 8 1 | 9 6 4 | 3 7 5 |
| 6 9 5 | 3 7 2 | 8 4 1 |
| 4 3 7 | 5 8 1 | 9 6 2 |
+-------+-------+-------+
| 8 5 6 | 2 4 7 | 1 9 3 |
| 1 2 4 | 6 9 3 | 7 5 8 |
| 3 7 9 | 8 1 5 | 4 2 6 |
+-------+-------+-------+
| 5 1 3 | 7 2 9 | 6 8 4 |
| 7 6 2 | 4 3 8 | 5 1 9 |
| 9 4 8 | 1 5 6 | 2 3 7 |
+-------+-------+-------+
Quote:
mhparker, rating post: (rating) 1.25: Harder Assassin. Actually, most recent Assassins seem to have become more difficult than they traditionally used to be. So a rating of "1.25" would be considered the norm now. Typical example: A59
CathyW: Strangely, I found this one easier than 58
Walkthrough by CathyW:
Strangely, I found this one easier than 58.

1. 10(4) r123c6 + r1c7 = {1234}
-> r1c45 see all cells of 10(4) thus min 5 -> r2c5 <> 9

2. 18(5) N4 must have 1,2; no 9 -> r3c12 of 12(3) <> 9

3. Outies N5: r37c5 = 10 (no 5)

4. Outies r12: r3c46 = 6 = [51]/{24}

5. Outies r89: r7c46 = 16 = {79} not elsewhere in N8/r7
-> r3c5 <> 1,3; r6c3 <> 3,5; r6c7 <> 1,3; r89c5 <> 1

6. Innies r34: r4c19 = 11 (no 1)

7. Innies r67: r6c19 = 9 -> r6c9 <> 9

8. Innies r2: r2c456 = 12 -> r2c4 = (1.7)

9. Innies r8: r8c456 = 15 = {168/258/348/456}. Analysis: r8c46 <> 3

10. Outies - Innies N1: r4c23 - r1c3 = 10
r1c3 max 7, r4c23 min 11, max 17 -> r4c2 <> 1

11. O-I N7: r6c23 - r9c3 = 8 -> r6c2 <> 8
r9c3 = (1.9), r6c23 min 9, max 17

12. Innies N4: r46c23 = 27 -> r6c23 min 10, max 16 -> r9c3 <> 1,9

13. O-I N3: r4c78 - r1c7 = 7 -> r4c8 <> 7
r1c7 = (1234), r4c78 min 8, max 11

14. O-I N9: r6c78 - r9c7 = 4 -> r6c8 <> 4
r9c3 = (1.9), r6c78 min 5, max 13

15. Innies N6: r46c78 = 16
Since r4c78 is min 8 -> r6c78 is max 8 (no 8,9) -> r7c7 <> 1,2

16. O-I N2: r3c5 - r1c37 = 4
-> r3c5 = (789), r1c37 = (1234)
-> r7c5 = (123)

17. O-I N8: r9c37 - r7c5 = 8

18. 1 locked to r4567c5 -> r6c46 <> 1

19. Outies c12: r258c3 = 11 (no 9)

20. Innies c123: r19c3 = 9 = [18/27/36/45]

21. Outies c89: r258c7 = 20 (no 1,2)

22. Innies c789: r19c7 = 5 = {14/23}

23. 25(4) r789c6 + r9c7 = 7{68}4 / 9{68}2 / 9{58}3
-> r9c7 <> 1 -> r1c7 <> 4
-> r89c6 = {58/68} 8 not elsewhere in N8/c6

24. Killer Pair r89c6 and 8(2) r89c5 -> r89c4 <> 5,6
-> 4 required for 10(4) thus locked to r123c6 not elsewhere in N2/c6 -> r3c6 <> 2, r1c5 <> 9
-> 4 locked to r89c4 not elsewhere in c4
-> KP 13(2) and 8(2) in c5 -> r456c5 <> 5,6

25. Split 6(2) r3c46:
If [24] -> r1c7 = 2 -> CONFLICT: No place for 2 in N1
-> r3c4 = 5, r3c6 = 1
-> r1c7 = (23) -> r9c7 = (23) not elsewhere in c7
-> r12c5 = {67} -> r1c4, r3c5 = (89)
-> r89c5 = {35} not elsewhere in N8/c5
-> 10(2) r67c7 = {46} not elsewhere in c7
-> 10(2) r34c7 = [91]
-> r3c5 = 8, r7c5 = 2, r1c4 = 9 .

Straightforward combinations and singles from here

:)
Walkthrough by Para:
CathyW wrote:
Will be interesting to see how Para solves this one.


Well here it is.

Walk-through Assassin 59

1. 10(4) at R1C6 = {1234}

2. R1C12, R34C7, R67C3 and R9C89 = {19/28/37/46}: no 5

3. R1C89 and R67C4 = {39/48/57}: no 1,2,6

4. R12C5, R34C3 and R9C12 = {49/56/67): no 1,2,3

5. R89C5 = {17/26/35}: no 4,8,9

6. 20(3) at R2C1 = {389/479/569/578}: no 1,2

7. 18(5) at R4C1 = {12348/12357/12456}: no 9; {12} locked for N4

8. 45 on R89: 2 outies: R7C46 = 16 = {79} -->> locked for R7 and N8
8a. Clean up: R6C3: no 3,5; R6C7: no 1,3; R89C5: no 1

9. 45 on R12: 2 outies: R3C46 = 6 = {24}/[51]: R3C4 = {245}; R3C6 = {124}

10. 45 on N2: 2 outies and 1 innie: R3C5 = R1C37 + 4: Min R3C5 = 7; Max R1C37 = 5 -->> R3C5 = {789}; R1C3 = {1234}

11. Killer Quad {1234} in R1C12 + R1C367 -->> locked for R1
11a. R1C89 = {57} -->> locked for R1 and N3
11b. Clean up: R1C12: no 3, R2C5: no 6,8,9; R4C7: no 3

12. 45 on N5: 2 outies: R37C5 = 10 = [73/82/91]: R7C7 = {123}

13. 45 on C123: 2 innies: R19C3 = 9 = [18/27/36/45]: R9C3 = {5678}

14. 45 on C789: 2 innies: R19C7 = 5 = {14/23}: R9C7 = {1234}

15. 25(4) = {2689/3589/4678}(other combo’s blocked, can only have one of {79})
15a. R9C7 = {234}; R89C6 = {58/68} -->> 8 locked for C6 and N8
15b. Killer Pair {56} in R89C5 and R89C6 -->> locked for N8
15c. 4 in N8 locked for C4
15d. Clean up: R1C7: no 4; R3C6: no 2

16. 4 in 10(4) at R1C6 locked for C6 and N2
16a. Clean up: R1C5: no 9
16b. Killer Pair {56} in R12C5 and R89C5 -->> locked for C5

Addition suggested by Mike:
17x. 13(2) at R9C1 = {49/58/67} = {4|5|6..}
17y. 15(3) at R8C1 = {159/168/249/258/267/348/357} = {1|2|3..): {456} blocked by 13(2) at R9C1
17. Distribution {123} in N7: 15(3) at R8C1 can only have one so R7C123 needs 2 of {123} -->> Killer Triple {123} in R7C123 + R7C5 -->> locked for R7
17a. Clean up: R6C7: no 7,8,9
17b. Killer Pair {24} in R19C7 + R67C7 -->> locked for C7
17c. Clean up: R34C7: no 6,8

18. 14(3) at R6C8 needs 2 of {4568} in R7C89: 14(3) = [1]{58}/[2]{48}/[3]{56} -->> R6C8 = {123}

19. 45 on N36: 3 innies: R1C7 + R6C78 = 9 = [162/243/261/342]: R6C7: no 2

20. R67C7 = {46} -->> locked for C7
20a. R1C19 = {23} -->> locked for C7
20b. R34C7 = {19} -->> locked for C7
20c. R2C7 = 8
20d. 13(3) at R2C7 = 8{14}(8{23} blocked by R1C7): R2C89 = {14} -->> locked for R2 and N3
20e. R34C7 = [91]

21. 45 on N2: R3C5 = R1C37 + 4 -->> R3C5 = {78} -->> R1C37 = 3 or 4 = [12/13] -->> R1C3 = 1
21a. R9C3 = 8; R3C6 = 1(hidden); R3C4 = 5; R12C5 = [67]; R3C5 = 8; R7C5 = 2
21b. R12C4 = [93]; R12C6 = [42]; R1C7 = 3; R7C46 = [79]; R9C7 = 2
21c. R8C6 = 8(hidden); R9C6 = 6; R6C78 = [42](step 19); R7C7 = 6; R3C89 = [62]
21d. R4C8 = 9; R34C3 = [76](last possible combo); R67C3 = [93]; R2C3 = 5
21e. R4C4 = 2; R4C56 = [47](last possible combo); R5C5 = 9(hidden); R6C5 = 1(hidden)

22. 45 on N1: 1 outie: R4C2 = 5

23. 45 on N4: 1 innie: R6C2 = 7
23a. R6C12 = [51](last remaining combo in 13(3) cage); R5C1 = 1(hidden)
23b. R6C46 = [85](last remaining combo in 16(4) cage); R5C46 = [63]; R6C19 = [36]
23c. R4C19 = [83]; R3C12 = [43]; R1C12 = [28]; R9C12 = [94](last possible combo)
23d. R2C12 = [69]; R5C23 = [24]; R8C123 = [762]; R89C4 = [41]; R9C89 = [37]
23e. R1C89 = [75]; R89C5 = [35]; R8C789 = [519]; R2C89 = [41]; R7C89 = [84]; R5C789 = [758]

And we are done.

greetings

Para
Walkthrough by Andrew:
I only finished Assassin 59 yesterday evening. I wasted a lot of time by making a totally illogical move and couldn't find where I'd gone wrong. It can be so difficult to find one's own mistakes! Thanks Ed for finding it for me.

Cathy's steps 11 and 12 were neat with steps 13 and 14 using the same principle as step 11. That's a technique that I don't think I've seen before. It works as follows

If there are 2 cells in one row RaCxy which total n more than another cell RbCx in a different row, then cell RaCy cannot contain n. Of course the same principle applies for CaRxy and CbRx. Maybe someone can post this technique as diagrams for the two cases? It should be worth putting into Ruud's killer guide.

The one surprising thing was that Cathy didn't use the killer quad in R1. Mike, since you used the same solution path, did you miss it?

My solution path started much more like Para's walkthrough which included the killer quad. Para's step 17 was a neat one.

Here is my walkthrough. In addition to Cathy's neat technique, which I'll try to remember for future use, I also missed the innies in rows 2 and 8.

1. R1C12 = {19/28/37/46}, no 5

2. R12C5 = {49/58/67}, no 1,2 3

3. R1C89 = {39/48/57}, no 1,2,6

4. R34C3 = {49/58/67}, no 1,2,3

5. R34C7 = {19/28/37/46}, no 5

6. R67C3 = {39/48/57}, no 1,2,6

7. R67C7 = {19/28/37/46}, no 5

8. R89C5 = {17/26/35}, no 4,8,9

9. R9C12 = {49/58/67}, no 1,2,3

10. R9C89 = {19/28/37/46}, no 5

11. R2C123 = {389/479/569/578}, no 1,2

12. 10(4) cage at R1C6 = {1234}, no 1,2,3,4 in R1C45, no 9 in R2C5

13. 18(5) cage in N4 = 12{348/357/456}, no 9, 1,2 locked for N4

14. 45 rule on R12 2 outies R3C46 = 6 = {24}/[51]

15. 45 rule on R89 2 outies R7C46 = 16 = {79}, locked for R7 and N8, clean-up: no 3,5 in R6C3, no 1,3 in R6C7, no 1 in R89C5

16. 45 rule on R1234 2 innies R4C19 = 11 = [29]/{38/47/56}, no 1, no 2 in R4C9

17. 45 rule on R6789 2 innies R6C19 = 9 = {18/27/36/45}, no 9 in R6C9

18. 45 rule on C12 3 outies R258C3 = 11 = {128/137/146/236/245}, no 9
18a. 1,2 only in R58C3 -> 8 can only be in R2C3 -> no 8 in R58C3

19. 45 rule on C89 3 outies R258C7 = 20 = {389/479/569/578}, no 1,2

20. 45 rule on C123 2 innies R19C3 = 9 = {18/27/36/45}, no 9

21. 45 rule on C789 2 innies R19C7 = 5 = {14/23}

22. 25(4) cage at R7C6, max R9C7 = 4 -> min R789C6 = 21, no 1,2,3
22a. 25(4) = {2689/3589/4678} (cannot be {1789/3679/4579} because 7,9 only in R7C6), no 1, clean-up: no 4 in R1C7 (step 21)
22b. 4 in {4678} must be in R9C7 -> no 4 in R89C6
22c. 8 locked in 25(4) cage in R89C6, locked for C6 and N8

23. 10(4) cage at R1C6, 4 locked in R123C6, locked for C6 and N2, clean-up: no 9 in R1C5, no 2 in R3C6 (step 14)

24. Killer pair 5/6 in R89C6 and R89C5, locked for N8

25. Killer pair 5/6 in R12C5 and R89C5, locked for C5

26. 45 rule on N9 2 outies R6C78 – 4 = 1 innie R9C7, max R9C7 = 4 -> max R6C78 = 8, no 8,9 in R6C7, no 7,8,9 in R6C8, clean-up: no 1,2 in R7C7

27. 45 rule on C1234 3 innies R456C4 = 16
27a. 45 rule on C6789 3 innies R456C6 = 15
27b. R456C5 = 14
27c. 45 rule on C5 5 innies R34567C5 = 24 -> R37C5 = 10 = [73/82/91] ->R3C5 = {789}, no 4 in R7C5
[Alternatively 45 rule on N5 2 outies R37C5 = 10 but I only saw that later]

28. 45 rule on N2 1 innie R3C5 – 4 = 2 outies R1C37, max R3C5 = 9 -> max R1C37 = 5 -> R1C3 = {1234}, clean-up: no 1,2,3,4 in R9C3 (step 20)
[At this stage I ought to have seen step 33.]

29. 4 in N8 locked in R89C4, locked for C4

30. 20(4) cage at R7C4 must contain 4 (step 29) = 4{169/178/259/367} (cannot be {2468/3458} because 5,6,8 only in R9C3)
30a. {3467} must use 7 in R7C4 -> no 7 in R9C3, clean-up: no 2 in R1C3 (step 20)

31. 45 rule on N7 2 outies R6C23 – 8 = 1 innie R9C3, min R9C3 = 5 -> min R6C23 = 13, no 3
31a. R9C3 = {568} -> R6C23 = 13,14,16 no 5,6 in R6C3 -> no 8 in R6C2
[Step 31a is doing it the hard way. With Cathy's technique I would have just said no 8 in R6C2 because R6C23 8 more than R9C3.]

32. 12(3) cage at R3C1, min R4C2 = 3 -> max R3C12 = 9, no 9

33. Killer quad 1/2/3/4 in R1C12, R1C3, R1C67, locked for R1, clean-up: no 8,9 in R1C89

34. Naked pair {57} in R1C89, locked for R1 and N3, clean-up: no 3 in R1C12, no 6,8 in R2C5, no 3 in R4C7
34a. R3C89 cannot be {79} -> no 1 in R4C8
34b. R3C89 cannot be {49/58/67} ({49} blocked because 17(3) cage cannot be {49}4) -> no 4 in R4C8

35. 18(4) cage at R1C3 = {1269/1359/1458/2358/2367} (cannot be {1278/3456} which clash with R12C5, cannot be {1368/1467} because R3C4 only contains 2,5, cannot be {2349} because [932] in R123C4 clashes with R123C6, cannot be {2457} because R1C4 only contains 6,8,9)
35a. {1458/2358} both require 8 in R1C4 -> no 8 in R2C4

36. 2 in C3 locked in R58C3
36a. R258C3 (step 18) = {128/236/245} = 2{18/36/45}, no 7

37. 7,9 in C3 locked in R346C3 -> R34C3 = {49/67}, no 5,8 and R67C3 = [75/93], no 4,8

38. 45 rule on N3 4 innies R1C7 +R3C789 = 20 = {1289/2369/2468} (cannot be {1469} which clashes with R3C6) = 2{189/369/468}, 2 locked for N3
38a. If {1289} R3C89 cannot be {89} -> 1,2 must be in R1C7 and R3C89 -> no 1 in R3C7 (it can still be 2 in the {2369} combination), clean-up: no 9 in R4C7
38b. R2C789 = {139/148/346}
[Mike has pointed out that after step 38 I missed 2 in R2 locked in R2C46 -> R3C4 = 5, R3C6 = 1.]

39. R2C123 (step 11) = {389/479/569} (cannot be {578} which clashes with R2C5) = 9{38/47/56}, 9 locked for R2 and N1, clean-up: no 1 in R1C12, no 4 in R4C3

40. R1C4 = 9 (hidden single in R1) -> R7C46 = [79], clean-up: no 1 in R7C5

41. 1 in N8 locked in R89C4, locked for C4
[At this stage I should have remembered 4 locked in R89C4 (step 29) -> R89C4 = {14}, locked for 20(4) cage. This was done in steps 46 and 47.]

42. 1 in N2 locked in R123C6, locked for C6 and 10(4) cage -> no 1 in R1C7, clean-up: no 4 in R9C7

43. Naked pair {23} in R19C7, locked for C7, clean-up: no 8 in R3C7, no 7,8 in R4C7, no 7 in R6C7, no 8 in R7C7

44. Naked pair {46} in R67C7, locked for C7 -> R2C7 = 8, R3C7 = 9, R4C7 = 1, clean-up: no 3 in R2C123 (step 39)

45. 9 in C3 locked in R46C3, locked for N4

46. Killer pair 2/3 in R7C5 and R89C5, locked for C5 and N8

47. R789C4 = 7{14} -> R9C3 = 8, R1C3 = 1 (step 20), clean-up: no 5 in R9C12, no 2 in R9C89
[R9C3 was also a hidden single in C3]

48. R8C6 = 8 (hidden single in N8)
48a. 8 in N9 locked in R7C89 -> 14(3) cage at R6C8 = 8{15/24}, no 3,6

49. Killer pair 7/8 in R12C5 and R3C5, locked for C5

50. 3 in R1 locked in R1C67 -> no 3 in R2C6

51. R2C7 = 8 -> R2C89 = 5 = {14}, locked for R2 and N3 -> R2C6 = 2, R1C7 = 3, R13C6 = [41], R3C4 = 5, R12C5 = [67], R2C4 = 3, R3C5 = 8, clean-up: R7C5 = 2 (step 27c), R9C7 = 2 (step 21)

52. Naked pair {35} in R89C5 -> R9C6 = 6, clean-up: no 7 in R9C12, no 4 in R9C89

53. Naked pair {26} in R3C89, locked for R3 -> R4C8 = 9, R4C5 = 4, clean-up: no 7 in R4C3 -> R4C3 = 6, R3C3 = 7, R2C3 = 5, R6C3 = 9, R7C3 = 3, no 1 in R9C9

54. R3C12 = {34} -> R4C2 = 5

55. Naked pair {49} in R9C12, locked for R9 and N7 -> R8C3 = 2, R5C3 = 4, R6C2 = 7, R89C4 = [41]

56. Naked pair {37} in R9C89, locked for R9 and N9 -> R89C5 = [35], R8C7 = 5, R5C7 = 7

57. R4C6 = 7 (hidden single in C6)

58. R8C7 = 5 -> R8C89 = 10 = [19] -> R8C12 = [76], R7C12 = [51], R2C12 = [69], R9C12 = [94], R3C12 = [43], R2C89 = [41], R7C89 = [84], R7C8 = 2 (cage sum), R67C7 = [46], R3C89 = [62]

59. Naked pair {35} in R5C68, locked for R5

60. R5C456 = [693] (only remaining permutation)

and the rest is naked singles
IOU by Andrew & sudokuEd:
About A59 WT Andrew wrote:
Cathy's steps 11 and 12 were neat with steps 13 and 14 using the same principle as step 11. That's a technique that I don't think I've seen before. It works as follows

If there are 2 cells in one row RaCxy which total n more than another cell RbCx in a different row, then cell RaCy cannot contain n. Of course the same principle applies for CaRxy and CbRx.
I liked this move too Andrew and Cathy - and think it will help our walk-throughs if we give it a name. My suggestion is IOU. If an Innie & Outie are in the same r, c or n, then the 3rd cell must be Unequal to the I/O difference. Hence, IOU. For example, Cathy's step 11 says
Cathy wrote:
11. O-I N7: r6c23 - r9c3 = 8
Since r6c3 (an Outie) is in the same column as r9c3 (an Innie), then the 3rd cell must be Unequal to the difference - in this case 8. From IOU -> r6c2 <> 8.

Another way of seeing this is that if r6c2 was 8 then the difference would be 0 which means that r6c3 = r9c3: which is impossible because they are in the same column.

Similarly, step 13 is
Cathy wrote:
13. O-I N3: r4c78 - r1c7 = 7
Since the difference is 7: IOU -> r4c8 <> 7

Perhaps with a walk-through we could explain the logic of this move the first time and after that just refer to it as IOU. So with Cathy's step 14,
Cathy wrote:
14. O-I N9: r6c78 - r9c7 = 4
then add: IOU -> r6c8 <> 4

Incidently, Assassin 60 has several of this move - including where the 1 innie (of a r or c) and 1 outie (of that r or c) are in the same nonet. This is a bit harder to see compared to the I/O of nonets.

Thanks for highlighting this trick Andrew. Good luck with Assassin 60! It's a corker!

Cheers
Ed


Last edited by Ed on Sun Dec 28, 2008 8:59 am, edited 3 times in total.

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Ed
PostPosted: Fri Jun 27, 2008 10:59 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 59v1.5 by Jean-Christophe (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3328:3328:3586:3586:3076:4613:4613:2567:2567:4361:4361:4361:3586:3076:4613:4623:4623:4623:2578:2578:3348:3586:4374:4613:2584:3865:3865:7195:2578:3348:4374:4374:4374:2584:3865:6947:7195:7195:7195:3879:3879:3879:6947:6947:6947:7195:4142:2607:5936:5936:5936:1843:2868:6947:4142:4142:2607:4409:5936:5179:1843:2868:2868:4415:4415:4415:4409:3139:5179:4933:4933:4933:1608:1608:4409:4409:3139:5179:5179:2639:2639:
Solution: 
+-------+-------+-------+
| 4 9 2 | 8 5 6 | 1 3 7 |
| 8 3 6 | 1 7 2 | 9 4 5 |
| 1 7 5 | 3 4 9 | 6 2 8 |
+-------+-------+-------+
| 7 2 8 | 9 1 3 | 4 5 6 |
| 9 5 4 | 6 2 7 | 3 8 1 |
| 3 6 1 | 4 8 5 | 2 7 9 |
+-------+-------+-------+
| 2 8 9 | 7 6 4 | 5 1 3 |
| 6 4 7 | 5 3 1 | 8 9 2 |
| 5 1 3 | 2 9 8 | 7 6 4 |
+-------+-------+-------+
Quote:
CathyW: I'm stuck
Andrew (in 2012): Another puzzle from my "unfinished backlog". There are so many 45s in this cage pattern that it's hard to know which ones to use. Once I'd found the right ones, there was still a lot of hard work. Mike and I both used forcing chains in N8; his was a more powerful one.
I'll rate my walkthrough at 1.75.
2021 revisit here
Walkthrough by mhparker:
CathyW wrote:
mhparker wrote:
I'll probably just quietly post my original walkthrough and leave it at that.
Please do post your walkthrough of the V1.5. I've already spent too much time today struggling (although eventually successfully!) on the A60 and Harry Potter is beckoning loudly! :wink:
Hope you're having a good time pottering around, if you know what I mean! :wink:

Here's my walkthrough for the A59 V1.5 as I originally wrote it nearly a week ago:


Assassin 59 V1.5 Walkthrough

1. 13/2 at R1C1: no 1,2,3

2. 14/4 at R1C3: no 9

3. 12/2 at R1C5: no 1,2,6

4. 10/2 at R1C8: no 5

5. 10/3 at R3C1: no 8,9

6. 13/2 at R3C3: no 1,2,3

7. 10/2 at R3C7: no 5

8. 10/2 at R6C3: no 5

9. 7/2 at R6C7: no 7,8,9

10. 11/3 at R6C8: no 9

11. 12/2 at R8C5: no 1,2,6

12. 19/3 at R8C7: no 1

13. 6/2 at R9C1 = {15/24} = {(4/5)..}

14. 10/2 at R9C8: no 5

15. Innies C123 = R19C3 = 5/2 = {14/23}

16. Innies C789 = R19C7 = 8/2: no 4,8,9

17. Outies N5 = R37C5 = 10/2: no 5

18. Split C5 innie cage at R456C5 = 11/3 = {128/137/146/236/245} (no 9)

19. Hidden killer quad on {6789} in C5, as follows:
19a. Both of 12/2 must each contain exactly 1 of {6789}
19b. R37C5 (step 17) must contain exactly 1 of {6789}
19c. -> 11/3 at R456C5 must contain exactly 1 of {6789}
19d. -> {245} combo blocked
19e. -> no 5 in R456C5

20. 5 in C5 locked in R1289C5
20a. -> one of the 2 12/2 cages at R1C5 and R8C5 must be {57}
20b. -> no 7 in R34567C5
20c. Cleanup: no 3 in R37C5 (step 17)
20d. 11/3 at R456C5 now restricted to {128/146/236} = {((6/8)..}

21. Outies R12 = R3C46 = 12/2: no 1,2,6
21a. no 3 in R3C6

22. Outies R89 = R7C46 = 11/2: no 1

23. Innies C1234 = R456C4 = 19/3
23a. {568} blocked by R456C5 (step 20d)
23b. 19/3 at R456C4 = {289/379/469/478} (no 1,5)

24. 5 in N5 locked in C6 -> no 5 elsewhere in C6
24a. Cleanup: no 7 in R3C4 (step 21), no 6 in R7C4 (step 22)

25. Innies R1234 = R4C19 = 13/2: no 1,2,3

26. Innies R6789 = R6C19 = 12/2: no 1,2,6

27. Outies C12 = R258C3 = 17/3 = {179/269/359/278/368/458/467}
27a. -> must contain exactly 1 of {1234}
27b. 10/2 at R6C3 must also contain exactly 1 of {1234}
27c. -> 17/3 at R258C3, 10/2 at R6C3 and R19C3 form naked killer quad on {1234} in C3
27d. -> no {49} combo in 13/2 at R34C3 = {58/67} = {(6/8)..}, {(7/8)..}
27e. -> {278} and {368} combos for 17/3 at R258C3 blocked by 13/2 at R34C3 (step 27d)
27f. -> 17/3 at R258C3 = {179/269/359/458/467}

28. Outies C89 = R258C7 = 20/3 = {389/479/569/578} (no 1,2)

29. Innies R2 = R2C456 = 10/3 = {127/136/145/235} (no 8,9)
29a. {12} only in R2C46
29b. -> no 7 in R2C46
29c. Cleanup: no 3,4 in R1C5

30. Innies R8 = R8C456 = 9/3 = {135/234} (no 6,7,8,9)
30a. ({126} blocked because none of these digits are present in R8C5)
30b. 3 locked in R8C456 for R8 and N8
30c. Cleanup: no 4,5 in R9C5; no 8 in R7C46 (step 22)

31. 3 in N7 now locked in 22/4 innies at R7C123+R9C3 = {2389/3469/3478/3568} (no 1)
31a. Cleanup: no 9 in R6C3, no 4 in R1C3 (step 15)

32. {458} combo for 17/3 at R8C1 blocked by 6/2 at R9C1
32a. -> 17/3 at R8C1 = {179/269/278/467} (no 5)

33. Either 9/3 at R8C456 = {135}, or...
33a. ...9/3 at R8C456 = {234} -> R7C46 (step 22) = [56]
33b. -> 5 in N8 locked in R7C4+R8C45
33c. -> no 5 in R9C4

34. I/O difference N2: R3C5 = R1C3 + R1C7 + 1
34a. min. sum of R1C3 + R1C7 = 3
34b. -> min. of R3C5 = 4 (no 1,2)
34c. Cleanup: no 8,9 in R7C5 (step 17)

35. 8 in N8 locked in R9 -> not elsewhere in R9
35a. Cleanup: no 2 in 10/2 at R9C8 = {19/37/46}

36. Common Peer Elimination (CPE): R7C46 see all candidate positions for {12} in C5
36a. -> no 1,2 in R7C46

37. Discontinuous complex Nice Loop with 2 strong links at discontinuity:
37a. (will use "=>" notation for strong links)
37b. Either R8C456 contains a 1, OR...
37c. ...contains a 2 (internal strong link)
37d. -> R7C46 <> {29}
37e. => R7C123 must contain a 9
37f. -> 17/3 at R8C1 cannot contain a 9, and thus also cannot contain a 1
37g. => R8C456 must contain a 1
37h. Conclusion: 9/3 at R8C456 = {135} (step 30); 1,5 locked for R8 and N8
37i. Cleanup: no 8 in R9C5, no 9 in R3C5 (step 17), no 6 in R7C6 (step 22)

38. 1 in C5 locked in N5 -> not elsewhere in N5
38a. -> 11/3 at R456C5 = {128/146} (no 3)

39. 3 in C5 now locked in R28C5
39a. -> one of the 2 12/2 cages at R1C5 and R8C5 must be {39}
39b. The other 12/2 cage is {57} (step 20a)
39c. -> no {48} combo for 12/2 at R1C5

40. {179} combo not available for 17/3 at R8C1 = {269/278/467} = {(2/4)..}
40a. -> {24} combo blocked for 6/2 at R9C1
40b. -> 6/2 at R9C1 = {15}, locked for R9 and N7
40c. no 9 in 10/2 at R9C8 = {37/46}; no 3,7 in R1C7 (step 16)

41. 9 in R9 locked in N8 -> not elsewhere in N8
41a. Cleanup: no 2 in R7C46 (step 22)

42. 11/2 at R7C46 (step 22) = {47}, locked for R7 and N8
42a. Cleanup: no 5 in R8C5; no 3,6 in R6C3; no 3 in R6C7; no 6 in R3C5 (step 17)

43. 12/2 at R8C5 = [39]
43a. -> 12/2 at R1C5 = {57}, locked for N2

44. Naked single (NS) at R8C6 = 1
44a. -> R8C4 = 5

45. 1 in C4 locked in R12C4
45a. -> no 1 in R1C3 (same cage)
45b. -> 5/2 at R19C3 (step 15) = {23}, locked for C3
45c. Cleanup: no 7,8 in 10/2 at R6C3 = [19/46]

46. 4 in R9 locked in R9C89
46a. -> 10/2 at R9C8 = {46}, locked for R9 and N9
46b. Cleanup: no 1 in R6C7, no 2 in R1C7 (step 16)

47. Hidden single (HS) in R9 at R9C7 = 7
47a. -> R1C7 = 1 (step 16)
47b. Cleanup: no 6 in R6C7; no 9 in 10/2 at R1C8; no 3,9 in 10/2 at R3C7 = {28/46}

48. Naked single (NS) at R7C6 = 4
48a. -> R9C6 = 8 (cage sum)
48b. Cleanup: no 4,8 in R3C4 (step 21)

49. R79C4 = [72] (naked singles)
49a. -> R7C5 = 6, R9C3 = 3
49b. -> R1C3 = 2, R3C5 = 4 (step 17)
49c. Cleanup: no 4 in R6C3, no 6 in R4C7, no 8 in 10/2 at R1C8 = {37/46}

50. 10/2 at R6C3 = [19]

51. 12/2 at R3C46 = [39]

52. R12C6 = [62]
52a. Cleanup: no 7 in 13/2 at R1C1, no 4 in 10/2 at R1C8

53. R12C4 = [81]
53a. Cleanup: no 5 in 13/2 at R1C1

54. HS in R1 at R1C5 = 5
54a. -> R2C5 = 7

55. Naked pair (NP) on {28} in R7/N7 at R7C12
55a. -> no 2,8 elsewhere in R7 and N7
55b. R6C2 = 6 (cage split)
55c. Cleanup: no 5 in R6C7, no 7 in R3C3

56. Hidden pair (HP) on {17} in N1 at R3C12
56a. -> R3C12 = {17} (no 5,6)
56b. R4C2 = 2 (cage split)
56c. Cleanup: no 8 in R3C7

57. NS at R7C2 = 8
57a. -> R7C1 = 2

58. 10/2 at R1C8 = {37}, locked for N3

59. 13/2 at R1C1 = {49}, locked for N1

60. 17/3 at R2C1 = {368} (no 5), locked for R2 and N1
60a. Cleanup: no 5,7 in R4C3

Now it's all naked and hidden singles to the end.
Andrew's 2012 Walkthrough:
Prelims

a) R1C12 = {49/58/67}, no 1,2,3
b) R12C5 = {39/48/57}, no 1,2,6
c) R1C89 = {19/28/37/46}, no 5
d) R34C3 = {49/58/67}, no 1,2,3
e) R34C7 = {19/28/37/46}, no 5
f) R67C3 = {19/28/37/46}, no 5
g) R67C7 = {16/25/34}, no 7,8,9
h) R89C5 = {39/48/57}, no 1,2,6
i) R9C12 = {15/24}
j) R9C89 = {19/28/37/46}, no 5
k) 10(3) cage R3C1 = {127/136/145/235}, no 8,9
l) 11(3) cage at R6C8 = {128/137/146/236/245}, no 9
m) 19(3) cage at R8C7 = {289/379/469/478/568}, no 1
n) 14(4) cage at R1C3 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R12 2 outies R3C46 = 12 = [39]/{48/57}, no 1,2,6, no 3 in R3C6

2. 45 rule on R1234 2 innies R4C19 = 13 = {49/58/67}, no 1,2,3

3. 45 rule on R6789 2 innies R6C19 = 12 = {39/48/57}, no 1,2,6

4. 45 rule on R89 2 outies R7C46 = 11 = {29/38/47/56}, no 1

5. 45 rule on R2 3 innies R2C456 = 10 = {127/136/145/235}, no 8,9, clean-up: no 3,4 in R1C5
5a. 7 of {127} must be in R2C5 -> no 7 in R2C46

6. 45 rule on R8 3 innies R8C456 = 9 = {135/234} (cannot be {126} because no 1,2,6 in R8C5), no 6,7,8,9, 3 locked for R8 and N8, clean-up: no 8 in R7C46 (step 4), no 4,5 in R9C5
6a. 19(3) cage at R8C7 = {289/469/478/568}
6b. Killer single 5 in R8C456 and 19(3) cage, locked for R8

7. 45 rule on C123 2 innies R19C3 = 5 = {14/23}

8. 45 rule on C789 2 innies R19C7 = 8 = {17/26/35}, no 4,8,9

9. 45 rule on C89 3 outies R258C7 = 20 = {389/479/569/578}, no 1,2

10. 45 rule on N5 2 outies R37C5 = 10 = {19/28/46}/[37], no 5, no 7 in R3C5
10a. 45 rule on C5 3 remaining innies R456C5 = 11 = {128/146/236} (cannot be {137/245} which clash with the pair of 12(2) cages in C5), no 5,7,9

11. 45 rule on C1234 3 innies R456C4 = 19 = {289/379/469/478} (cannot be {568} which clashes with R456C5), no 1,5
11a. 5 in N5 only in R456C6, locked for C6, clean-up: no 7 in R3C4 (step 1), no 6 in R7C4 (step 4)

12. 45 rule on N2 1 innie R3C5 = 2 outies R1C37 + 1
12a. Min R1C37 = 3 -> min R3C5 = 4, clean-up: max R7C5 = 6 (step 10)
12b. 1,2 in C5 only in R4567C5, CPE no 1,2 in R6C46
12c. 8 in N8 only in R9C456, locked for R9, clean-up: no 2 in R9C89

13. 45 rule on N14 3(1+2) innies R1C3 + R6C23 = 9
13a. Max R6C23 = 8, no 8,9 in R6C23, clean-up: no 1,2 in R7C3
13b. Min R16C3 = 3 -> max R6C2 = 6

14. 45 rule on N36 3(1+2) innies R1C7 + R6C78 = 10
14a. Min R16C7 = 3 -> max R6C8 = 7

15. 45 rule on C12 3 outies R258C3 = 17 = {179/269/278/359/368/458/467}
15a. Killer quad 1,2,3,4 in R19C3, R258C3 and R67C3, locked for C3, clean-up: no 9 in R34C3
15b. R258C3 = {179/269/359/458/467} (cannot be {278/368} which clash with R34C3)

16. 3 in N7 only in R7C123 + R9C3
16a. 45 rule on N7 4 innies R7C123 + R9C3 = 22 = {2389/3469/3478/3568}, no 1, clean-up: no 4 in R1C3 (step 7)
16b. R7C12 cannot be {59/68} (5 of {3568} must be in R7C12) -> no 2 in R6C2

17. 45 rule on N47 3 innies R4C23 + R9C3 = 13
17a. Min R49C3 = 7 -> max R4C2 = 6

18. 45 rule on N3 2 outies R4C78 = 1 innie R1C7 + 8, IOU no 8 in R4C8

19. 45 rule on N7 2 outies R6C23 = 1 innie R9C3 + 4, IOU no 4 in R6C2

20. 45 rule on N9 2 outies R6C78 = 1 innie R9C7 + 2, IOU no 2 in R6C8

[This was how far I got when this puzzle first appeared. Some steps have been rewritten in my current walkthrough style.]

21. 5 in N2 only in 14(4) cage at R1C3, R12C5 = {57} and R3C46 (step 1) = [57]
21a. 14(4) cage at R1C3 = {1238/1256/1346/2345} (cannot be {1247} which contains 7 but not 5, locking-out cages), no 7
[Note. This elimination wouldn’t have worked if there had still been 7 in R3C4.]

22. R258C7 (step 9) = {389/479/569/578}, R19C7 (step 8) = {17/26/35} -> combined cage R19C7 + R258C7 = {17}{389}/{17}{569}/{26}{389}/{26}{479}/{26}{578}/{35}{479}
22a. R34C7 = {19/28/46} (cannot be {37} which clashes with combined cage), no 3,7

23. 5 in N5 only in R456C6
23a. 45 rule on C6789 3 innies R456C6 = 15 = {159/258/357/456}
23b. Consider placements for 9 in N8
9 in 17(4) cage at R7C4 = {1259/1349}, 1 locked for N8 -> no 9 in R3C5 (step 10)
or R89C5 = [39] -> no 9 in R3C5
or 9 in 20(4) cage at R7C6, locked for C6 -> no 1 in R456C6 -> 17(4) cage at R3C5 cannot be 9[215/314/413] because R34C5 = [91] clashes with R37C5 (step 10), CCC
-> no 9 in R3C5, clean-up: no 1 in R7C5 (step 10)
23c. R37C5 = [46/64/82]
23d. R12C5 = [57/75/93] (cannot be [84] which clashes with R37C5), no 4,8
23e. R89C5 = [39/57] (cannot be [48] which clashes with R37C5), no 4,8
23f. Naked quad {3579} in R1289C5, locked for C5
23g. 1 in C5 only in R456C5, locked for N5, clean-up: no 9 in R456C6 (step 23a)
23h. 9 in N5 only in R456C4, locked for C4, clean-up: no 2 in R7C6 (step 4)

24. R3C5 = R1C37 + 1 (step 12)
24a. Max R3C5 = 8 -> max R1C37 = 7, no 7 in R1C7, clean-up: no 1 in R9C7 (step 8)

25. 5 in N8 only in 17(4) cage at R7C4 and R89C5 = [57]
25a. 17(4) cage at R7C4 = {1358/1457/2348/2357/2456} (cannot be {1268} because 6,8 only in R9C4, cannot be {1367} which contains 7 but not 5, locking-out cages)
25b. 4 of {1457} must be in R9C3, 6,8 of {2348/2456} must be in R9C4 -> no 4 in R9C4

26. 9 in C4 only in R456C4 (step 11) = {289/379/469}
26a. Hidden killer triple 6,7,8 in 14(4) cage at R1C3, R456C4 and 17(4) cage at R7C4 for C4, R456C4 contains one of 6,7,8, 17(4) cage contains one of 6,7,8 -> 14(4) cage must contain one of 6,8 (7 has been eliminated) -> 14(4) cage (step 21a) = {1238/1256/1346}, CPE no 1 in R1C6

27. 45 rule on N8 2 outies R9C37 = 1 innie R7C5 + 4
27a. R7C5 = {246} -> R9C37 = 6,8,10 = [42/26/35/37/46], no 3 in R9C7, clean-up: no 5 in R1C7 (step 8)

28. R37C5 (step 23c) = [46/64/82], R456C5 (step 10a) = {128/146}
28a. R7C5 = {246} -> 23(4) cage at R6C4 = 17(3)+6/19(3)+4/21(3)+2 = {179/458}6/{568}4/{489/678}2 (cannot be {278}6/{289}4 which clash with R456C5 = {128} when R7C5 = {46}, CCC, cannot be {359}6/{379}4/{579}2 because no 3,5,7,9 in R6C5), no 2,3 in R6C456
28b. 5 of {458}6 must be in R6C6, 4 of {489}2 must be in R6C5 (cannot be [984]2 which clashes with R37C5), no 4 in R6C6
28c. 5 of {568}4 must be in R6C6, 6 of {678}2 must be in R6C5 (cannot be [768]2/[867]2 which clash with R37C5), no 6 in R6C6
28d. R456C6 (step 23a) = {258/357} (cannot be {456} = {46}5 which clashes with 23(4) cage = {458}6/{568}4, CCC), no 4,6

29. R37C5 (step 23c) = [46/64/82], R456C5 (step 10a) = {128/146}
29a. R3C5 = {468} -> 17(4) cage at R3C5 = 4+13(3)/6+11(3)/8+9(3) = 4{139/157/256}/6{137/245}/8{135/234} (cannot be 4{238}/6{128} which clashes with R456C5 = {128} when R3C5 = {46}, CCC, cannot be 8{126} which clashes with R456C5 = {146} when R3C5 = 8, CCC), no 8 in R4C456
29b. 17(4) cage = 4{139/256}/6{137}/8{135/234} (cannot be 4{157} which clashes with {179/458}6 (step 28a) using R37C5, cannot be 6{245} which clashes with {568}4 (step 28a) using R37C5)
29c. 6 of 4{256} must be in R4C4 (cannot be 4[625] which clashes with R37C5), no 6 in R4C5
29d. 4 of 8{234} must be in R4C5 (cannot be 8[423] which clashes with R37C5), no 4 in R4C4

30. 17(4) cage (step 29b) = 4{139/256}/6{137}/8{135/234}, 23(4) cage at R6C4 (step 28a) = {179/458}6/{568}4/{489/678}2 -> combined cage 17(4) + 23(4) = 4{139/256}+{179/458}6/8{135/234}+{489/678}2 (cannot be 6{137}+{568}4 which clashes with 15(3) cage at R5C4 = {249} = [942]) -> no 6 in R3C5, no 7 in R4C46, no 4 in R7C5 (step 23c)
30a. 23(4) cage at R6C4 = {179/458}6/{489/678}2
30b. 6 of {678}2 must be in R6C5 (cannot be [687]2 which clashes with R37C5), no 6 in R6C4
30c. From combined cage 17(4) + 23(4), 15(3) cage at R5C4 = {159/249/267/348}

31. R7C5 = {26} -> R9C37 = 6,10 (step 27a) = [42/37/46], no 2 in R9C3, no 5 in R9C7, clean-up: no 3 in R1C3 (step 7), no 3 in R1C7 (step 8)

32. R9C37 = [42/37/46] (step 31), R9C5 = {79}, R9C89 = {19/37/46} -> combined cage R9C35789 = [42]7{19}/[42]9{37}/[37]9{46}/[46]7{19}/[46]9{17}, 4,7 locked for R9, clean-up: no 2 in R9C12
32a. Naked pair {15} in R9C12, locked for R9 and N7, clean-up: no 9 in R9C89
32b. 9 in R9 only in R9C56, locked for N8, clean-up: no 2 in R7C4 (step 4)
32c. 2 in R9 only in R9C467, CPE no 2 in R8C6
[Cracked. The rest is fairly straightforward.]

33. R7C46 = [47/56/74]
33a. R89C5 = [39] (cannot be [57] which clashes with R7C46)
33b. 7 in N8 only in R7C46 = {47}, locked for R7 and N8 -> R8C6 = 1, R8C4 = 5 (step 6), clean-up: no 3,6 in R6C3, no 3 in R6C7

34. 1 in N2 only in R12C4, locked for 14(4) cage at R1C3 -> R1C3 = 2, R9C3 = 3 (step 7), clean-up: no 8 in R1C89, no 7 in R6C3, no 8 in R7C3, no 7 in R9C89

35. R9C7 = 7 (hidden single in R9), R1C7 = 1 (step 8), R7C6 = 4, R9C6 = 8 (cage sum), R7C4 = 7, R9C4 = 2 (cage sum), R7C5 = 6, R3C5 = 4 (step 23c), R7C3 = 9, R6C3 = 1, R6C5 = 8, clean-up: no 9 in R1C89, no 9 in R3C7, no 6,9 in R4C7, no 4 in R6C19 (step 3), no 6 in R6C7

36. Naked pair {12} in R45C5, locked for N5
36a. Naked triple {357} in R456C6, locked for C6 and N5 -> R3C6 = 9, R3C4 = 3 (step 1), R12C6 = [62], R12C4 = [81], clean-up: no 5,7 in R1C12, no 4 in R1C89
36b. Naked pair {37} in R1C89, locked for R1 and N3 -> R12C5 = [57]

37. Naked pair {28} in R7C12, locked for R7 and N7, R6C2 = 6 (cage sum), clean-up: no 7 in R3C3, no 7 in R4C9 (step 2), no 5 in R6C7

38. R6C19 (step 3) = {39} (only remaining combination, cannot be {57} which clashes with R6C6), locked for R6 -> R6C4 = 4, R6C6 = 5 (cage sum), R45C6 = [37], R6C7 = 2, R7C7 = 5, R6C8 = 7, R1C89 = [37], R7C89 = [13], R6C19 = [39], clean-up: no 8 in R34C7, no 4 in R4C1 (step 2)

39. R34C7 = [64], clean-up: no 9 in R4C1 (step 2), no 7 in R4C3

40. 45 rule on N3 1 remaining outie R4C8 = 5, R4C2 = 2, R4C5 = 1, R4C4 = 9 (cage sum)

and the rest is naked singles.

Rating Comment. I'll rate my walkthrough at 1.75. I used heavy analysis and a short forcing chain.


Last edited by Ed on Sun Jul 06, 2008 10:26 am, edited 1 time in total.

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Ed
PostPosted: Fri Jun 27, 2008 11:13 pm 
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Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 60 by Ruud (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:1536:1536:5890:5890:5890:5890:5382:3591:3591:1536:5642:5642:3340:3340:3854:5382:5382:3591:5906:5642:2324:2324:3340:3854:3854:4121:4121:5906:2844:2324:3358:3358:3616:3616:3874:4121:5906:2844:2844:3358:4392:4392:3616:3874:3874:5906:4398:4398:3888:4392:2610:2610:4148:4148:4150:4150:4398:3888:3888:2610:5180:5180:4148:5183:4150:3393:5186:5186:2628:5180:4166:4166:5183:5183:3393:3393:5186:2628:2628:4166:4166:
Solution: 
+-------+-------+-------+
| 1 3 5 | 6 8 4 | 9 2 7 |
| 2 6 9 | 3 1 7 | 8 4 5 |
| 8 7 4 | 2 9 5 | 3 1 6 |
+-------+-------+-------+
| 6 2 3 | 1 5 8 | 4 7 9 |
| 4 1 8 | 7 6 9 | 2 5 3 |
| 5 9 7 | 4 2 3 | 1 6 8 |
+-------+-------+-------+
| 7 4 1 | 8 3 6 | 5 9 2 |
| 3 5 2 | 9 7 1 | 6 8 4 |
| 9 8 6 | 5 4 2 | 7 3 1 |
+-------+-------+-------+
Quote:
mhparker, rating post: (rating) 1.5: Hard Assassin, having a significantly longer and/or narrower solution path, and/or requiring more advanced techniques. The A60 was definitely one of these.
CathyW: Ruud has certainly ramped up the difficulty this week
Andrew: I particularly liked Para's (walk-through) which was so direct.
Walkthrough by rcbroughton:
Plugged away at this one to a solution:


a. Cage 21(3) n3 no 1/2/3
0b. Cage 9(3) n124 no 7/8/9
0c. Cage 11(3) n4 no 9
0d. Cage 10(3) n568 and 10(3) n89 - no 8/9
0r. Cage 20(3) n9, 20(3) n7, 20(3) n8 - no 1,2

1. 6(3) n1={123} locked for n1

2. 22(3) n1 no 1/2/3/4 - must use 9 - locked for n1

3. 45 Rule on n3 - innies r3c789 total 10 = {127}/{136}/{145}/{235} - no 8,9
3a. 16(3) n36 means r3c89 can't be {12}/{13}/{14}/{15}/{23}/{27}
3b. r3c7 to can't be 4, 5, 6 or 7

4. 45 Rule on n7 - innies r789c3 total 9 = {126}/{135}/{234} - no 7/8/9
4a. 13(3) n78 means r89c3 can't be {12}
4b. no 6 at r7c3

5. 45 Rule on n3 - outies r4c9 minus innies r3c7 equals 6 -> r4c9 = 7/8/9

6. 45 Rule on n7 - outies r9c4 minus innies r7c3 equals 4 - no 1/2/3/4 in r9c4

7. 45 Rule on c98 - innies r27c8 total 13 ={49}/{58}/{67} - no 3 at r7c8

8. 45 Rule on c123 - outies r39c4 minus innies r1c3 equals 2 - no 5/6 r3c4

9. 45 Rule on c6-9 - innies r15c6 total 13 = {49}/{58}/{67} - no 1/2/3

10. 45 Rule on n9 - innies r7c9 r9c7 total 9 - no 1/9 at r7c9

11. 45 Rule on r9876 - innies r6c15 total 7 - no 7/8/9

12. 15(3) n23 - {456} not valid
12a. r3c7 = 1/2/3 - can't be at r23c6

13. 9(3) n124 - {234} must have 4 at r3c3 so no 4 at r3c4 r4c3
13b. {126}/{135} - can't have 5/6 in r4c3

14. 17(3) n5 only combo with 1 requuires a 7 - only occurs in r5c5 - so no 1 at r5c5

15. 17(3) n47 - 1 requires {79}, 2 requires {69} or{78} - no 1/2 r6c23

16. 45 Rule on n4 - outies r3c1 r7c3 minus innies r4c3 equals 6 - no 6 at r3c1

17. 45 Rule on r89 - innies r8c27 total 11 - no 1/9 at r8c2

18. 45 Rule on n124 - outies r3c7 r7c3 total 4 - no 4/5 at r7c3

19. 45 Rule on n6985 - innies r4c9 r9c4 total 14 - no 8/9 at r9c4
Or the same elimination looking at innies/outies on n7 - as in step 6

20. 45 Rule on r12 - outies r3c2567 total 24
[Edit - don't need the extra 20a here]20b. no possibility with 1/2/3 at r3c5

21. 13(3) n2 - only combo with 9 is {139} - 9 must be at r3c5 - no 9 at r2c45

22. 17(3) n47 again - {467}/{458} can't be placed - no 4
22a. {359}/{368} need 3 at r3c3 - no 3 at r6c23

23. 45 Rule on n2 - outies r1c3 r3c7 minus innies r3c4 equals 6 - no 6 at r1c3

24. innies on n1 - r3c13 r1c3 = h17(3) =4{58}/4{67}

25. 45 Rule on n2 - outies r134c3 r3c7 total 15
25a. can't have [63] [41] at r34c3 because of cage sum for 9(3) n124
25b. h17(3) from 24 means r13c3 can't be [56] [74] [75]
25c. for r134c3=13 - [562]/[751]/[742] - blocked by 25b [841]/[463] blocked by 25a25d. can't make total 13 in r134c3 ( and [463]) - no 2 at r3c7

26. 45 Rule on n3 - outies r4c9 minus innies r3c7 equals 6 - no 8 at r4c9

27. 16(3) n36 - can't make a combo with 7 at r3c89

28. 45 Rule on n124 - outies r3c7 r7c3 total 4 - no 2 at r7c3

29. 45 Rule on n7 - outies r9c4 minus innies r7c3 equals 4 - no 6 at r9c4

30. 20(3)n8 - {578} blocked by r9c4 - remaining combos must use 9 - locked for n8

31. 13(3) n78 - can't place {346} because of r9c4 - ={157}/{247}/{256} - no 3

32. 45 Rule on r1 - innies r1c12789 total 22
32a. r1c12 totals 3, 4, 5
32b. r1c89 total 5,6,7,8,9,10,11,12,13
32c. r1c89 can't be {58} {47} {56} {37} {46} {45} - no valid combo in 14(3)
32d. can't make a combo with 8 at r1c7

preamble to step 33
14(3) n3
21(3)n3 blocks {158}/{347}
h10(3) blocks {149}
h10(3) and 21(3) blocks {356}
14(3) = {167}/{239}/{248}/{257}

33. 45 Rule on r1 - outies r2c1789 total 19
33a. r2c789 totals 16(no 3),17(no 2),18(no 1)
33b. r2c78 can only be: 12={75}/{84},14={68}/{95},15={78},16={79},17={89}33c. can't make combos with 7,8,9 at r2c9

33. 45 Rule on r1 - innies r1c7 minus outies r2c19 equals 2 - no 4 at r1c7
[should have been 34 - oops]

34. 45 Rule on c1 - innies r12789c1 total 22=
34a. r12c1 totals 3,4,5
34b. r789c1 totals 19,18,17
34c. r89c1 can't be {68} {48} {37} {46} {67} {45} because of cage sum in 20(3)n7
34d can't make a combo with 9 at r7c1

35. innies on r123 = r3c13489 = 21(5)
35a. c89 can only total 7 or 9
35b. c34 can only total 8,7 or 6
35c. {12369} -> can't be placed because of r3c1
35d. {12378} -> can't be placed because of r3c1 has only 7/8
35e. {12459} -> can't be placed - no 9
35f. {12468} -> [842]{16}
35g. {12567} -> [761]{25}
35h. {13458} -> blocked by r3c7
35i. {13467} - ditto
35j. {23457} -> [742]{35}
35k. no 4/5 r3c1, no 5 r3c3, no 3 r3c4, no {34} r3c89

36. Hidden single 3 at r3c7
36a. 15(3)=3{48}/{57} - no 6,9

37. 4 locked in c3 of n1

38. 9(3) n1 now = {126{/{234} - must use 2. -> no 2 at r4c4

39. 16(3) n3 now = {16}9/{25}9 - 9 placed at r4c9

40 13(3)n7={157}/{256} - must use 5 - no 5 at r9c12

41. 23(4)c1 - on;y combos with 1 also need a 9 - no 1 at r5c1

42 20(3)n7 - only combo with a 6 needs a 5 - no 6 at r8c1

43. from step 28 - r7c3 = 1
43a. 17(3)=1{79} - {79} locked for n4, r6

44. 3 locked in r45 for c3 - locked for n4

45. 23(4)c1= {2678}/{4568} - must use 68 locked for n4 and c1

45. 11(3)n4 now={128}/{245} - must use 2 locked for n4
45a. naked single 3 at r4c3 -> 9(3) = [423]

46. 16(3)n36 = {16}9 - {16} locked for n3, r3

47. hidden triple {456} at r456c1 -lcked for n4, r1
47a. -> 23(4) = 8{456}

48. hidden single 8 at r5c3

Rest is simple eliminations


Still stumped with the Reject Version

Rgds
Richard
Walkthrough by mhparker:
Hi folks,

I found this Assassin comparable in difficulty to last week's A59 V1.5, so I was surprised to see two walkthroughs posted already (congratulations Richard and Cathy!). Maybe it's because this Assassin was more susceptible to intensive innie/outie work. In my case, however, I ignored much of this information, and used a couple of interesting chains to gain a foothold instead. Therefore, I've decided to post my walkthrough, too.

Now for the A60RP, if Richard hasn't finished it already... :wink:


Assassin 60 Walkthrough

1. 6/3 at R1C1 = {123}, locked for N1

2. 21/3 at R1C7 = {489/579/678} (no 1,2,3)

3. 22/3 at R2C2 = {(58/67)9} (no 4)
3a. 9 locked for N1

4. 9/3 at R3C3 = {126/135/234} (no 7,8,9)
4a. needs 2 of {123}, only in R3C4+R4C3
4b. -> R3C4 = {123}, R4C3 = {123}

5. 11/3 at R4C2 = {128/137/146/236/245} (no 9)

6. 10/3 at R6C6 and R8C6 = {127/136/145/235} (no 8,9)

7. 20/3 at R7C7, R8C1 and R8C4 = {389/479/569/578} (no 1,2)

8. Innies N3: R3C789 = 10/3 = {127/136/145/235} (no 8,9)

9. Outies N124: R3C7+R7C3 = 4/2
9a. -> R3C7 = {123}, R7C3 = {123}
9b. -> min. R6C23 = 14
9c. -> no 1,2,3,4 in R6C23

10. I/O difference N3: R4C9 = R3C7 + 6
10a. -> R4C9 = {789}

11. Innies C89: R27C8 = 13/2 = {49/58/67} (no 3)

12. Outies R12: R3C2567 = 24/4
12a. max. R3C7 = 3
12b. -> min. R3C256 = 21
12c. -> no 1,2,3 in R3C56
12d. Cleanup: no 9 in R2C45

13. Killer Hidden Triple (KHT) on {123} in R3 at R3C4789
13a. -> {145} combo blocked for 10/3 at R3C789 (step 8)
13b. -> no 4 in R3C89

14. 15/3 at R2C6 cannot be {456} due to R3C7
14a. cannot reach cage sum with 2 of {123}
14b. -> must contain exactly 1 of {123}, which must go in R3C7
14c. -> no 1,2,3 in R2C6

15. I/O difference R12: R3C25 = R2C6 + 9
15a. -> no 9 in R3C2, no 4 in R3C5
15b. Cleanup: no 8 in R2C45

16. 9 in R3 locked in N2 -> not elsewhere in N2

17. 9 in R1 locked in N3 -> not elsewhere in N3
17a. Cleanup: no 4,5 in R1C7

18. 9 no longer available to 23/4 at R1C3
18a. -> 23/4 = {2678/3578/4568} (no 1)
18b. 8 locked for R1

19. 8 in N3 locked in R2 -> not elsewhere in R2
19a. Cleanup: no 5 in R3C2, no 4 in R3C6 (otherwise 15/3 cage sum unreachable)

20. 4 in R3 locked in N1 -> not elsewhere in N1

Now for a couple of chains (no eliminations yet):

21. if R3C7 = 2, then...
21a. -> R7C3 = 2 (step 9)
21b. -> 9/3 at R3C3 = {135}
21c. -> R3C89 <> {5..}

22. if R3C7 = 3, then...
22a. -> R3C6 <> 9 (combinations 15/3)
22b. -> R3C5 = 9 (strong link R3)
22c. -> R2C45 = {13}
22d. -> R3C4 = 2
22e. -> R3C89 <> {2..}

23. Above 2 chains (steps 21 and 22) together block {235} combo in R3C789 (step 8)
23a. -> R3C789 = {127/136} (no 5) (see also step 13a)
23b. 1 locked for R3 and N3

24. 1 in R1 locked in R1C12
24a. -> no 1 in R2C1

25. 1 in R2 locked in 13/3 at R2C4 = {139/148/157} (no 2,6)

26. {28} in R2 locked in R1 outies at R2C1789 = 19/4 = {2458/2368} (no 7)
26a. {23} only in R2C19
26b. -> no 6 in R2C9

27. {79} in 21/3 at R1C7 now only in R1C7
27a. -> {579} combo blocked
27b. -> 21/3 at R1C7 = {489/678} (no 5)
27c. 7 only in R1C7
27d. -> no 6 in R1C7

28. 5 in N3 locked in 14/3 at R1C8 = {257/356} (no 4,8,9)

29. Hidden single (HS) in R1/N3 at R1C7 = 9
29a. -> R2C78 = {48} (no 6), 4 locked for R2
29b. -> R2C19 = [25] (step 26)
29c. Cleanup: no 7,8 in R3C5 (see step 25); no 4,6 in R7C8; no 8 in R3C2; no 9 in R3C6

30. HS in R3/N2 at R3C5 = 9
30a. -> R2C45 = {13} (no 7), 3 locked for N2

31. Naked single (NS) at R3C4 = 2
31a. Cleanup: no 5 in R3C3; no 6,7 in R3C6; no 8 in R4C9 (step 10)

32. 2 in R1/N3 locked in split 9/2 at R1C89 = {27} (no 3,6)
32a. 7 locked for R1 and N3

33. HS in N2 at R2C6 = 7
33a. -> R3C67 = [53]
33b. -> R4C9 = 9 (step 10, but also obtainable via cage-split of 16/3)
33c. Cleanup: no 2,4 in R69C7

34. HS in R1/N1 at R1C3 = 5

35. HS in R3/N1 at R3C1 = 8

36. HS in R3/N1 at R3C3 = 4
36a. -> R4C3 = 3

37. HS in R3/N1 at R3C2 = 7

38. HS in C6 at R5C6 = 9
38a. Cleanup: no 4,8 in R56C5

39. Outie N47: R9C4 = 5
39a. -> no 8,9 in R89C3

40. Innie N7: R7C3 = 1
40a. -> R6C23 = [97]

41. R2C23 = [69]

42. HS in C3 at R5C3 = 8
42a. -> R45C2 = {12}, locked for C2 and N4

43. R1C12 = [13]

44. Naked triple (NT) on {456} in C1 at R456C1 -> no 4,5,6 elsewhere in C1

45. 20/3 at R8C4 = {389/479}
45a. 9 only in R8C4
45b. -> R8C4 = 9
45c. no 6 in R89C5

46. 20/3 at R8C1 = {389/479}
46a. 9 only in R9C1
46b. -> R9C1 = 9

47. HS in C8 at R7C8 = 9
47a. -> R2C8 = 4 (step 11)
47a. -> R2C7 = 8

48. Outie C45: R1C6 = 4

49. HS in C6 at R4C6 = 8
49a. -> no 6,7 in R45C7

50. Outies C6: R69C7 = 8/2 = [17]
50a. -> R67C6 = {36}; R89C6 = {12}, 2 locked for N8
50b. Cleanup: no 4 in R8C5

51. Innie N9: R7C9 = 2
51a. -> R6C89 = {68}, 6 locked for R6 and N6

(Pending naked/hidden singles now...)

52. 15/3 at R6C4 = {348} (no 6,7)
52a. 8 locked in R7C45 for R7 and N8

Now all naked singles to end
Walkthough by Para:
Hi all

Finally finished it. Had to write my whole walk-through over because word crashed on me before i saved anything. So it works a bit backwards. It opens with the moves that finally broke it open for me(because they were there from the beginning), and ends with the easier bits.

Walk-through Assassin 60


1. 6(3) at R1C1 = {123} -->> locked for N1

2. 21(3) at R1C7 = {489/579/678}: no 1,2,3

3. 22(3) at R2C2 = {589/679}: no 4; 9 locked for N1

4. 9(3) at R3C3 = {126/135/234}: no 7,8,9
4a. Needs one of {456}, goes in R3C3: R3C4 + R4C3: no 4,5,6

5. 11(3) at R4C2 = {128/137/146/236/245}: no 9

6. 10(3) at R6C6 and R8C6 = {127/136/145/235}: no 8,9

7. 20(3) at R7C7, R8C1 and R8C4 = {389/479/569/578}: no 1,2

8. 45 on N124: 2 outies: R3C7 + R7C3 = 4 = {13/22}: no 4,5,6,7,8,9

9. 45 on N3: 3 innies: R3C789 = 10 = {127/136/145/235}: no 8,9
9a. 45 on N3: 1 innie and 1 outie: R4C9 = R3C7 + 6 -->> R4C9 = {789}

10. 45 on R12: 4 outies: R3C2567 = 24 = {1689/2589/2679/3489/3579} = {7/8}({3678} blocked: 9 locked in these R3C256 for R3): Needs on of {123} in R3C7 -->> R3C56: no 1,2,3
10a. R3C7 + R7C3 = {22} -->> R3C4 + R4C3 = {13} -->> R3C3 = 5: R3C2567: {2589} blocked.

11. R3C789: no {127}: no 7
11a. Explanation: R3C789 = [1]{27} clashes with 16(3) cage at R3C8
11b. R3C789 = [2]{17}: blocked by R3C2567: when R3C7 = 2, R3C256 = {679}

12. Hidden Killer Pair {78} in R3C1 + R3C2567 for R3(R3C2567 needs one of {78} and only other place for {78} in R3 is R3C1) -->> R3C1 = {78}

13. 45 on N1: 3 innies: R1C3 + R3C13 = 17 = {458/467} -->> R13C3 = {45/46}: no 7,8; 4 locked for C3

14. 45 on N7: 3 innies: R789C3 = 9 = {126/135/234}: no 7,8,9
14a. 45 on N7: 1 innie and 1 outie: R9C4 = R7C3 + 4 -->> R9C4 = {567}

15. 13(3) at R8C3 = {157/256}(only remaining combinations) = {15}[7]/{25}[6]/{26}[5] -->> R89C3 = {15/25/16}: no 3

16. Killer Pair {56} in R13C3 + R89C3 locked for C3

17. 45 on C12: 4 outies : R2567C3 = 25 = {1789} -->> R7C3 = 1, R256C3 = {789}
17a. R9C4 = 5(step 14a); R3C7 = 3(step 8); R4C9 = 9(step 9a), R4C3 = 3(hidden)
17b. R89C3 = {26} -->> locked for C3 and N7

18. 11(3) at R4C2 = {128} (only possible combination as it needs one of {78} in R5C3) -->> R5C3 = 8; R45C2 = {12} -->> locked for C2 and N4
18a. R1C2 = 3

19. 23(4) at R3C1 = {4568} (last possible combination): no 7,9; R3C1 = 8; R456C1 = {456} -->> locked for C1 and N4
19a. R6C23 = {79} -->> locked for R6
19b. R789C1 = {379} -->> locked for N7
19c. R789C2 = {458} -->> locked for C2

20. R3C2567 = 3{579}: R3C256 = {579} -->> locked for R3
20a. R3C3 = 4; R3C4 = 2; R1C3 = 5; R2C2 = 6(hidden)

21. 6 in N2 locked in R1C456 -->> 23(4) at R1C3 = 5{468} -->> R1C456 = {468} -->> locked for R1 and N2
21aa. R3C89 = {16} -->> locked for N3 (did this but not on paper)
21a. R1C1 = 1(hidden); R2C1 = 2
21b. R1C789 = {279} -->> locked for N3

22. 15(3) at R2C6 = 3{57}(last remaining combination) -->> R23C6 = {57} -->> locked for C6 and N2
22a. R3C5 = 9; R3C2 = 7; R2C3 = 9; R23C6 = [75]; R6C23 = [97]; R5C6 = 9(hidden)

23. 21(3) at R1C7 needs 2 of {458} in R2C78 -->> 21(3) = {489} -->> R1C7 = 9; R2C78 = {48} -->> locked for N3
23a. R2C9 = 5

24. 20(3) at R8C4 = {389/479} -->> R8C4 = 9; R89C5 = {3478}

25. 20(3) at R8C1 = {389/479} -->> R9C1 = 9
25a. R7C8 = 9(hidden)

26. 45 on C89: 1 innie: R2C8 = 4; R2C7 = 8

27. 1 in N8 locked for C6 and 10(3) cage at R8C6
27a. 10(3) at R8C6 = {127/136} -->> R89C6 = {12/13}: no 4,6; R9C7 = {67}

28. 45 on N69: 1 outie and 2 innies: R4C6 = R69C7: Min R69C7 = 7 -->> R4C6 = 8 -->> R69C7 = [17/26]: R6C7 = {12}

29. 10(3) at R6C6 = {136}: no {24}
29a. R6C7 = 1; R9C7 = 7; R1C6 = 4(hidden)

30. 45 on N9: 1 innie: R7C9 = 2
30a. R1C89 = [27]; R6C5 = 2(hidden); R5C5 = 6; R67C6 = [36]
30b. R6C4 = 4; R1C45 = [68]; R4C5 = 5(hidden)

31. 45 on R6789: 1 innie: R6C1 = 5
31a. R45C1 = [64]

And the rest is all singles

greetings

Para
Walkthrough by CathyW:
Edit: Thanks to Ed and Mike - hopefully this one is error free. I think I'd previously left in an invalid step from WT1. Thankfully the end result is the same. :)
At the third attempt:

1. 6(3) N1 = {123} not elsewhere in N1

2. 22(3) N1 = {589/679} 9 not elsewhere in N1

3. Innies N1 r1c3 + r3c13 = 17 = {458/467} -> r3c3 = (456) -> r3c4, r4c3 of 9(3) = (123)

4. Innies N3: r3c789 = 10 = {127/136/145/235}

5. Outies – Innies N3: r4c9 – r3c7 = 6
-> r3c7 = (123), r4c9 = (789) -> r23c6 = (4…9)

6. O-I r123: r4c39 – r3c1 = 4; r4c39 min 8, max 12

7. O-I N2: r1c3 + r3c7 – r3c4 = 6 -> r1c3 <> 6
r1c3 + r3c7 = r3c4 + 6 = 7, 8 or 9 = [52/43], [53/71], [81/72]

8. Innies N9: r7c9 + r9c7 = 9 = [81]/{27/36/45}

9. Innies N7: r789c3 = 9 = {126/135/234}

10. O-I N7: r9c4 – r7c3 = 4
r9c4 = (5…9) -> r7c3 <> 6

11. Innies r89: r8c27 = 11 = [29]/{38/47/56}

12. Innies r6789: r6c15 = 7 = {16/25/34}

13. Innies c89: r27c8 = 13 = {49/58/67}

14. Innies N5689: r4c9 + r9c4 = 14 = [95/86/77]
-> r9c4 <> 8,9 -> r7c3 <> 4,5 -> r6c23 = (5…9)

15. Innies c6789: r15c6 = 13 = {49/58/67}

16. Outies N124: r3c7 + r7c3 = 4 = {13/22}

17. O-I r12: r3c25 – r2c6 = 9 -> r3c25 min 13, max 17 -> r2c6 <> 9
a) r2c6 = 4 -> r3c25 = {58/67} ({49} not an option}
b) r2c6 = 5 -> r3c25 = [59]/{68}
c) r2c6 = 6 -> r3c25 = [69]/{78}
d) r2c6 = 7 -> r3c25 = [79]
e) r2c6 = 8 -> r3c25 = [89]
For all options r3c2 <> 9 -> 9 locked to r2c23, not elsewhere in r2.
-> r7c8 <> 4
-> 9 locked to r1c789 -> r1c456 <> 9 -> r5c6 <> 4

18. 23(4) r1c3456 = {2678/3578/4568} no 1 and must have 8 -> r1c789 <> 8
-> 8 locked to r2c789, not elsewhere in r2
-> r3c2 <> 5

19. O-I c123: r39c4 – r1c3 = 2
a) r1c3 = 4 -> r39c4 = 6 = [15]
b) r1c3 = 5 -> r39c4 = 7 = [16/25]
c) r1c3 = 7 -> r39c4 = 9 = [27/36]
d) r1c3 = 8 -> r39c4 = 10 = [37]

20. O-I N4 r3c1+r7c3-r4c3 = 6 -> r3c1 <> 6

21. From step 6:
a) r3c1 = 4 -> r4c39 = 8 = [17]
b) r3c1 = 5 -> r4c39 = 9 = [18/27]
c) r3c1 = 7 -> r4c39 = 11 = [29/38]
d) r3c1 = 8 -> r4c39 = 12 = [39]

22. 21(3) N3: If {489} r1c7 = 9 -> r1c7 <> 4; if {579} r1c7 = 9 -> r1c7 <> 5

23. 17(3) N5: r5c5 <> 1 since no 7 or 9 in r6c5

24. From step 9: split 9(3) r789c3 = {126/135/234} – must have two of 123; r4c3 = (123)
Killer combination on 1, 2, 3 -> r5c3 <> 1,2,3 -> r45c2 <> 7,8

25. 15(3) r3c7+r23c6 = 1[59/68], 2[49/58]/{67}, 3[48]/{57} -> r3c6 <> 4

26. Outies N1234: r4c9+r7c3 = 10 = [73/82/91]

27. Outies N1247: r3c7+r9c4 = 8 = [17/26/35]

28. 21(3) N3 = {489/579/678} -> Combinations {158/347} blocked for 14(3) N3
-> 14(3) = {149/167/239/248/257/356}

29. 16(3) r7c12+r8c2 = {169/178/268/349/358/457}
Combination {259/367} blocked by split 9(3) r789c3

30. Outies c89: r1278c7 = 28 = {4789/5689} -> r45c7 <> 8, 9
-> split 11(2) r8c27 = [29/38/47/56/65/74] (r8c2 <> 8)

31. 20(3) N9 = {479/569/578} -> split 9(2) N9 <> {45}, 16(4) N9 <> {45…} but must have one of 4,5 -> 16(4) = {1249/1258/1348/2347/2356} ({1357 blocked by 20(3))

32. O-I c12: r25c3 – r6c2 = 8 -> r25c3 min 13, max 17

33. Outies r12: r3c2567 = 24 (9 locked to r3c56) = [6{89}1]/[6{79}2]/[7{69}2]/[7953]/[8{69}1]/[8{59}2] ([7593] not possible as couldn’t make up 15(3) r23c6+r3c7)

34. Innies N5: r4c6+r6c46 = 15

35. Outies c12: r2567c3 = 25 = {1789/2689/3589/3679} (4 eliminated from r5c3) -> r45c2 of 11(3) <> 5,6 -> 11(3) N4 = {128/137/146/236/245}

36. 23(4) r1c3456 = {2678/3578/4568} – Must have one of 3,6 within N2 -> 13(3) N2 <> {346}

37. From step 16:
a) r3c7 = 1, r7c3 = 3 -> r4c9 = 7, r9c4 = 7
NT {689} r3c256 (step 33) -> r3c89 = {45}
CONFLICT – no candidates would be left in r3c3
-> r3c7 <> 1, r7c3 <> 3 -> r9c4 <> 7 -> r4c9 <> 7

b) r3c7 = 2, r7c3 = 2 -> r4c9 = 8, r9c4 = 6, r3c4+r4c3 = {13} -> r3c3 = 5 -> 22(3) N1 = {679} … OK

c) r3c7 = 3, r7c3 = 1 -> r4c9 = 9, r9c4 = 5, -> r3c256 = [795] -> r23c6 = [75] -> r2c23 = {69}

Conclusion: either remaining option for split 4(2) r3c7 + r7c3, 22(3) N1 = {679}
-> split 17(3) N1 = {458}

38. 9(3) r3c34+r4c3 = 4{23}/5{13} -> either r3c4 or r4c3 = 3 -> r4c4 <> 3

39. From step 17:
a) r2c6 = 4 -> r3c25 = {67}
b) r2c6 = 5 -> r3c25 = [68]
c) r2c6 = 6 -> r3c25 = [78]
d) r2c6 = 7 -> r3c25 = [79]
-> r3c5 <> 5 -> r2c45 <> 7 -> split 24(4) r3c2567 = [6{79}2]/[7{69}2]/[[7953]
-> r3c56 <> 8 -> HS r3c1 = 8
(Andrew pointed out: Step 39d also gives r2c6 <> 5,6 since 39b and 39c no longer apply.)

40. NP {45} r13c3 -> 4,5 not elsewhere in c3 -> 13(3) r89c3+r9c4 = {26}5 -> r4c9 = 9, r3c7 = 3, r7c3 = 1
-> 2,6 not elsewhere in N7/c3 -> r4c3 = 3, r6c23 = {79} not elsewhere in N4/r6 -> r5c3 = 8, r45c2 = {12} -> r1c2 = 3 -> r12c1 = {12}, r456c1 = {456}, r789c1 = {379}, r789c2 = {458}

41. UR r2c2 = 6 -> r3c2 = 7, r2c3 = 9, r6c2 = 9, r6c3 = 7, …
(Andrew commented: It looks as if there’s also UR r3c2 = 6 -> r2c23 = {79}, r6c23 = {79} -> r3c2 = 7.)

Relatively straightforward cage combos and singles from here

PS I've finished 'pottering' as well! Did someone mention work?!
Walkthrough by Andrew:
Ruud wrote:
The solving skills of the regular crowd has improved tremendously since I first posted these Assassins.
Definitely!
CathyW wrote:
Did anyone else try Mike's A61X?
Andrew wrote:
I started it last night but it was nearly bedtime so I didn't get much further than the preliminary steps and some easy 45s.

Must find time to work through the posted A60 walkthroughs and decide whether to post my one. Then I'll have a proper try at A61X.
That was written in the Assassin 61 thread on 1st August. I must admit I got sidetracked, first by A61X which I thought I was about to finish but then reached an impossible position (I've since managed to solve it. I'd eliminated a combination that contained the solution. :oops: ) and then by A62.

I eventually found time to work through Richard's and Mike's walkthroughs yesterday followed by Para's and Cathy's ones today. All contained excellent stuff and took different routes to reach the key eliminations from R3C7 and R7C3.

I'll have to make an effort to be first to post a walkthrough for some future Assassin. Then others will have to go through my walkthrough first before they post their ones! :wink:

I particularly liked Para's which was so direct. He said that he'd done a major rewrite after Word had crashed before he'd saved his walkthrough. Maybe it was a good thing that Word crashed on him! :wink: I also lost my partial walkthrough but that was just carelessness on my part. However my reconstruction was pretty well how I had done it first time.

Mike (steps 21 and 22) and Cathy (step 37) had chains based on R3C7 (and R7C3) which were used to achieve results away from R3C7 although Cathy also got an elimination from R3C7 out of it. I hope that my step 28, which produced an elimination from R3C7 and R7C3 is acceptable as a contradiction move. Maybe in hindsight I should have continued looking for other moves before I used that one.

Anyway, having gone through the other walkthroughs, here is how I solved Assassin 60.

1. 6(3) cage in N1 = {123}, locked for N1

2. 22(3) cage in N1 = {589/679} = 9{58/67}, 9 locked for N1
2a. 45 rule on N1 3 innies R1C3 + R3C13 = 17 = 4{58/67}

3. 9(3) cage at R3C3 = {126/135/234}, no 7,8,9
3a. R3C3 = {456} -> no 4,5,6 in R3C4 + R4C3

4. 21(3) cage in N3 = {489/579/678}, no 1,2,3

5. 11(3) cage in N4 = {128/137/146/236/246}, no 9

6. 10(3) cage at R6C6 = {127/136/145/235}, no 8,9

7. 20(3) cage in N7 = {389/479/569/578}, no 1,2

8. 20(3) cage in N8 = {389/479/569/578}, no 1,2

9. 10(3) cage at R8C6 = {127/136/145/235}, no 8,9

10. 20(3) cage in N9 = {389/479/569/578}, no 1,2

11. 45 rule on N7 3 innies R789C3 = 9 = {126/135/234}, no 7,8,9

12. 45 rule on N3 3 innies R3C789 = 10 = {127/136/145/235}, no 8,9

13. 45 rule on N9 2 innies R7C9 + R9C7 = 9 = {27/36/45}/[81], no 1,9 in R7C9

14. 45 rule on R89 2 innies R8C27 = 11 = [29]/{38/47/56}, no 1,9 in R8C2

15. 45 rule on C89 2 innies R27C8 = 13 = {49/58/67}, no 3

16. 45 rule on R1 1 innie R1C7 – 2 = 2 outies R2C19, min R2C19 = 3 -> min R1C7 = 5, max R1C7 = 9 -> max R2C19 = 7 -> max R2C9 = 6

17. 45 rule on C1 2 outies R19C2 – 4 = 1 innie R7C1, max R19C2 = 12 -> max R7C1 = 8

18. 45 rule on N5689 2 innies R4C9 + R9C4 = 14 = {59/68/77}

19. 45 rule on N124 2 outies R3C7 + R7C3 = 4 = {13/22}
19a. Max R7C3 = 3 -> min R6C23 = 14, no 1,2,3,4

20. 45 rule on R6789 2 innies R6C15 = 7 = {16/25/34}, no 7,8,9
20a. Max R6C5 = 6 -> min R5C56 = 11, no 1

21. 45 rule on C6789 2 innies R15C6 = 13 = {49/58/67}, no 1,2,3

22. 45 rule on N7 1 outie R9C4 – 4 = 1 innie R7C3 -> no 8,9 in R9C4, clean-up: no 5,6 in R4C9 (step 18)

23. Killer triple 1/2/3 in R4C3 and R789C3 (step 11), locked for C3

24. Min R5C3 = 4 -> max R45C2 = 7, no 7,8

25. 45 rule on C89 4 outies R1278C7 = 28 = {4789/5689} = 89{47/56}, 8,9 locked for C7, no 3, clean-up: no 8 in R8C2 (step 14)

26. 45 rule on R12 4 outies R3C2567 = 24, max R3C7 = 3 -> min R3C256 = 21, no 1,2,3
26a. Min R3C5 = 4 -> max R2C45 = 9, no 9

27. R3C4 = {123} -> R3C789 (step 12) must contain two of 1,2,3 = {127/136/235} (cannot be {145}), no 4
[I suppose this is called a Hidden Killer Triple. I’m just used to writing the logic as I spot it.]

28. R3C7 + R7C3 (step 19) = {13/22}
28a. If R3C7 + R7C3 = {22} => R3C4 + R4C3 = {13} => R3C3 = 5 => R3C789 = {127} ({136} clashes with R3C4 = {13}) => R3C89 = {17} = 8 => R3C4 = 3, R3C2567 = 24 (step 26), R3C34 = [53] = 8 -> R3C1 = 5 clashes with R3C3 -> R3C7 + R7C3 cannot be {22}
28b. R3C7 + R7C3 = {13}, clean-up: no 6 in R9C4 (step 22), no 8 in R4C9 (step 18)
[Step 28a was originally a contradiction for the 4 outies from N4 but then I saw the contradiction along R3 which is more direct and easier to present.]

29. 16(3) cage at R3C8 = {169/259/367}
29a. {367} must have 7 in R4C9 -> no 7 in R3C89
29b. R3C789 = {136/235} = 3{16/25}, 3 locked for R3 and N3

30. 13(3) cage at R8C3 = {157/247/256} (cannot be {346} because no 3,4,6 in R9C4), no 3

31. R3C789 contains 1/2 (step 29b) -> 14(3) cage in N1 must contain 1/2 = {149/158/167/239/248/257}

32. R3C2567 (step 26) = {1689/3489/3579/3678} (cannot be {4569/4578} which don’t contain 1,3)

33. Killer triple 4/5/6 in R3C2567 (step 32), R3C3 and R3C89 (step 29b), locked for R3

34. 45 rule on R123 2 outies R4C39 – 4 = 1 innie R3C1, R3C1 = {78} -> R4C39 = 11,12 = {29/39} -> R4C3 = {23}, R4C9 = 9, R9C4 = 5 (step 18), R7C3 = 1 (step 22), R3C7 = 3 (step 28b), clean-up: no 4,6 in R7C9 (step 13)
34a. R7C3 = 1 -> R6C23 = 16 = {79}, locked for R6 and N4
34b. R9C4 = 5 -> R89C3 = 8 = {26}, locked for C3 and N7 -> R4C3 = 3

35. R3C789 = 3{16/25} (step 29b)
35a. R4C3 = 3 -> R3C34 = 6 = [42/51]
35b. R3C89 cannot be {25} which clashes with R3C34 -> R3C89 = {16}, locked for R3 and N3 -> R3C34 = [42] -> R1C3 + R3C1 = [58] (step 2a) -> R2C2 = 6 (hidden single in N1), R5C3 = 8
35c. R5C3 = 8 -> R45C2 = 3 = {12}, locked for C2 and N4 -> R1C2 = 3

36. 5 in R3 locked in R3C56, locked for N2

37. R3C7 = 3 -> R23C6 = 12 = [75] (only remaining permuation) -> R2C3 = 9, R3C25 = [79], R6C23 = [97]

38. R456C1 = {456}, locked for C1

39. 21(3) cage in N3 = {489} (only remaining combination because the remaining 7 is in the same cell as the 9) -> R1C7 = 9, R2C78 = {48}, locked for R2 and N3
39a. 14(3) cage in N3 = {257} -> R2C9 = 5, R1C89 = {27}, locked for R1 -> R12C1 = [12]

40. R5C6 = 9 (hidden single in C6) -> R56C5 = 8 = {26/35]/[71], no 4

41. 1 in N8 locked in R89C6, locked for C6, no 1 in R9C7
41a. 10(3) cage at R8C6 = 1{27/36}, no 4
41b. {127} has 7 in R9C7 -> no 2 in R9C7
41c. {136} has 6 in R9C7 -> no 6 in R89C6

42. Grouped X-Wing in 1, R3C89 = {16}, 1 in N9 locked in R89C89, 1 locked for C89
[Alternatively 1 in C7 locked in R456C7, locked for N6]

43. 20(3) cage in N9 = {479/569/578}
43a. {479} has 9 in R7C8 -> no 4 in R7C8
43b. {569} has 9 in R7C8 -> no 6 in R7C8

44. Killer pair 6/7 in 20(3) cage (step 43) and R9C7, locked for N9

45. 16(4) cage in N9 = {1249/1258/1348}
45a. Killer pair 8/9 in 16(4) and 20(3) cages in N9, locked for N9

46. 16(3) cage at R6C8 = {268/358} = 8{26/35}, no 4
46a. 8 locked in R6C89, locked for R6 and N6
46b. R7C9 = {23} -> no 2,3 in R6C89

47. 14(3) cage at R4C6 = 2{57}/6{17}8{15/24} -> no 4 in R4C6, no 6 in R45C7

48. 3 in N6 locked in R5C89, locked for R5
48a. 15(3) cage = {357} (only remaining combination), locked for N6
48b. 5 in N6 locked in R45C8, locked for C8

49. R6C89 = {68} -> R7C9 = 2 -> R1C89 = [27], R5C9 = 3
49a. Naked pair {68} in R6C89, locked for R6

50. 14(3) cage at R4C6 = {248} (only remaining combination) -> R4C6 = 8, R45C7 = {24}, locked for C7 -> R2C78 = [84], R6C7 = 1
50a. R6C7 = 1 -> R67C6 = 9 = [36] (only remaining permutation) -> R1C6 = 4, R6C4 = 4, R6C1 = 5, R6C5 = 2, R5C5 = 6 (cage sum), R1C45 = [68], R45C1 = [64], R4C5 = 5 (hidden single in N5) and the rest is naked singles and cage sums
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Ed
PostPosted: Fri Jun 27, 2008 11:17 pm 
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Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 60 RP by Ruud (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2304:2304:3330:3330:3330:4869:4869:5639:5639:2304:5642:5642:3852:3852:3342:4869:4869:5639:5138:5642:4372:4372:3852:3342:3342:5145:5639:5138:3100:4372:3870:3870:2848:2848:5145:5145:5138:3100:3100:3870:3368:3368:2848:5931:5145:3629:4142:4142:3632:3368:4402:4402:5931:5931:3629:3629:4142:3632:3632:4402:3644:3644:5931:6463:3629:4161:4161:6979:6979:3644:4678:4678:6463:6463:6463:4161:4161:6979:6979:4678:4678:
Solution: 
+-------+-------+-------+
| 2 4 1 | 5 7 3 | 9 6 8 |
| 3 6 7 | 4 9 8 | 5 2 1 |
| 5 9 8 | 6 2 1 | 4 3 7 |
+-------+-------+-------+
| 9 7 3 | 1 5 6 | 2 8 4 |
| 6 1 4 | 9 8 2 | 3 7 5 |
| 8 5 2 | 7 3 4 | 6 1 9 |
+-------+-------+-------+
| 1 2 9 | 3 4 7 | 8 5 6 |
| 7 3 5 | 8 6 9 | 1 4 2 |
| 4 8 6 | 2 1 5 | 7 9 3 |
+-------+-------+-------+
Quote:
mhparker, rating post: (rating) 3.0: "Ruudiculous", requiring a team effort and massive hypotheticals to solve, if it can be solved at all....(possibly) A60RP could be considered examples of this.
rcbroughton: This is a real tough nut.
mhparker: My only real suggestions are that we document the misery, be thankful for every candidate we've managed to pick off, pack our bags and go home**!
Ruud: A60RP may be unsolvable at this moment, but a lot of progress has been made in the Killer community, so I expect this puzzle to fall at the hands of a skilled player at some time in the future.
Afmob (6 months later): This one was definitely harder than A74 Brick Wall since even finding those hypotheticals was difficult especially those moves between step 9 and 14 took me a long time to find...Rating: Hard 2.5. I had to use some hypotheticals and massive combo analysis but those hypotheticals were of medium length
Andrew (in 2015): This is the hardest puzzle I've done for a long time.
Mike's rating definitions included "3.0: "Ruudiculous", requiring a team effort and massive hypotheticals to solve, if it can be solved at all. The A50V2 and (possibly) A60RP could be considered examples of this." I'm not sure whether Mike meant that A60RP was harder or easier than A50V2. In my opinion, having done both of these puzzles in recent months, A60RP is a lot harder!
Walkthrough by Afmob with obvious moves left out:
Another unsolvable off the list :lol:. This one was definitely harder than A74 Brick Wall since even finding those hypotheticals was difficult especially those moves between step 9 and 14 took me a long time to find.
Now every Killer below 2.0 should help me to relax :wink:.

Since I try to shorten up my walkthroughs there might be some obvious moves I left out (e.g. step 4: R4C6 <> 7 because of R8C7) on purpose because they're not needed to solve this assassin.

A60 RP Walkthrough:

1. R6789
a) Innies R89 = 4(2) = {13} locked for R8
b) 27(4) = 9{378/468/567} -> 9 locked between R9+N8 -> R9C45 <> 9
c) 14(3): R7C78 <> 1,3 because R8C7 = (13)
d) Innies N9 = 13(2) <> 1,2,3
e) Innies+Outies: 4 = R5C8 - R6C5
-> R5C8 = (56789), R6C5 = (12345)

2. C6789
a) Innies C89 = 7(2) <> 7,8,9; R2C8 <> 4,6
b) Innies+Outies: 4 = R8C5 - R5C6
-> R8C5 <> 4, R5C6 = (12345)
c) 13(3) @ N5 must have 6,7,8 xor 9 and it's only possible @ R5C5 -> R5C5 = (6789)
d) 14(3): R7C7 <> 2,4 because R7C8+R8C7 <= 9
e) Outies = 17(3): R6C5 <> 5 because Outies would be >= 18

3. C123
a) 22(3) = 9{58/67} -> 9 locked for N1
b) Innies = 17(4) -> R34C3 @ 17(3) <= 14 -> R3C4 <> 1,2

4. N5689
a) Outies = 8(1+1) <> 8,9; R3C8 <> 5,7

5. R6789+C6789
a) Outies R6789 = 17(3): R5C6 <> 5 because Outies would be >= 18
b) Innies+Outies R6789: 4 = R5C8 - R6C5 -> R5C8 <> 9
c) Innies+Outies C6789: 4 = R8C5 - R5C6 -> R8C5 <> 9
d) Killer pair (13) locked in 18(4) + 16(4) for R9
e) 27(4) = 69{48/57} -> 6 locked between R9+N8 -> R9C45 <> 6

6. N23
a) Innies+Outies: -8 = R1C3 - R3C48
-> R1C3 <> 8 because R3C48 <= 15

7. C123
a) Innies+Outies: R18C3 = R3C4 -> R3C4 <> 4 because R8C3 <> 1,3
b) 14(4): R6C1 <> 7 because 7{124} blocked by Killer pair (24) of 25(4)
c) 14(4): R6C1 <> 5 since 5{234} blocked by Killer pair (24) of 25(4) and
5{126} forces 25(4) = {4579} -> no candidate for R8C3

8. R789
a) 14(4) <> 7 because if 14(4) = {1247}:
- 25(4) = {2689} locked for N9 -> 14(4) = 2{47}1 -> R8C3 = 5; 1 locked in 18(4) for R9
- 16(4) = {2356} -> R8C4 = 6; 6 locked in 25(4) for R9 -> no 6 in 27(4) (step 5e)
b) Innies N7 = 20(5) = 13{259/268/457}: R78C3 <> 4 because
- R78C3 = {47} (because 7 only possible there) -> R7C12+R8C2 = {135} -> no combo for 14(4)
c) R3C8 <> 4 (step 4b)
d) 18(4) must have 1 xor 3 -> 18(4) <> 13{59/68}, <> {2457} and
<> {3456} because it's blocked by Killer triple (456) of Innies N9 = 13(2)

9. C123
a) Innies+Outies N7: -6 = R6C1 - R78C3: R7C3 <> 1 because
- only combos with 1 are R78C3 = 1[6/7] (since R78C3 >= 7)
- i) R78C3 = [16] -> Innies C123 = 17(4) = {2456} -> R34C3 @ 17(3) must be {45}
- R13C3 = 2[4/5] -> <> [25] since it's a Killer pair of 9(3) @ N1
- R13C3 = [24] -> Innies N1 = 14(3) = [284] -> no combo for 17(3) since it must be [485]
- ii) R78C3 = [17] -> Innies C123 = 17(4) = {2357} -> R34C3 @ 17(3) must be {35}
- R13C3 = 2[3/5] -> <> [25] (Killer pair of 9(3))
- R13C3 = [23] -> Innies N1 = 14(3) = [293] -> no combo for 17(3) since it must be [395]

b) 1 locked in 14(4) for N7 -> R6C1 <> 1; 14(4) = 1{238/256/346}
c) 14(4) <> 5 because:
- R7C12 = {25} (since (56) is a Killer pair of 25(4)) and R8C2 = 1
- Innies N7 = 20(5) must be {12359} -> not possible because R8C3 <> 3,9
-> 14(4) = 13{28/46}

d) Innies+Outies N7: -6 = R6C1 - R78C3: R7C3 <> 2 because
- only possible combos are R78C3 = 2[6/7]
- [27] impossible since it's a Killer pair of 25(4)
- R78C3 = [26] -> Innies C123 = {1367} -> because of Innies N1 = 14(3) and 17(3)
R134C3 must be [371]
- Innies N1 = 14(3) = [347]
- 20(3) = 4{79}, {79} locked for C1+N4
- {58} locked in 25(4) -> no combo for 25(4)

10. R789 !
a) 16(4) must have 1 xor 3 (step 5d, 8d) -> 16(4) <> 13{48/57}
b) 16(4) <> {1456} since it would force 25(4) = {2689} (2 only possible @ 25(4) + 18(4) for R89)
- 6 locked in 18(4) + 25(4) for R89 -> no 6 in 27(4) (step 5e)
c) ! 14(3) @ N9 <> 2 because
- 14(3) @ N9 = [923] -> Innies N7 = {13457}, R2C8 = 5 (step 2a) -> 14(4) = 6{34}1
- 6 locked in 25(4) = {2689} + 27(4) for R89
- 18(4) = {1458} -> 3 locked in 16(4) @ R9 = {2347} -> R8C3 = 7 -> R3C8 = 1 (step 4b)
- {48} locked for C8 and R9C9 = 1
- 23(4) @ N6 must have {67} -> impossible because 23(4) <> 1 and R6C8 <> 2,8

e) R2C8 <> 5 (Innies C89 = 7(2))
f) ! 2 locked in 18(4) + 16(4) for R89
g) 25(4) = 47{59/68} -> 4,7 locked for N7
h) R3C8 <> 1 (Outies N5689 = 8(1+1))
i) 14(4): R6C1 <> 6 because R7C12+R8C2 <> 4
j) Innies N7 = 20(5): R7C3 <> 5 because 9 only possible there

11. R123
a) Innies = 22(4): R3C13 <> 6 because
- possible combos are 6{259/349/358} since R3C8 = (23)
- {2569} -> R3C13 = {56} blocked by Killer pair (56) of 22(3)
- {3469} -> R3C13 = {46} -> Innies N1 = 14(3) = 46{?}
- {3568} -> R3C13 = {56} blocked by Killer pair (56) of 22(3) and
- R3C13 = {68} leaves no combo for Innies N1 = 14(3)
b) Innies N1 = 14(3): R3C1 <> 3 because
- I N1 = {356} blocked by Killer pair of (56) of 22(3)
- I N1 = [437] -> I R123 = 22(4) = 37{48} not possible because R3C8 <> 4,8
- I N1 = [734] -> I R123 = [3496] (not 34{78} because R3C8 <> 7,8) -> 17(3) = 49[?]
c) Innies N1 = 14(3) <> 6
- {356} blocked by Killer pair (56) of 22(3) -> I N1 = [671]
- I R123 = 71{59/68} -> 71{59} impossible because R3C8 <> 5,9
- I R123 = [7186] -> 17(3) = 18[?]
d) Innies N1 = 14(3) = {158/248/347} because {257} blocked by Killer pair (57) of 22(3)
e) 20(3) <> 3 because
- 20(3) = {389} -> R3C1 = 8 and 3 locked in 14(4) for C2 -> 9(3) = {126}
- no combo for 22(3)

12. C123
a) Consider combos of 20(3) -> R89C1 <> 5
- i) 20(3) = {479} -> R89C1 = (568) -> 25(4) = {4678} <> 5
- ii) 20(3) = 5{69/78} -> 5 locked for C1 -> R89C1 <> 5
b) Innies C1 = 25(6) <> 5 (I C1 = {123568}) because R89C1 would be {68}
-> blocked by Killer pair (68) of 14(4)
c) 5 locked in 20(3) @ C1 -> 20(3) <> 4

13. R123+C123 !
a) Innies N1 = 14(3): R3C3 <> 3 because
- R3C13 = [73] -> Innies R123 = 22(4) = 37{48} -> not possible because R3C8 <> 4,8
b) ! Innies N1 = 14(3) <> 3 because
- I N1 = [374] -> I R123 = 74[92/83/56]
- I R123 = [7492] -> 17(3) = 49[?]
- I R123 = [7483] -> 17(3) = [485] -> I C123 = 17(4) = 345[?]
- I R123 = [7456] -> 17(3) = [458] -> no combo for 20(3) (R3C1 = 7 but R4C3 = 8)
c) Innies N1 = 14(3) = 8{15/24} -> 8 locked for R3+N1
d) 22(3) = {679} locked for N1
e) 9(3): R1C2 <> 1 because R12C1 <> 5
f) 17(3): R4C3 <> 1 because R3C3 <> 7,9
g) Innies+Outies N23: -8 = R1C3 - R3C48; R1C3 = (1245)
-> R3C4 <> 5 because R3C8 <> 4,7,8
h) 17(3): R4C3 <> 4,5 because R3C3 <> 3,6,7,9 and R3C4 <> 4,5,8

14. C123 !
a) 3 locked in 9(3) + 14(4) for C12
b) ! 9(3) <> {135} because
- R12C1 = {13} and R3C1 = 8 -> 20(3) = 8{57} locked for C1
- R67C1 = (246) -> only {46} possible @ R14(4) -> no 2 @ C1
c) 9(3) = {234} locked for N1
d) Hidden Single: R7C1 = 1 @ C1
e) R8C2 = 3, R8C7 = 1
f) 1 locked in R13C3 for C3

15. R789
a) 3 locked in 18(4) for R9
b) 14(3) = 1{49/58/67} -> R7C7 <> 5,6
c) 16(3) <> {457} because R7C3 = (689)
d) 16(3): R6C3 <> 4 because 3 only possible there

16. C123 !
a) ! Innies+Outies C1: -5 = R17C2 - R89C1; R1C2 = (24)
-> R7C2 <> 8 because R89C1 @ 25(4) can't be {69/78/89}
b) 14(4): R6C1 <> 2
c) 2 locked in 9(3) @ C1 -> R1C2 <> 2
e) R1C2 = 4
e) ! Innies C1 = 19(3) <> {469} because R89C1 @ 25(4) can't be {69}
f) Innies C1 = 19(3) = {478} locked for C1
g) R3C1 = 5, R1C3 = 1, R3C3 = 8
h) 20(3) = {569} -> 6,9 locked for N4
i) 17(3) = 8[63/72] -> R3C4 <> 3,9; R4C3 <> 7
j) Innies = 17(4) = 18[26/35] -> R8C3 <> 2
k) Hidden Single: R7C2 @ N7 = 2 -> R6C1 = 8, R5C3 = 4 @ N4

17. C123
a) 12(3) = {147} -> {17} locked for C2+N4
b) 16(3) = {259} -> R6C2 = 5, R6C3 = 2, R7C3 = 9
c) 25(4) = {4678} -> R9C2 = 8, R9C3 = 6
d) R8C3 = 5 -> R3C8 = 3 (Outies N5689 = 8(1+1))
e) 17(3) = {368} -> R3C4 = 6, R4C3 = 3
f) 16(4) = {1258} -> R8C4 = 8, 2 locked for R9

18. R789
a) 27(4) = {5679} -> 5 locked for R9, 6 locked for R8+N8
b) 18(4) = {2349} locked for N9, R9C9 = 3
c) 14(3) @ N8 = {347}
d) 5,9 locked in R789C6 for C6
e) Innies N9 = 13(2): R7C9 <> 5,7
f) 7 locked in R79C7 for C7

19. C456
a) 11(3): R4C6 <> 7,8 because R45C7 >= 5
b) 8 locked in R45C5 for C5
c) 13(3) @ R1 = 1{39/57}
d) Killer pair (79) locked in 13(3) @ R1 + 15(3) for N2
e) 19(4) must have 1 xor 2 and R2C8 = (12) -> R1C67+R2C7 <> 2
f) 13(3) @ R2C6 must have 3,8 and it's only possible @ R2C6 -> R2C6 = (38)
g) 13(3) @ R2C6 = 1{39/48} -> R3C6 = 1
h) 3 locked in R12C6 for C6+N2
i) 13(3) @ R1 = {157}, {57} locked for R1+N2
j) 13(3) @ N5 = 2{38/47} -> R5C6 = 2
k) 9 locked in 15(3) = {159}

20. N58
a) 11(3) = 2{36/45} -> R4C7 = 2, R5C7 = (35)
b) 17(3) = {467} because (39) only possible @ R6C7
c) Naked triple (467) locked in R467C6 for C6
d) 27(4) = {5679} -> R8C6 = 9, R9C6 = 5, R9C7 = 7, R8C5 = 6
e) 14(3): R6C4 <> 3 because 3{47} blocked by R7C6 = (47)

21. Rest is singles.

Rating: Hard 2.5. I had to use some hypotheticals and massive combo analysis but those hypotheticals were of medium length

(Archive Note) A couple of typos have been corrected.
Walkthrough by Andrew:
As with several other difficult puzzles, where I've used a lot of forcing chains, I've repeated the results from previous steps which I hope will make my walkthrough easier to follow (but also makes it longer).

Prelims

a) 9(3) cage at R1C1 = {126/135/234}, no 7,8,9
b) 22(3) cage at R2C2 = {589/679}
c) 20(3) cage at R3C1 = {389/479/569/578}, no 1,2
d) 11(3) cage at R4C6 = {128/137/146/236/245}, no 1
e) 14(4) cage at R6C1 = {1238/1247/1256/1346/2345}, no 9
f) 27(4) cage at R8C5 = {3789/4689/5679}, no 1,2

Steps resulting from Prelims
1a. 22(3) cage at R2C2 = {589/679}, 9 locked for N1
1b. 27(4) cage at R8C5 = {3789/4689/5679}, CPE no 9 in R9C45

2. 45 rule on N89 2 innies R8C27 = 4 = {13}, locked for R8

3. 45 rule on N9 2 innies R7C9 + R9C7 = 13 = {49/58/67}, no 1,2,3

4. 45 rule on C89 2 innies R27C8 = 7 = {16/25/34}, no 7,8,9
4a. Max R7C8 + R8C7 = 9 -> min R7C7 = 5
4b. 14(3) cage in N9 can only contain one of 1,3 -> no 1,3 in R7C8, clean-up: no 4,6 in R2C8

5. 45 rule on R6789 1 outie R5C8 = 1 innie R6C5 + 4, no 1,2,3,4 in R5C8, no 6,7,8,9 in R6C5

6. 45 rule on N5689 2(1+1) outies R3C8 +R8C3 = 8 = [17]/{26/35}/[44], no 8,9, no 5,7 in R3C8

7. 45 rule on C123 2 innies R18C3 = 1 outie R3C4
7a. Min R18C3 = 3 -> min R3C4 = 3
7b. Max R18C3 = 9 -> no 8 in R1C3

8. 45 rule on C6789 1 outie R8C5 = 1 innie R5C6 + 4, no 6,7,8,9 in R5C6, no 4 in R8C5

9. 13(3) cage in at R5C5 must contain one of 6,7,8,9 -> R5C5 = {6789}

10. 45 rule on R6789 4 outies R5C568 = 17 must contain one of 1,2,3,4 -> R5C6 = {1234}, no 5, clean-up: no 9 in R8C5 (step 8)

11. 45 rule on C6789 3 outies R568C5 = 17 must contain one of 1,2,3,4 -> R6C5 = {1234}, no 5, clean-up: no 9 in R5C8 (step 5)

[Now looking at interactions with the hidden 17(3) cages]
12. 13(3) cage at R5C5 = {139/238/247/346} (cannot be {148} because neither of R5C568 or R568C5 can be [818])
12a. 15(3) cage at R4C4 = {159/168/258/267/456} (cannot be {249/348/357} which clash with 13(3) cage), no 3
12b. R5C568 and R568C5 = {179/278/359/368/467} (cannot be {269} because 13(3) cage only contains one of 2,6,9, cannot be {458} because 13(3) cage can only contain one of 4,5,8)
[Note that the two hidden 17(3) cages cannot have the same combination because they each have two cells in the 13(3) cage.]

13. 14(3) cage at R7C7 = {149/158/167/239/347/356} (cannot be {248/257} because R8C7 only contains 1,3)
13a. 18(4) cage at R8C8 = {1269/1278/1458/1467/2349/2358/2367} (cannot be {1359/1368/2457/3456} which clash with 14(3) cage at R7C7)

14. 18(4) cage at R8C8 (step 13a) = {1269/1278/1458/1467/2349/2358/2367}
14a. 16(4) cage at R8C3 = {1249/1258/1267/1456/2347/2356} (cannot be {1348/1357} which clash with 18(4) cage)
14b. Killer pair 1,3 in 16(4) cage and 18(4) cage, locked for R9
14c. 27(4) cage at R8C5 = {4689/5679}, CPE no 6 in R9C45

15. 45 rule on R1 5(4+1) outies R2C1789 + R3C9 = 18
15a. Min R2C1789 = 10 -> max R3C9 = 8

16. R27C8 = 7 (step 4), R3C8 + R8C3 = 8 (step 6), 25(4) cage at R8C1 = {2689/4579/4678}, 16(4) cage at R8C3 (step 14a) = {1249/1258/1267/1456/2347/2356}
16a. 14(4) cage at R6C1 cannot be 7{24}1 which clashes with 25(4) cage -> no 7 in R6C1
16b. 14(4) cage cannot be 5{26}1 (because 25(4) cage = {4579} => R78C3 cannot be {38} because no 3,8 in R8C3), cannot be 5{24}3 which clashes with 25(4) cage) -> no 5 in R6C1
16c. 45 rule on N7 2 innies R78C3 = 1 outie R6C1 + 6
16d. Analysing 14(4) cage for each remaining value of R6C1
R6C1 = 1=> R7C3 = 1 (hidden single in N7), R8C3 = 6, R8C2 = 3, R7C12 = 10 = {28}
R6C1 = 2 => R7C12 + R8C2 = 12 = {138/147/345} (cannot be {156} which clashes with 25(4) cage)
and taking this further
R6C1 = 2 => R7C12 + R8C2 = 12 cannot be {147} => 25(4) cage {2689} => R78C3 = [35], R8C2 = 1 => R8C7 = 3 => 3 in R9 only in R9C45 => 16(4) cage at R8C3 with 5 in R8C3 = {2356} = [56]{23} => R6C1 = 2, R7C12 = {47}, R9C45 = {23} block all combinations for 25(4) cage
R6C1 = 2 => R7C12 + R8C2 = 12 cannot be {345} = {45}3 which clashes with 14(3) cage at R7C1 = {149/158} while R7C12 + R8C1 = {345} => R78C3 = [17] => R3C8 + R8C3 = [17] => R27C8 = {25}/[34] blocks 14(3) cage = {167})
-> when R6C1 = 2, R7C12 + R8C2 = {138} => R78C3 = {26}
R6C1 = 3 => R7C3 = 3 (hidden single in N7), R8C3 = 6, R8C2 = 1, R7C12 = 10 = {28}
R6C1 = 4 => R7C12 + R8C2 = 10 = {136} (cannot be {127} which clashes with 25(4) cage, cannot be {235} because R78C3 cannot be {19} since no 1,9 in R8C3) => R78C3 = [82]
R6C1 = 6 => R7C12 + R8C2 = 8 = {134} (cannot be {125} because R78C3 cannot be {39} since no 3,9 in R8C3) => R78C3 = {57}
R6C1 = 8 => R7C12 + R8C2 = 6 = {123}, R78C3 = 14 = [86/95]
16e. Summary R78C3 = [16]/{26}/[36]/{57}/[82/86/95], no 4 in R78C3, clean-up: no 4 in R3C8 (step 6)
R7C12 + R8C2 = {123/134/136/128/138/238}, no 5,7 in R7C12

17. 20(3) cage at R3C1 = {389/479/569/578}, 25(4) cage at R8C1 (step 16) = {2689/4579/4678}
17a. Consider placements for 4 in N7
4 in R7C12 + R8C2 (step 16e) = {134} => R6C1 = 6, 25(4) cage = {2689} => R89C1 = {28/29} (R89C1 cannot be {89} which clashes with 20(3) cage), 2 locked for N7
or 4 in 25(4) cage = {4579/4678}
-> no 2 in R9C23

18. 14(4) cage at R6C1 = {1238/1346}, 27(4) cage at R8C5 (step 14c) = {4689/5679}
18a. Consider combinations for 25(4) cage at R8C1 (step 16) = {2689/4579/4678}
25(4) cage = {2689/4579} => caged X-Wing for 9 in 25(4) cage and 27(4) cage for R89, no other 9 in R89
or 25(4) cage = {4678} => R78C3 = [95] (hidden pair in N7) => R8C4 + R9C45 = 11 => no 9 in R8C4
-> no 9 in R8C4

[All I can see now is a heavier step, and it gets even harder later.]
19. 25(4) cage at R8C1 (step 16) = {2689/4579/4678}, R7C9 + R9C7 (step 3) = {49/58/67}, R8C5 = R5C6 + 4 (step 8), 16(4) cage at R8C3 (step 14a) = {1258/1267/1456/2347/2356}, 27(4) cage at R8C5 (step 14c) = {4689/5679}
19a. Consider placements for 9 in N7
R7C3 = 9 => 9 in N8 only in R89C6, locked for 27(4) cage => no 9 in R9C7
or R8C1 = 9 and 4 in R7C12, locked for R7 => no 4 in R7C9 => no 9 in R9C7
or R8C1 = 9 and 4 in 25(4) cage = {4579} => R9C123 = {457}, locked for R9 and N7, R8C3 = {26} => 27(4) cage = {5679} (cannot be {4689} = [64]{89}, R8C3 = 2 which together with R9C23 = {57} block all combinations for 16(4) cage, cannot be [84]{69} which clashes with R5C6 + R8C5 = [48], IOD clash) = {57}{69}, 5,7 locked for R8 => 5,7 in N9 must be in R7C789 but R7C78 cannot contain both of 5,7 => R7C9 = {57} => R7C9 = 7, R9C7 = 6
or 9 in R9C789, locked for R9
-> no 9 in R9C7, clean-up: no 4 in R7C9

20. 27(4) cage at R8C5 (step 14c) = {4689/5679}, 9 locked for C6 and N8
20a. Max R67C6 = 15 -> min R6C7 = 2

21. From step 16d R7C3 can only contain 9 for R6C1 = 8, R7C12 + R8C2 = {123} -> R7C123 cannot contain both of 6,9 -> R8C13 + R9C123 must contain at least one of 6,9
21a. R8C13 + R9C123 contains at least one of 6,9, 27(4) cage at R8C5 (step 14c) = {4689/5679} contains both of 6,9 -> 18(4) cage at R8C8 cannot contain more than one of 6,9 -> 18(4) cage (step 13a) = {1278/1458/1467/2349/2358/2367} (cannot be {1269} which contains both of 6,9)
21b. 14(3) cage at R7C7 (step 13) = {149/158/167/239/356} (cannot be {347} which clashes with 18(4) cage)

22. 25(4) cage at R8C1 (step 16) = {2689/4579/4678}, 16(4) cage at R8C3 (step 14a) = {1258/1267/1456/2347/2356}, 27(4) cage at R8C5 (step 14c) = {4689/5679}
22a. Consider placements for 8 in N8
8 in R7C456 => 8 in N7 only in 25(4) cage = {2689/4678} => caged X-Wing for 6 in 25(4) cage and 27(4) cage, no other 6 in R89 => 16(4) cage = {1258/2347}
or 8 in 16(4) cage = {1258} or
8 in 27(4) cage = {4689} => caged X-Wing for 4 and/or 6 in 25(4) cage and 27(4) cage, no other cage in R89 can contain both of 4,6 => 16(4) cage = {1258/1267/2347/2356}
-> 16(4) cage = {1258/1267/2347/2356}

23. 16(4) cage at R8C3 (step 22a) = {1258/1267/2347/2356}
23a. Consider placements for 2 in N9
R7C8 = 2
or 2 in 18(4) cage at R8C8 => caged X-Wing for 2 in 16(4) cage and 18(4) cage, no other 2 in R89 => 2 in N7 only in R7C123, locked for R7
-> no 2 in R7C456
23b. 2 in N8 only in R8C4 + R9C45, locked for 16(4) cage -> no 2 in R8C3, clean-up: no 6 in R3C8 (step 6)

24. R78C3 (step 16e) = [16/26/36]/{57}/[86/95] -> no 6 in R7C3
24a. R78C3 cannot total 10 -> no 4 in R6C1 (step 16c)
24b. No 4 in R6C1 -> no 6 in R7C12 + R8C2 (from the analysis in step 16d) -> no 6 in R7C12

25. 27(4) cage at R8C5 (step 14c) = {4689/5679}
25a. 6 in N7 only in R8C13 + R9C123
25b. Grouped X-Wing for 6 in R8C13 + R9C123 and 27(4) cage, no other 6 in R89

26. 14(4) cage at R6C1 (step 18) = {1238/1346}, 20(3) cage at R3C1 = {389/479/569/578}
26a. Consider combinations for 25(4) cage at R8C1 (step 16) = {2689/4579/4678}
25(4) cage = {2689}, 2 locked for C1 and N7 => 14(4) cage = {1346} => R6C1 = 6, R78C3 (step 16d) = {57}, locked for C3, R7C8 = 2 (hidden single in R7) => R78C7 = [93], R8C2 = 1, 1 in C1 only in R12C1 => 9(3) cage at R1C1 = {135}, locked for N1 => 22(3) cage at R2C2 = {679}, locked for N1 => R1C3 + R3C13 = {248}, 2 locked for C3 => 2 in N4 must be in R456C2 but R456C2 cannot contain all of 2,5,8 because 12(3) cage at R4C2 can only contain one of 2,5,8 => either R45C1 must contain at least one of 5,8 and/or R456C3 must contain 8 => R3C1 = 8 (hidden single in N1) => 20(3) cage cannot be {479}
or 25(4) cage = {4579} => 20(3) cage cannot be {479}
or 25(4) cage = {4678} => 20(3) cage cannot be {479} (because R89C1 = {68} clashes with 14(4) cage at R6C1) -> 20(3) cage = {389/569/578} (cannot be {479}, no 4

27. 14(4) cage at R6C1 (step 18) = {1238/1346}
27a. Consider combinations for 20(3) cage at R3C1 (step 26a) = {389/569/578}
20(3) cage = {389/578}, locked for C1
or 20(3) cage = {569}, locked for C1 => 14(4) cage = {1238}, CPE no 8 in R89C1
-> no 8 R89C1

28. R18C3 = R3C4 (step 7)
28a. Min R18C3 = 6 -> min R3C4 = 6
28b. Max R18C3 = 9, min R8C3 = 5 -> max R1C3 = 4

29. 45 rule on N1 3 innies R1C3 + R3C13 = 14 = {158/167/248/347} (cannot be {257/356} which clash with 22(3) cage at R2C2)
29a. 1 of {158/167} must be in R1C3 -> no 1 in R3C3

30. R1C3 + R3C13 (step 29) = {158/167/248/347}, 20(3) cage at R3C1 (step 26a) = {389/569/578}, 14(4) cage at R6C1 (step 18) = {1238/1346}, 25(4) cage at R8C1 (step 16) = {2689/4579/4678}
30a. Consider placements for 9(3) cage at R1C1 = {126/135/234}
[Analysed in order {126/234/135} for clarity, since {135} is the most complicated]
9(3) cage = {126} with 2 in R12C1 => no 2 in R89C1
or 9(3) cage = {126} with 2 in R1C2 => R12C1 = {16}, locked for C1 => 14(4) cage = {1238}, CPE no 2 in R89C1
or 9(3) cage = {234} => 1 in C1 only in R67C1, locked for 14(4) cage => R8C2 = 3, R8C7 = 1 => R7C78 = 13, no 2 in R7C8 => 2 in R7 only in R7C123, locked for N7, no 2 in R89C1
or 9(3) cage = {135} => R1C3 + R3C13 = {248} => R3C1 = 8, R13C3 = {24}, locked for C3 => 4 in N4 only in R456C2
then consider the possibilities for 4 in R456C2 in more detail
4 in 12(3) cage at R4C2 = {147}, locked for N4 => 20(3) cage = 8{39}, locked for C1 => 25(4) cage = {4579/4678} (cannot be {2689} because R89C1 = {26} clashes with 14(4) cage), no 2 in R89C1
or 12(3) cage = {246}, locked for N4 => 14(4) cage = {1238}, CPE no 2 in R89C1
or 12(3) cage = {345}, locked for N4 => 20(3) cage = 8{39}, locked for C1 => 25(4) cage = {4579/4678} (cannot be {2689} because R89C1 = {26} clashes with 14(4) cage), no 2 in R89C1
or 4 in R6C2 => R67C3 = 12 => R78C3 cannot be 12 (CCC) => no 6 in R6C1 (step 16c) => 14(4) cage = {1238}, CPE no 2 in R89C1
-> no 2 in R89C1
30b. 25(4) cage = {4579/4678}, 4 locked for N7
30c. 14(4) cage = {1238} (only remaining combination), no 6 in R6C1
30d. R78C3 (step 24) = [16/26/36/86/95] (cannot be {57} which clashes with 25(4) cage), no 5,7 in R7C3, no 7 in R8C3, clean-up: no 1 in R3C8 (step 6)
30e. 2 in N7 only in R7C123, locked for R7, clean-up: no 5 in R2C8 (step 4)

31. 16(4) cage at R8C3 (step 22a) = {1258/1267/2356}, no 4

32. 45 rule on R123 2 innies R3C18 = 1 outie R4C3 + 5
32a. Max R3C18 = 11 -> max R4C3 = 6
32b. Min R3C18 = 6, R3C8 = {23} -> no 3 in R3C1
32c. Min R3C18 = 7 -> min R4C3 = 2

33. 25(4) cage at R8C1 (step 30b) = {4579/4678}
33a. Analysing 20(3) cage at R3C1 (step 26a) = {389/569/578}
20(3) cage = {389/569}, no 7
or 20(3) cage = {578}, locked for C1 => 9 in C1 only in 25(4) cage = {4579} => R89C1 = {49} => 6 in C1 only in 9(3) cage at R1C1 = {126}, locked for N1 => 22(3) cage at R2C2 = {589} => R3C1 = 7
-> no 7 in R45C1

34. 45 rule on N3 2 innies R3C78 = 1 outie R1C6 + 4
34a. Min R3C78 = 5, max R3C8 = 3 -> min R3C7 = 2

35. R27C8 (step 4) = [16/25/34], R3C8 + R8C3 (step 6) = [26/35}, R7C9 + R9C7 (step 3) = {58/67}/[94], 25(4) cage at R8C1 (step 30b) = {4579/4678}
35a. 14(3) cage at R7C7 (step 21b) = {149/158/167/356}
35b. Analysing combinations containing 6
7 of {167} must be in R7C3
14(3) cage = {356}, locked for N9 => R7C9 = 9, 25(4) cage at R8C1 = {4579}, locked for N7, R8C3 = 6 => R3C8 = 2, no 2 in R2C8 => no 5 in R7C8 => {356} = [563]
-> no 6 in R7C7

36. R27C8 (step 4) = [16/25/34], R8C5 = R5C6 + 4 (step 8), R3C8 + R8C3 (step 6) = [26/35}, R5C568 (step 12b) = {179/278/359/368/467}, 14(3) cage at R7C7 (step 21b) = {149/158/167/356}, R7C9 + R9C7 (step 3) = {58/67}/[94], 25(4) cage at R8C1 (step 30b) = {4579/4678}, 27(4) cage at R8C5 (step 14c) = {4689/5679}
36a. Consider placements for R8C5
R8C5 = 5 => R9C7 = {67} => R7C9 + R9C7 = {67}, locked for N9 => 14(3) cage = {149/158}
or R8C5 = 6 => R8C3 = 5 => 25(4) cage = {4678} => R7C3 = 9 (hidden single in N7) => R7C9 + R9C7 = {58/67} => 14(3) cage = {149/158/167} (cannot be {356} which clashes with R7C9 + R9C7)
or R8C5 = 7 => R9C7 = {56} => 14(3) cage = {149/158/167} (cannot be {356} which clashes with R9C7)
or R8C5 = 8 => R5C6 = 4 => R5C568 = {467} = [647], R6C5 = 3 (cage sum), 27(4) cage = {4689} => R9C7 = 4 => R7C9 = 9, 9 in N7 only in 25(4) cage = {4579}, locked for N7 => R8C3 = 6, R3C8 = 2, 2 in N9 only in R89C9, locked for C9, R5C8 + R7C9 = [79] = 16 => R6C89 = 7 = {16} (cannot be {25/34} which are blocked by 2 in R3C8 and in R89C9 and by 3 in R6C5) => R27C8 cannot be [16] => 14(3) cage cannot be {167/356} (note that there isn’t any valid permutation for R27C8 in this path, but the important point is that it eliminates 6 from R7C8 in this path)
or R8C5 = 8 => R5C6 = 4 => R5C568 = {467} = [746] => 14(3) cage = {149/158}
-> 14(3) cage = {149/158/167} -> R8C7 = 1
36b. 7,8,9 only in R7C7 -> R7C7 = {789}
[Things get easier from here.]

37. R8C2 = 3
37a. 1 in N7 only in R7C123, locked for R7
37b. 3 in R7 only in R7C456, locked for N8
37c. Min R7C45 = 7 -> max R6C4 = 7

38. 16(4) cage at R8C3 (step 31) = {1258/1267}
38a. 5 of {1258} must be in R8C3 -> no 5 in R8C4 + R9C45

39. 9(3) cage = {135/234} (cannot be {126} because naked quad {1268} in R1267C1 clashes with 20(3) cage at R3C1), no 6, 3 locked for C1 and N1)
39a. 20(3) cage at R3C1 (step 26a) = {569/578}, 5 locked for C1
39b. 7 of {578} must be in R3C1 -> no 8 in R3C1

40. 20(3) cage at R3C1 (step 39a) = {569/578}, 25(4) cage at R8C1 (step 30b) = {4579/4678}
40a. Hidden killer pair 6,9 only in 20(3) cage at R3C1 and 25(4) cage for C1, 25(4) cage cannot contain both of 6,9 -> 20(3) cage = {569}, locked for C1, 9 also locked for N4
40b. Killer pair 5,6 in 22(3) cage at R2C2 and R3C1, locked for N1
40c. Naked pair {47} in R89C1, locked for N7

41. 45 rule on N1 3 innies R1C3 + R3C13 (step 29) = {158/167} (cannot be {248} because R3C1 only contains 5,6) -> R1C3 = 1, R3C3 = {78}
41a. Min R3C34 = 13 -> max R4C3 = 4

42. Naked pair {23} in R12C1, locked for C1, R1C2 = 4 (cage sum)

43. 14(4) cage at R6C1 (step 30c) = {1238} -> R7C2 = 2, R7C1 = 1 (hidden single in N7) -> R6C1 = 8

44. R18C3 = R3C4 (step 7)
44a. R1C3 = 1, R8C3 = {56} -> R3C4 = {67}
44b. R1C3 + R3C13 (step 41) = {158/167}
44c. 17(3) cage at R3C3 = {278/368} (cannot be {467} which clashes with R3C13, CCC) -> R3C3 = 8, R4C3 = {23}

45. R7C3 = 9 -> 25(4) cage at R8C1 (step 30b) = {4678} -> R9C23 = [86], R8C3 = 5, R3C8 = 3 (step 6), clean-up: no 1 in R5C6 (step 8), no 4 in R7C8 (step 4), no 5,7 in R7C9 (step 3), no 4 in R9C7 (step 3)
45a. R2C3 = 7 -> R23C2 = 15 = {69}, locked for C2 and N1 -> R3C1 = 5

46. 45 rule on N47 4(3+1) innies R45C1 + R4C3 + R8C3 = 23
46a. R45C1 = {69} = 15, R8C3 = 5 -> R4C3 = 3, R3C4 = 6 (cage sum), R23C2 = [69]

47. R7C3 = 9 -> R6C23 = 7 = [52], R5C3 = 4, clean-up: no 6 in R5C8 (step 5), no 8 in R8C5 (step 8)

48. 16(4) cage at R8C3 (step 38) = {1258} (only remaining combination) -> R8C4 = 8, R9C45 = {12}, locked for R9
48a. 27(4) cage at R8C5 (step 14c) = {5679} (only remaining combination), no 4, 6 locked for N8

49. Naked quad {5678} in R7C789 + R9C7, locked for N9, 7 also locked for C7
49a. R9C9 = 3 (hidden single in N9)

50. 45 rule on N3 1 remaining innie R3C7 = 1 outie R1C6 + 1 -> R1C6 = 3, R3C7 = 4, R12C1 = [23]
50a. R1C3 = 1 -> R1C45 = 12 = {57] (only remaining combination), locked for R1 and N2
50b. R3C7 = 4 -> R23C6 = 9 = [81], R3C5 = 2, R9C45 = [21]
50c. Naked pair {49} in R2C45, locked for R2

51. R3C9 = 7 -> 22(4) cage at R1C8 = {1678} (only possible combination) -> R2C9 = 1, R1C89 = {68}, locked for R1, R1C7 = 9, R2C78 = [52], R79C7 = [87], R7C89 = [56]

52. R5C6 = 2 -> R56C5 = 11 = [74/83]
52a. Naked pair {78} in R5C58, locked for R5 -> R45C2 = [71]

53. R4C7 = 2 (hidden single in C7) -> 11(3) cage at R4C6 = {236} (only remaining combination, cannot be {245} because 4,5 only in R4C6) -> R4C6 = 6, R45C1 = [96]

54. R5C4 = 9 (hidden single in N5) -> R4C45 = 6 = [15], R1C5 = 7, R5C5 = 8, R6C5 = 3 (cage sum)

and the rest is naked singles.
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Ed
PostPosted: Fri Jun 27, 2008 11:24 pm 
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Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 60 RP-Lite by mhparker (July 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:2560:3330:3330:3330:4869:4869:6151:6151:2560:4874:4874:3084:3084:2574:4869:4869:6151:2322:4874:5908:5908:3084:2574:2574:3353:6151:2322:4124:5908:4894:4894:5152:5152:3353:3353:2322:4124:4124:4894:3880:3880:5152:5419:3353:5677:4654:4654:2608:3880:2866:2866:5419:5419:5677:5677:4654:2608:2608:2866:4412:4412:5419:5951:5677:4161:4161:6467:6467:4412:5190:5190:5951:5951:5951:4161:4161:6467:6467:5190:5190:
Solution: 
+-------+-------+-------+
| 2 1 3 | 4 6 5 | 7 9 8 |
| 7 6 9 | 1 8 2 | 3 4 5 |
| 5 4 8 | 9 3 7 | 1 6 2 |
+-------+-------+-------+
| 1 7 6 | 8 5 3 | 9 2 4 |
| 3 5 4 | 6 2 9 | 8 7 1 |
| 8 9 2 | 7 4 1 | 6 5 3 |
+-------+-------+-------+
| 9 3 7 | 2 1 4 | 5 8 6 |
| 6 2 1 | 5 9 8 | 4 3 7 |
| 4 8 5 | 3 7 6 | 2 1 9 |
+-------+-------+-------+
Quote:
mhparker, rating post: (rating) 1.75: Very hard Killer, but still not hard enough to require a team effort to solve. Does not require any hypotheticals. The A60RP-Lite could maybe deserve such a rating.
CathyW: :brickwall: It's no good. I'm not getting any further
sudokuEd: Finally found a nice solution to RP Lite - after nearly 3 weeks.
Para: I solved the A60RP-Lite with "only" those (normal Killer) techniques. That doesn't mean it isn't hard. Using those techniques can make a puzzle awfully difficult as well. You just need the right insight to use those techniques appropriately
Andrew: It's easy to forget that each of these ratings represent a range. Obviously, from what people say about A60RP-Lite, it must be at the very top of the 1.75 rating range
Andrew in 2011: I'll rate my walkthrough for A60 RP-Lite at Easy 1.75 ... I certainly don't want to disagree with Mike's rating definitions. Maybe my solving path is a bit easier (than Para and Ed's), even though it's a longer one?
Adjusted Walkthrough by Para:
Hi all

Guess Cathy just beat me to it. But here's mine. It is a bit adjusted because the original was a big mess. loads of steps through each other. I missed far too many things early on to make it easy to read.

Walk-through Assassin 60RP-Lite

1. 10(3) at R1C1, R2C6 and R6C4 = {127/136/145/235}: no 8,9

2. 19(3) at R2C2 and R4C4 = {289/379/469/478/568}: no 1

3. 9(3) at R3C1 = {126/135/234}: no 7,8,9

4. 23(3) at R3C3 = {689}

5. 13(4) at R3C8 = {1237/1246/1345}: no 8,9; 1 locked in 13(4) cage -->> R56C8: no 1

6. 20(3) at R4C6 = {389/479/569/578}: no 1,2

7. 11(3) at R6C6 = {128/137/146/236/245}: no 9

8. 45 on N5689: 2 outies: R3C8 + R8C3 = 7 = {16/25/34}: no 7,8,9

9. 45 on R123: 1 outie and 2 innies: R4C3 + 5 = R3C18: Max R3C18 = 11 -->> Max R4C3 = 6: R4C3 = 6; R3C18 = {56} -->> locked for R3
9a. Naked Pair {89}at R3C34 -->> locked for R3
9b. Clean up: R8C3 = {12}(step 8)
9c. 13(4) at R3C8 = {1246/1345}; R3C8 = {56}; R4C89 + R5C9 = {124/134}: no 5,6,7 -->> 1,4 locked for N6

10. 9(3) at R3C1 needs one of {56} in R3C1: 9(3) = 5{13}/6{12} = {2|3..},{2|5..},{3|6..} -->> R45C1 = {12/13}: no 4,5; 1 locked for C1 and N4

11. 10(3) at R1C1 can’t have {23},{25} or {36} in R12C1 because of 9(3) at R3C1
11a. 10(3) at R1C1 = {27}[1]/{35}[2]/{45}[1] -->> R12C1 = {27/35/45} = {2|5..}: no 6, R1C2 = {12}
11b. Killer Pair {25} in R12C1 + 9(3) at R3C1 -->> locked for C1
11c. 5 in C1 locked for N1
11d. 1 in N1 locked for R1
11e. 1 in R6 locked for N5

12. 45 on C123: 3 innies: R138C3= 12 = [381]/{1[9]2} -->>R1C3 = {123};1 locked for C3; R13C3 = {3|9..}
12a. 19(3) at R2C2 = {469/478}({289} blocked by R3C3; {379} blocked by R13C3): no 2,3; 4 locked for N1; 6 only in R2C2 -->> R2C2: no 9
12b. 9 in N1 locked for C3

13. 10(3) at R1C1 = {27}[1]/{35}[2]: 2 locked for N1; R12C1 = {27/35}= {2|3}
13a. Killer Pair {23} in R12C1 + 9(3) cage at R3C1 -->> locked for C1

14. 45 on R89: 2 innies: R8C27 = 6 = {15/24} = {1|2..}: no 3,6,7,8,9
14a. Killer Pair {12} in R8C27 + R8C3 -->> locked for R8

15. 45 on N7: 1 outie and 2 innies: R6C1 = R78C3: R6C1 = {4789} -->> R78C3 = [52/71/72/81]: [31] blocked by R1C3 -->> R6C1: no 4; R7C3: no 2,3,4
15a. 4 in C1 locked for N7
15b. Clean up: R8C7: no 2(step 14)

16. 18(3) at R6C2 = [927]/{38}[7]/{37}[8]/[945] -->> R6C23 = {37/38}/[92/94]: no 5

17. 5 in N4 locked within 16(3) cage at R4C2 -->> 16(3) = {259/358/457} = {3|7|9..}

18. Combining 15 and 16
18a. R6C123 = [794]/[892]/[9]{38}: [9]{37} blocked by 16(3) at R4C2: 9 locked for R6 and N4: R6C2 = {389}; R6C3 = {2348}
18b. Clean up: R7C3 = {57}(step 16); 16(3) at R4C2 = {358/457}: no 2
18c. CPE: R7C2: no 9 -->> sees all 9’s in R6

19. Combination analysis 22(4) at R6C1 from step 15: When R6C1 = 7: no {25}(must be in R78C3); When R6C1 = 8, no {17}(must be in R78C3); When R6C1 = 9, no {27}(must be in R78C3)
19a. 22(4) = [7]{68}[1]/[8932]/[8635]/[9481]: R7C1 = {4689}; R7C2 = {368}; 8 locked within 22(4) cage -->> R89C1: no 8
19b. 8 in C1 locked within 22(4) cage -->> R7C2: no 8
19c. 22(4) = [7861]/[8932]/[8635] -->> R7C1: no 4; R6C1: no 9
19d. R6C2 = 9(hidden)
19e. Clean up: R6C3 = {24}
19f. R6C13 = [74/82] = {2|4..},{2|7..},{4|8..},{7|8..}

20. 4 in N7 locked within 23(4) cage at R8C1 -->> 23(4) = {2489/3479/4568}: no 1
20a. 1 in N7 locked for R8
20b. Clean up: R8C2: no 5
20c Naked Pair {12} in R8C23 -->> locked for N7
20d. 23(4) = {3479/4568} = {3479}/{46}{58} -->> R9C2: no 6

21. 22(4) at R6C1 = [7861]/[8932] -->> R7C1: no 6
21a. R7C123 = [865/937] = {6|7..}
21b. 17(3) at R7C7 = {39}[5]/{48}[5]/{58}[4]: {67}[4] blocked by R7C123: 5 locked in 17(3) cage for N9; R7C78 = {34589} = {39/48/58} = {8|9..}
21c. Killer Pair {89} in R7C1 + R7C78 -->> locked for R7

22. 45 on N9: 2 innies: R7C9 + R9C7 = {17/26}: no 3,4,8,9

23. 45 on R6789: 1 innie and 1 outie: R5C8 = R6C5 + 3 -->> R5C8 = {56789}; R6C5 = {23456}

24. 45 on N5: 3 innies: R4C6 + R6C46 = 11 = {128/137/146}(1 in N5 locked within these cells): no 5,9

25. 45 on R6789: 4 innies: R6C589 + R7C9 = 18
25a. R6C89 can’t have both {23} because of 13(4) at R3C8(step 9c)
25b. R6C589 can’t have both {24},{27},{48},{78} because of R6C13(step 19f)
25c. R6C5 can’t be 3 less than any of the other innies(step 23; R5C8 sees all other innies)
25d. R7C9 = 1: R6C589 = [6]{38} (others blocked by 25b and 25c)
25e. R7C9 = 2: R6C589 = [2]{68}/[3]{58}/[6]{37}(others blocked by 25b and 25c)
25f. R7C9 = 6: R6C589 = [4]{35}(others blocked by 25b and 25c)
25g. R7C9 = 7: R6C589 = [2]{36} (others blocked by 25a, 25b and 25 c)
25h. Conclusion: R6C5 = {2346}: no 5; R6C89 = {35/36/37/38/58/68}: no 2
25i. Clean up: R5C8: no 8(step 23)

26. 5 in R6 locked for N6
26a. Clean up: R6C5: no 2
26b. R6C589 = [6]{38}/[3]{58}/[6]{37}/[4]{35} -->> R6C89: no 6; R7C9: no 7
26c. Clean up: R9C7: no 1

27. 20(3) at R4C6 = {389}/{479}: no 6 -->> 9 locked within 20(3) cage in R45C7 -->> locked for C7 and N6
27a. Clean up: R6C5: no 6
27b. R6C589 = [3]{58}/[4]{35} -->> R6C89 = {35/58} = {3|8..}: no 7; R7C9: no 1; 5 locked within R6C89 for R6
27c. Clean up: R9C7: no 7
27d. Naked Pair {26} in R7C9 + R9C7 within N9

28. Killer Triple {348} within R6C13 + R6C5 + R6C89 -->> locked for R6

29. 11(3) at R6C6 = [173/164/263/623] -->> R6C6: no 7; R7C6 = {34}

30. 1 in R7 locked for N8 and 10(3) cage at R6C4
30a. R6C6 = 1(hidden)
30b. Clean up: R6C7: no 2(step 29)

31. Hidden Triple {124} in N6 within R4C89 + R5C9 -->> R4C89 + R5C9 = {124}
31a. 13(4) at R3C8 = {1246} -->> R3C8 = 6
31b. R3C1 = 5; R5C8 = 7; R6C7 = 6; R7C6 = 4; R9C7 = 2; R7C9 = 6
31c. R7C2 = 3; R7C13 = [97] (step 21a); R6C3 = 2; R8C23 = [21]
31d. R6C1 = 8; R6C4 = 7; R6C5 = 4(hidden); R1C23 = [13]; R2C2 = 6(hidden)
31e. R4C2 = 7(hidden); R3C2 = 4; R23C3 = [98]; R3C4 = 9; R5C23 = [54]
31f. R9C23 = [85]

32. Naked Pair {58} in R7C78 -->> locked for R7 and N9
32a. R8C7 = 4; R89C1 = [64]

33. 45 on N5: 1 innie: R4C6 = 3

34. R45C1 = [13]; R5C9 = 1(hidden); R9C8 = 1(hidden)

35. 15(3) at R5C5 = {29}4 -->> R5C56 = {29} -->> locked for N5 and R5
35a. R45C7 = [98]; R5C4 = 6; R9C4 = 3; R7C78 = [58]; R1C7 = 7; R12C1 = [27]

36. 45 on N3: 1 innie + 1 outie: R1C6 = R3C7 + 4 -->> R1C6 = 5; R3C7 = 1
36a. R23C6 = [27]; R2C78 = [34]; R1C89 = [98]; R23C9 = [52]; R4C89 = [24]
36b. R6C89 = [53]; R5C56 = [29]; R1C45 = [46]; R3C5 = 3; R7C45 = [21]
36c. R89C6 = [86]; R2C45 = [18]; R4C45 = [85]; R8C4 = 5; R89C5 = [97]
36d. R8C8 = 3; R89C9 = [79]

greetings

Para
ALT ending by sudokuEd:
mhparker wrote:
manual work... involved in the production of the RP Lite.....roughly 8 hours.
We must be bonkers to make these versions. 8-)

Finally found a nice solution to RP Lite - after nearly 3 weeks. Congratulations to Para in getting it (and a walk-through!) soooo quickly. =D> Haven't had a chance to look at Cathy's way.

Para wrote:
11. 10(3) at R1C1 can’t have .....{25} ..... in R12C1 because of 9(3) at R3C1
Para wrote:
12. 6 only in R2C2 -->> R2C2: no 9
Missed these on early attempts which make a big difference. But my final solution, on the whole, mirrored Para's to step 24.

Finished it quite differently though. End of Para's step 24 here (first use of a bookmark from Sudoku Solver: :) Paste these marks into Sudoku Solver :D :D )
Code:
-------------------------------.-------------------------------.-------------------------------.
| 2357      12        13        | 23456789  23456789  23456789  | 23456789  23456789  23456789  |
| 2357      4678      4789      | 123456789 123456789 1234567   | 123456789 123456789 123456789 |
| 56        47        89        | 89        12347     12347     | 12347     56        12347     |
:-------------------------------+-------------------------------+-------------------------------:
| 123       34578     6         | 2345789   2345789   3478      | 35789     1234      1234      |
| 123       34578     34578     | 23456789  23456789  23456789  | 356789    56789     1234      |
| 78        9         24        | 123467    23456     1234678   | 235678    235678    235678    |
:-------------------------------+-------------------------------+-------------------------------:
| 89        36        57        | 1234567   1234567   1234567   | 34589     34589     1267      |
| 4679      12        12        | 3456789   3456789   3456789   | 45        346789    346789    |
| 4679      3578      3578      | 123456789 123456789 123456789 | 1267      12346789  12346789  |
'-------------------------------.-------------------------------.-------------------------------'

Alternate step 25.
[1] in r7c9 -> 7 in r9c7 (h8(2)n9)-> 7 for n6 in 21(4) = [1]{578} only
[2] in r7c9 -> 6 in r9c7 (h8(2)n9) -> 6 in n6 in 21(4) = [2]{568} only
[6] in r7c9 -> 21 (4) = [6]{357} ([6]{258} blocked by clash with r6c5: i/o r6789)
[7] in r7c9 -> 21(4) = [7]{356} ([7]{239} clashes with 13(4) in n6)
-> no 2 r6c89, no 9 r5c8
-> 9 in n6 only in 20(3)
which leads to the solution very quickly through the h11(3) in n5, then the 11(3) cage at r6c6.

Now, onto Assassin 61.

Cheers
Ed
2011 Walkthrough by Andrew:
Many thanks to Mike for another nice variant and to Ed for encouraging me to belatedly try this puzzle. I enjoyed working on this one, even though it is a hard puzzle. So many 45s in this cage pattern; I seemed to keep on spotting them.

Mike (mhparker) wrote:
P.S. Don't be put off by the word "Lite" in the name - it ain't that easy!
Maybe that’s why I didn’t try it back in 2007.

Para's steps 8 and 9 are neat! They are slightly quicker than my 4 innies for R123 because his way immediately reduces the 4 innies to two naked pairs. Para's step 25c is a nice observation!

Ed's alternative to Para's step 25 was simpler and easier to follow.


Here is my walkthrough for A60 RP-Lite.

Prelims

a) 10(3) cage in N1 = {127/136/145/235}, no 8,9
b) 19(3) cage in N1 = {289/379/469/478/568}, no 1
c) 10(3) cage at R2C6 = {127/136/145/235}, no 8,9
d) 9(3) cage at R3C1 = {126/135/234}, no 7,8,9
e) 23(3) cage at R3C3 = {689}
f) 19(3) cage in N5 = {289/379/469/478/568}, no 1
g) 20(3) cage at R4C6 = {389/479/569/578}, no 1,2
h) 10(3) cage at R6C4 = {127/136/145/235}, no 8,9
i) 11(3) cage at R6C6 = {128/137/146/236/245}, no 9
j) 13(4) cage at R3C8 = {1237/1246/1345}, no 8,9

1. 13(4) cage at R3C8 = {1237/1246/1345}, CPE no 1 in R56C8

2. 8,9 in C1 only in R6789C1, CPE no 8,9 in R78C2

3. 45 rule on R89 2 innies R8C27 = 6 = {15/24}
3a. Max R8C7 = 5 -> min R7C78 = 12, no 1,2 in R7C78

4. 45 rule on C123 1 outie R3C4 = 2 innies R18C3 + 5
4a. Min R18C3 = 3 -> min R3C3 = 8
4b. 6 in 23(3) cage only in R34C3, locked for C3
4c. R3C4 = {89} -> R18C3 = 3,4 = {12/13}, 1 locked for C3

5. 45 rule on C1 4 innies R6789C1 = 1 outie R1C2 + 26
5a. Max R6789C1 = 30 -> max R1C2 = 4
5b. Min R6789C1 = 27, no 1,2 in R6789C1

6. 45 rule on R123 4 innies R3C1348 = 28 = {4789/5689}, no 1,2,3
6a. 8,9 only in R3C34 -> R3C34 = {89}, locked for R3 and 23(3) cage at R3C3 -> R4C3 = 6
6b. R3C18 = [47/56/65], no 4 in R3C8
[I never spotted 45 rule on N5689 2 outies R3C8 + R8C3 = 7, which would have immediately reduced R3C18 to a naked pair; step 9 gave me the save result.]

7. 9(3) cage at R3C1 = {126/135/234}
7a. R3C1 = {456} -> no 4,5 in R45C1

8. 13(4) cage at R3C8 = {1237/1246/1345}
8a. R3C8 = {567} -> R4C89 + R5C9 = {1234}, 1 locked for N6

9. 45 rule on N47 1 outie R3C1 = 1 remaining innie R8C3 + 4, no 4 in R3C1, no 3 in R8C3
9a. 9(3) cage at R3C1 = {126/135}, 1 locked for C1 and N4
9b. Killer pair 1,2 in R8C27 and R8C3, locked for R8
9c. 1 in N1 only in R1C23, locked for R1
9d. 1 in R6 only in R6C456, locked for N5

10. 19(3) cage in N1 = {379/469/478} (cannot be {289} which clashes with R3C3, cannot be {568} which clashes with R3C1), no 2,5
10a. 5 in N1 only in R123C1, locked for C1

11. R3C1348 (step 6) = {5689} (only remaining combination) -> R3C18 = {56}, locked for R3

12. 13(4) cage at R3C8 = {1246/1345}, 4 locked for N6

13. 45 rule on N5 3 innies R4C6 + R6C46 = 11 = {128/137/146/236/245}, no 9
13a. 8 of {128} must be in R4C6 -> no 8 in R6C6

14. 45 rule on N3 2 innies R3C78 = 1 outie R1C6 + 2
14a. Min R3C78 = 6 -> min R1C6 = 4
14b. Max R3C78 = 11 -> no 7 in R3C7
14c. Max R3C78 = 10 -> no 9 in R1C6

15. 45 rule on N9 2 innies R7C9 + R9C7 = 8 = {17/26/35}, no 4,8,9

16. Hidden killer triple 5,6,7 in R12C1, R3C1 and R6789C1 for C1, R12C1 contains one of 5,6,7, R3C1 = {56} -> R6789C1 must contain one of 6,7
16a. Max R6789C1 = 28 (cannot be {6789} which contains both of 6,7 and there’s no 5 in R6789C1) -> max R1C2 = 2 (step 5)

17. 10(3) cage in N1 = {127/145/235} (cannot be {136} which clashes with 9(3) cage at R3C1), no 6

18. 45 rule on N7 2 innies R78C3 = 1 outie R6C1
18a. Max R78C3 = 9 -> no 9 in R7C3
18b. R78C3 cannot be [41/42] = 5,6 (because no 5,6 in R6C1) -> no 4 in R7C3

19. 45 rule on C89 2 innies R27C8 = 12 = {39/48/57}, no 1,2,6

20. 19(3) cage in N5 = {289/379/469/478/568}
20a. 2 of {289} must be in R45C4 (R45C4 cannot be {89} which clashes with R3C4) -> no 2 in R4C5
20b. 6 of {568} must be in R5C4 -> no 5 in R5C4

21. 45 rule on C6789 1 innie R5C6 = 1 outie R8C5, no 2 in R5C6
21a. 9 in C6 only in R589C6, R5C6 = R8C5 -> 9 must be in R8C56 + R9C6, locked for N8

22. 45 rule on R6789 1 outie R5C8 = 1 innie R6C5 + 3, no 2,3 in R5C8, no 1,7,8,9 in R6C5

23. 1 in R6 only in R6C46 -> R4C6 + R6C46 (step 13) = {128/137/146}, no 5
23a. 4 of {146} must be in R4C6 -> no 4 in R6C46
23b. 19(3) cage in N5 (step 20) = {289/379/469/568} (cannot be {478} which clashes with R4C6 + R6C46
23c. 6 of {469} must be in R5C4 -> no 4 in R5C4

24. 15(3) cage in N5 = {249/258/357/456} (cannot be {267/348} which clash with
R4C6 + R6C46)
24a. R5C8 = R6C5 + 3 (step 22)
24b. 15(3) cage = {249/357/456} (cannot be {258} which would make R5C568 {28}8/{58}5), no 8, clean-up: no 8 in R8C5 (step 21)

25. 20(3) cage at R4C6 = {389/479/578} (cannot be {569} because no 5,6,9 in R4C6), no 6

26. 18(3) cage at R6C2 = {279/378/459}
26a. 5 of {459} must be in R7C3 -> no 5 in R6C23
26b. 5 in N4 only in 16(3) cage = {259/358/457}

27. 45 rule on N4 3 remaining innies R6C123 = 1 outie R3C1 + 14
27a. R3C1 = {56} -> R6C123 = 19,20 = {289/478/389/479} (cannot be {379} which clashes with 16(3) cage)
27b. 3 of {389} must be in R6C23 (R6C23 cannot be {89} because 18(3) cage only contains one of 8,9) -> no 3 in R6C1
27c. 18(3) cage at R6C2 (step 26) = {279/378/459}
27d. 8 of {378} must be in R6C23 (R6C23 cannot be {37} because R6C123 doesn’t contain both of 3,7) -> no 8 in R7C3

28. R78C3 = R6C1 (step 18)
28a. Min R6C1 = 4 -> no 2 in R7C3 (R78C3 cannot be [22])

29. 11(3) cage at R6C6 = {128/137/146/236/245}
29a. 4 of {245} must be in R7C6 -> no 5 in R7C6

30. Hidden killer pair 8,9 in R7C1 and 23(4) cage for N7, 23(4) can only contain one of 8,9 (23(4) cage cannot be {1589/2489} because then R8C23 = [51/42] = 6 clashes with R8C27, CCC) -> R7C1 = {89}

[The next step has been available for a long time but I’ve only just spotted it.]

31. R3C1 = R8C3 + 4 (step 9) -> R3C1 + R8C3 = [51/62]
31a. 45 rule on N1 3 innies R1C3 + R3C13 = 16 = {169/259/358} (cannot be {268} = [268] which clashes with R3C1 + R8C3)
31b. 10(3) cage in N1 (step 17) = {127/235} (cannot be {145} which clashes with R1C3 + R3C13), no 4, 2 locked for N1
31c. Killer pair 1,3 in 10(3) cage and R1C3, locked for N1
31d. Killer pair 2,3 in R12C1 and 9(3) cage at R3C1, locked for C1
[Cathy and Para both used the slightly simpler
45 rule on C3 3 innies R138C3 = 12 = {129/138} = [192/291/381]
19(3) cage in N1 (step 10) = {469/478} (cannot be {379} which clashes with R13C3), no 3, 4 locked for N1
I’d seen those innies but not the interaction with the 19(3) cage.]

32. 4 in C1 only in R689C1, CPE no 4 in R78C2, clean-up: no 2 in R8C7 (step 3)
32a. 2 in R8 only in R8C23, locked for N7

33. 19(3) cage in N1 (step 10) = {469/478}
33a. 6 of {469} must be in R2C2 -> no 9 in R2C2
33b. 9 in N1 only in R23C3, locked for C3
33c. 18(3) cage at R6C2 (step 26) = {279/378/459}
33d. 9 of {279/459} must be in R6C2 -> no 2,4 in R6C2

34. R78C3 = R6C1 (step 18)
34a. R78C3 = [52/71/72] (cannot be [31] which clashes with R1C3, cannot be [32/51] = 5 because no 5,6 in R6C1) -> no 3 in R7C3
34b. R78C3 = [52/71/72] = 7,8,9 -> R6C1 = {789}

35. 4 in C1 only in R89C1, locked for N7
35a. 23(4) cage in N7 = {3479/4568}, no 1
35b. Killer pair 5,7 in R7C3 and 23(4) cage, locked for N7
35c. Naked pair {12} in R8C23, locked for R8 and N7
35d. Naked pair {12} in R18C2, locked for C2

36. 22(4) cage at R6C1 = {1678/2389}, 8 locked for C1

37. 18(3) cage at R6C2 (step 26) = {279/378/459}
37a. R7C3 = {57} -> no 7 in R6C23
37b. R6C123 (step 27a) = {289/389/479} (cannot be {478} = [784] because 18(3) cage doesn’t contain both of 4,8), 9 locked for R6 and N4, clean-up: no 2 in 16(3) cage in N4 (step 26b)

38. R8C23 = {12} = 3
38a. 45 rule on N7 3 remaining innies R7C123 = 19 = {379/568}

39. 17(3) cage in N9 = {359/458} (cannot be {368} because R8C7 only contains 4,5, cannot be {467} = {67}4 which clashes with R7C123), no 6,7, 5 locked for N9, clean-up: no 5 in R2C8 (step 19), no 3 in R7C9 + R9C7 (step 15)
39a. Killer pair 3,8 in R7C123 and 17(3) cage in N9, locked for R7

40. R78C3 = [52/71/72] = 7,8,9 -> R6C1 = {789} (step 34b)
40a. R78C3 = [52] = 7 => R6C1 = 7 or R78C3 = [71/72], CPE no 7 in R5C3 + R89C1

41. 23(4) cage in N7 (step 35a) = {3479/4568}
41a. R89C1 = {46/49} -> R9C23 = {37}/{58}, no 6,9 in R9C2

42. R6C2 = 9 (hidden single in C2)
42a. 18(3) cage at R6C2 (step 26) = {279/459} -> R6C3 = {24}

43. 10(3) cage at R6C4 = {127/136/145/235}
43a. 3 of {136} must be in R6C4 -> no 6 in R6C4

44. R5C8 = R6C5 + 3 (step 22)
44a. 21(4) cage at R5C8 = {1389/1578/2379/2568/3567} (cannot be {1569} which clashes with 20(3) cage at R4C6)
44b. 2 of {2379} must be in R7C9 (R6C89 cannot be {23} which clashes with 13(4) cage at R3C8)
2 of {2568} must be in R7C9 (cannot be 5{28}6/8{25}6 because R6C589 cannot be 2{28}/5{25})
-> no 2 in R6C89

[After I saw Ed’s post I think there’s a neat way to reduce the number of combinations for 21(4) cage at R5C8.
R7C9 + R9C7 (step 15) = {17/26} -> combinations with 1 in R7C9 must also contain 7 and combinations with 2 in R7C9 must also contain 6.
I eliminated {2379} in the next step, effectively using this method although I didn’t see it this way at the time, but didn’t spot that it can also be used to eliminate {1389} which, as Ed said, places 9 in N6 in 20(3) cage at R4C6 and the solving path finishes a lot more quickly.]

45. 21(4) cage at R5C8 = {1389/1578/2568/3567} (cannot be {2379} because R7C9 = 2, R9C7 = 6 (step 15), R6C7 = 6 (hidden single in N6) clashes with R9C7)
45a. 21(4) cage = {1389} => R6C7 = 6 (hidden single in N6) or 21(4) cage = {1578/2568/3567}, 5 locked for N6 -> no 5 in R6C7
45b. 20(3) cage at R4C6 (step 25) = {389/479/578}
45c. 4 of {479} must be in R4C6, 7 of {578} must be in R45C7 (R45C7 cannot be {58} which clashes with 21(4) cage at R5C8) -> no 7 in R4C6
[With hindsight the first part of step 45 can be improved by converting it to a short forcing chain
Consider the placement of 6 in N6
R6C7 = 6 => no 6 in R9C7 => no 2 in R7C9 (step 15)
or 6 in 21(4) cage at R5C8
-> 21(4) cage can only contain 2 if it also contains 6
-> 21(4) cage at R5C8 = {1389/1578/2568/3567} (cannot be {2379} which contains 2 but not 6)
Similarly the 21(4) cage can only contain 1 if it also contains 7, because the 1 can only be in R7C9.]

46. R4C6 + R6C46 (step 23) = {128/137/146}
46a. R4C6 = {348} -> no 3 in R6C46

47. 10(3) cage at R6C4 = {127/145}, no 6, CPE no 1 in R9C4

48. 11(3) cage at R6C6 = {128/137/146/236}
48a. 3,8 of {128/236} only in R6C7 -> no 2 in R6C7
48b. 3 of {137} only in R6C7 -> no 7 in R6C7

49. 2 in N6 only in 13(4) cage at R3C8 (step 12) = {1246} (only remaining combination), no 3 -> R3C8 = 6, R3C1 = 5, R45C1 = {13} (step 9a), locked for C1 and N4
49a. Naked pair {27} in R12C1, locked for C1 and N1 -> R1C2 = 1, R1C3 = 3, R3C2 = 4, R67C1 = [89], R8C2 = 2, R7C2 = 3 (step 36), R7C3 = 7 (step 38a), R6C3 = 2 (step 42a), R8C3 = 1, clean-up: no 3,9 in R2C8 (step 19), no 1 in R9C7 (step 15)
49b. Naked pair {57} in R45C2, locked for C2 and N4 -> R5C3 = 4, R9C2 = 8, R9C3 = 5, R2C2 = 6, R2C3 = 9 (step 33), R3C34 = [89]

50. R6C5 = 4 (hidden single in R6)
50a. 15(3) cage in N5 (step 24b) = {249/456}, no 3,7, clean-up: no 3,7 in R8C5 (step 21)
50b. 2 of {249} must be in R5C5 -> no 9 in R5C5

51. R4C6 + R6C46 (step 23) = {137} (only remaining combination) -> R4C6 = 3, R6C46 = {17}, locked for R6 and N5, R45C1 = [13]
51a. 20(3) cage at R4C6 (step 25) = {389} (only remaining combination) -> R45C7 = {89}, locked for C7 and N6

52. R5C89 = [71] (hidden pair in N6), R45C2 = [75], clean-up: no 6 in R5C56 (step 50a), no 5 in R7C8 (step 19), no 7 in R9C7 (step 15)
52a. R6C56 = [29], R45C7 = [98], R5C4 = 6, R8C5 = 9 (step 21)

53. Naked pair {45} in R78C7, locked for C7 and N9 -> R7C8 = 8, R8C8 = 3, R2C8 = 4, R4C89 = [24], R6C8 = 5, R1C8 = 9, R9C8 = 1

54. 21(4) cage at R5C8 (step 45) = {3567} (only remaining combination) -> R7C9 = 6, R6C9 = 3, R9C7 = 2, R1C7 = 7, R12C1 = [27]

55. R1C3 = 3 -> R1C45 = 10 = [46]

56. Naked pair {37} in R9C45, locked for R9 and N8
56a. R8C3 = 1, R9C45 = {37} = 10 -> R8C4 = 5 (cage sum)

and the rest is naked singles.


Rating Comment. I'll rate my walkthrough for A60 RP-Lite at Easy 1.75 because of step 45, which I think is my hardest step; I also used a couple of short forcing chains.

I know that in Mike's original rating definition post he said "1.75: Very hard Assassin, but still not hard enough to require a team effort to solve. Does not require any hypotheticals. The A60RP-Lite could maybe deserve such a rating." I certainly don't want to disagree with Mike's rating definitions.

Maybe my solving path is a bit easier, even though it's a longer one? I didn’t have anything as heavy as Para's step 19 or step 25; Ed’s alternative to the latter is simpler.
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Ed
PostPosted: Fri Jun 27, 2008 11:39 pm 
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Location: Sydney, Australia
Unsolvables thread
Quote:
Glyn: Perhaps we need to collect all these unsolvables together, as we have a much smaller sample set of Killers to work with than vanilla Sudokus.
Para, in Unsolvable thread lead-in: On Glyn's suggestion, i browsed the forum for all puzzles (at that point) considered unsolvable....Here is the first list of unsolvables
Unsolvables 1-9 with links to current walkthroughs:
Para wrote:
Glyn wrote:
Perhaps we need to collect all these unsolvables together, as we have a much smaller sample set of Killers to work with than vanilla Sudokus.


Hi all

On Glyn's suggestion, i browsed the forum for all puzzles (at that point) considered unsolvable.

Here is the first list of unsolvables(from this forum).

1) Sept 28th 2006: SudokuEd's UTA 1

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=802#p802

PS:

3x3:d:k:6913:6913:5122:5122:5122:5134:5134:5134:5134:5138:6913:6913:6913:5122:1792:1792:2320:2320:5138:4373:4373:4373:5654:5654:5654:1305:1305:5138:4373:4390:4390:6439:5941:5941:5941:5941:5915:4390:4390:6439:6439:6439:3882:5931:5941:5915:5915:5915:5915:6439:3882:3882:5931:3911:3894:3894:2362:2362:4355:4355:5931:5931:3911:1343:1343:2370:2370:5966:4355:4355:4432:3911:5704:5704:5704:5704:5966:5966:5966:4432:4432:

Image

This puzzle only has a walk-through from Richards solver. I think it also needs a human effort.

2) Dec 29th 2006: SudokuEd's Bullseye 3(aka Assassin 30V2.1)

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=849#p849

PS:

3x3::k:4352:4352:4098:4098:5124:1797:1797:1543:1543:4352:5642:5642:5642:5124:6670:6670:6670:1543:2578:5642:1556:1556:5124:5399:5399:6670:3354:2578:5642:1556:10782:10782:10782:5399:6670:3354:2852:2852:2852:10782:769:10782:4650:4650:4650:2605:7982:3119:10782:10782:10782:4147:5940:2357:2605:7982:3119:3119:4922:4147:4147:5940:2357:4415:7982:7982:7982:4922:5940:5940:5940:3655:4415:4415:2378:2378:4922:1869:1869:3655:3655:

Image

No Tag effort reported on this one.

3) Feb 17th 2007: SudokuEd's CDKV3

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=938#p938

PS:

3x3::k:2048:2817:2817:3075:3075:3075:4870:4870:4870:2048:10:4619:4619:13:3854:4870:16:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:37:3366:3366:40:2857:2857:43:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:64:3905:5433:67:3643:3643:70:6727:3905:3905:3905:3915:3915:3915:6727:6727:6727:

Image

There hasn't been any mention of anyone solving or trying to solve this one.

4) Feb 22nd 2007: Ruud's Assassin39V3

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=946#p946

PS:

3x3::k:5632:5632:5632:5635:5635:4357:4357:4357:4360:3849:3850:5632:5632:5635:4110:2575:2575:4360:3849:3850:5140:5140:5635:4110:4110:4360:4360:3849:5140:5140:5662:5662:5662:4110:4360:3363:2852:2085:2085:2855:5662:3113:1322:1322:3363:2852:6702:4655:2855:3113:3113:6963:6963:3637:6702:6702:4655:4655:6714:6963:6963:2877:3637:6702:3648:3648:4655:6714:6212:6212:2877:3637:6702:2377:2377:2377:6714:6714:6212:6212:6212:

Image

I think after the struggle of A39V2 people were kind of tired of the cage pattern.

5) April 28th 2007: Para's Assassin 44V1.5

Solved: Walk-throughs at: http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=960#p960

PS:

3x3::k:4608:4608:4608:4608:5380:5125:5125:5125:4360:5385:5385:5385:2060:5380:6158:6158:4360:4360:3090:3090:3090:2060:3094:3094:6158:4360:5402:4123:3090:5917:4126:4126:4126:6158:5154:5402:4123:5917:5917:3367:3367:3367:5154:5154:5402:4123:5917:3887:4144:4144:4144:5154:4916:5402:4123:5687:3887:3385:3385:1595:4916:4916:4916:5687:5687:3887:3887:5380:1595:3397:3397:3397:5687:4937:4937:4937:5380:5197:5197:5197:5197:

Image

This one might be a bit out of place, but as it was judged difficult and no WT at all, i am going to put it in. This probably be the first to be solved out of this set.

6) May 11th 2007: Ruud's Assassin 50V2

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=994#p994

PS:

3x3::k:3072:3072:4098:4098:5636:2565:2565:3335:3335:3072:4618:4618:5636:5636:5636:5647:5647:3335:2834:4618:4618:2837:11542:4119:5647:5647:2330:2834:5916:2837:2837:11542:4119:4119:5154:2330:5916:5916:11542:11542:11542:11542:11542:5154:5154:2605:5916:2863:2863:11542:4658:4658:5154:2357:2605:3639:3639:2863:11542:4658:6204:6204:2357:4159:3639:3639:5698:5698:5698:6204:6204:4167:4159:4159:3146:3146:5698:1357:1357:4167:4167:

Image

There was a tryfurcated opening for this puzzle by Glyn. But i guess there must be a nicer way through it... some day.

Quote:
mhparker: Wow, (JSudoku 1.3b1) can do the Assassin 50 V2!
However, before you all get too excited, here are the stats and the HUGE solver log (including a finned jellyfish found near the start!) in TT. It's clearly way off the scale of anything that can be reasonably posted on any forum without ending up on the "Unsolvables" list. :twisted:
JSudoku solver log: here
mhparker: However, before you all get too excited, here are the stats and the HUGE solver log (including a finned jellyfish found near the start!) in TT. It's clearly way off the scale of anything that can be reasonably posted on any forum without ending up on the "Unsolvables" list. :twisted:
Glyn: at last my tryfurcation can be consigned to history
Andrew (in 2015): When I started on this puzzle again, having not got very far when it first appeared, I wondered how far I'd get. However I found that it's about the same difficulty as Assassin 39 V2, which I did recently, but a bit shorter walkthrough; then I realised that I can omit my heaviest steps, so I've also posted a simplified walkthrough.


7) July 20th 2007: Ruud's Assassin 60RP

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=1129#p1129

PS:

3x3::k:2304:2304:3330:3330:3330:4869:4869:5639:5639:2304:5642:5642:3852:3852:3342:4869:4869:5639:5138:5642:4372:4372:3852:3342:3342:5145:5639:5138:3100:4372:3870:3870:2848:2848:5145:5145:5138:3100:3100:3870:3368:3368:2848:5931:5145:3629:4142:4142:3632:3368:4402:4402:5931:5931:3629:3629:4142:3632:3632:4402:3644:3644:5931:6463:3629:4161:4161:6979:6979:3644:4678:4678:6463:6463:6463:4161:4161:6979:6979:4678:4678:

Image

There has been made a start to this puzzle, except this hasn't led to much yet.

8) July 27th 2007: Para's Special Killer-X 4V2

PS:

3x3:d:k:2560:3585:3842:3842:5124:3589:3589:4615:2568:3593:2560:3585:3842:5124:3589:4615:2568:5137:3602:3593:11540:3585:5124:4615:11540:5137:3866:3602:3602:3593:11540:5124:11540:5137:3866:3866:5156:5156:5156:5156:11540:5161:5161:5161:5161:5165:5165:3887:11540:5169:11540:3379:4404:4404:5165:3887:11540:2873:5169:3643:11540:3379:4404:3887:2624:2873:3906:5169:2884:3643:2630:3379:2624:2873:3906:3906:5169:2884:2884:3643:2630:

Image

Horrible result of an atempt to create a suitable Killer-X 4V2.

9) Aug 1st 2007: Mhparker's Assassin61XV3

PS: 3x3:d:k:4096:4096:4354:4354:2820:5125:5125:5127:5127:4096:5130:4354:4354:2820:5125:5125:3088:5127:6674:5130:5130:4629:2820:4631:3088:3088:4378:6674:6674:4629:4629:4631:4631:4631:4378:4378:4644:6674:6674:3111:3111:3111:7722:7722:4378:4644:4644:5423:5423:5423:3378:3378:7722:7722:4644:3383:2872:5423:3642:3378:2620:3901:7722:2623:3383:2872:4162:3642:4164:2620:3901:2887:2623:3383:4162:4162:3642:4164:4164:3901:2887:

Image

Another horrible creation while trying to make a suitable V2 for an assassin.

A62V2 might need to be added to this list. But as no real attempt at that one has been made, so i'll keep it off. But i think a list of 9 puzzles suits Sudoku.

So the challenge is up to remove these puzzles from the list.

greetings

Para


Andrew wrote:
Para

Good stuff collecting them together!

I'm sure it's well known on this forum that I believe that all Assassins, except for a few easier early ones, and all forum puzzles ought to have at least one posted walkthrough if they can be solved manually. Your message should encourage that to happen!

I have solved CDKV3 with a couple of hints from Ed. I was planning to post my walkthrough after doing the ones for SampuZ4V1 and SampuZ5 but it got delayed because of our recent move. You have reminded me that I must do it soon.

I started your A44V1.5 but didn't get very far. I can post my starting moves if anyone is interested and would like to pick it up, either as a tag or just to finish it off.

I also started your Special Killer-X 4V2 and got further than with A44V1.5 before I got stuck. Again I can post my starting moves ...

I assume that the walkthroughs should be posted to the threads for those puzzles. Do you also want a short message posting in this thread to say that there is now a walkthrough? Alternatively should we inform you by PM so that the solved puzzle can be removed from your message?


Para wrote:
Hi

I think when we got a walk-through i'll just put a similar message under each puzzle like Mike Mepham did, that there is a solution path.

Yes it is probably more suitable that each puzzle gets it's walk-through posted in their original thread. Except maybe number 8 and 9 because they were posted in a thread to show a point and there is already a Tag going on. So they might need a seperate thread otherwise their will be different Tag solutions running through eachother. And that would be very confusing.

greetings

Para
Unsolvables 10-14 with links to current walkthroughs:
sudokuEd wrote:
Great idea Para (and Glyn). Brings back lots of memories looking through that list.

I realize I missed Mike's Assassin 61X V3 from the ratings page so will add that in. Sorry Mike.

Can add one more to the list. This one didn't have a pic - but should be a real good solvable puzzle. Better to call it not-yet-solved.

I've done a step count for each of the 10 unsolvable puzzles if anyone is interested. I hope that using step counts will help us to design puzzles at the desired level. In tt.

1) Sept 28th 2006: SudokuEd's UTA 1 (Did Not Solve (DNS) 42 steps No more progress)

2) Dec 29th 2006: SudokuEd's Bullseye 3 (aka Assassin 30V2.1) (DNS 30 steps)

3) Feb 17th 2007: SudokuEd's CDKV3 (117)

4) Feb 22th 2007: Ruud's Assassin39V3 (DNS 62 steps no more progress)

5) April 28th 2007: Para's Assassin 44V1.5 (101)

6) May 11th 2007: Ruud's Assassin 50V2 (DNS 24 No progress)

7) July 20th 2007: Ruud's Assassin 60RP (DNS 28 No progress)

8) July 27th 2007: Para's Special Killer-X 4V2 (DNS 8 No progress)

9) Aug 1st 2007: Mhparker's Assassin61XV3 (DNS 17 steps No more progress)

10) October 22 2006: sudokuEd's Assassin 21V2 (136)

Cheers
Ed

10) October 22 2006: sudokuEd's Assassin 21V2

solved: Walk-throughs now at viewtopic.php?p=807#p807

PS:
3x3::k:3584:3584:4098:5123:5123:5123:5638:5638:5638:3584:4098:4098:5123:4109:4109:4109:3344:5638:3584:6675:6675:6675:4118:4118:792:3344:3344:3867:6675:4637:4637:6943:4118:792:1570:1570:3867:6675:4637:6943:6943:6943:4650:5163:3372:2605:2605:1327:3888:6943:4650:4650:5163:3372:2358:2358:1327:3888:3888:2619:2619:5163:4670:7743:2358:5185:5185:5185:5185:4165:4165:4670:7743:7743:7743:2379:2379:5185:4165:4670:4670:

Image


Para wrote:
Hi all

Let's add another, because there hasn't been any activity on this puzzle at all.

11) August 26th 2007: Ruud's Assassin 65V3

Solved: Walk-throughs at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=1205#p1205

PS:

3x3::k:4352:4352:4354:4099:4099:4099:2310:3591:3591:4352:3082:4354:4354:3853:2310:2310:3856:3591:3858:3082:3082:10261:3853:3095:3856:3856:5658:3858:3082:4637:10261:3853:3095:5921:5658:5658:3858:4637:4637:10261:10261:10261:5921:5921:3628:4397:4397:4637:2608:2865:10261:5921:4148:3628:4397:4663:4663:2608:2865:10261:4148:4148:3628:3903:4663:3649:3649:2865:5444:5444:4148:3143:3903:3903:3649:3147:3147:3147:5444:3143:3143:

Image

greetings

Para


mhparker wrote:
Para wrote:
Let's add another, because there hasn't been any activity on this puzzle at all.

Ditto:

12) September 18th 2007: Ruud's Assassin 68V3

Solved: Walk-through at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=1220#p1220

PS:

3x3::k:4608:4608:4608:4099:3588:4357:5638:5638:5638:6665:4608:4099:4099:3588:4357:4357:5638:6161:6665:4608:5908:4629:4629:4629:4632:5638:6161:6665:6665:5908:5908:4629:4632:4632:6161:6161:3364:3364:4390:5908:4629:4632:4650:2347:2347:3885:3364:4390:4390:2353:4650:4650:2347:4149:3885:3885:3896:3896:2353:3387:3387:4149:4149:2879:4416:3896:4674:4674:4674:3387:4934:2887:2879:4416:4416:2123:2123:2123:4934:4934:2887:

Image
Quote:
mhparker: Let's add another, because there hasn't been any activity on this puzzle at all.
Jean-Christophe, in JSudoku thread this forum: Proud to announce JSudoku could also solve Ruud's Assassin 68V3
Andrew in 2015: Harder than puzzles I've done recently, including Assassin 50 V2. After working hard and making progress in one area, I had to start thinking hard and work in other areas.


Para wrote:
Here's another one for the record books. Got lost after my initial post, it was mentioned there.
It's kinda funny that some monsters avoid this thread by us being so stubborn in wanting to solve the really brutal ones. Like A48-Hevvie and the Brick Wall.

13) August 3rd 2007: Ruud's A62V2

Solved: Walk-through at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=1135#p1135

PS:

3x3::k:7168:7168:3074:4099:4099:4869:4869:1799:1799:7168:7168:3074:3074:4099:2830:4869:5136:1799:4626:4626:4626:3074:4886:2830:5136:5136:4122:4626:5660:4886:4886:4886:5920:5920:5136:4122:2084:5660:5660:4647:5920:5920:3626:3626:4122:2084:4910:4647:4647:3889:3889:3889:3626:6197:2084:4910:4910:2873:3889:4667:6197:6197:6197:2367:4910:5697:2873:4419:4667:4667:4934:4934:2367:2367:5697:5697:4419:4419:4667:4934:4934:

Image

greetings

Para


Ruud wrote:
14) January 11, 2008: A85 (original version)

Solved:Tag solution at http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?p=1337#p1337

PS:

3x3::k:5376:5376:4098:4098:1284:2821:2821:2567:2567:5376:4618:4618:4618:1284:6158:6158:6158:2567:2578:4618:3348:3348:5654:4375:4375:6158:3354:2578:4618:3348:5654:5654:5654:4375:6158:3354:1828:1828:8486:8486:8486:8486:8486:1323:1323:2605:7470:4399:4144:4144:4144:2611:5940:3125:2605:7470:4399:4399:4144:2611:2611:5940:3125:3903:7470:7470:7470:3395:5940:5940:5940:4423:3903:3903:2378:2378:3395:2381:2381:4423:4423:

Image


This is the original version posted on the website before it was replaced on January 12 with a more suitable version.

Ruud
(remaining) Unsolvables
8) July 27th 2007: Para's Special Killer-X 4V2
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:2560:3585:3842:3842:5124:3589:3589:4615:2568:3593:2560:3585:3842:5124:3589:4615:2568:5137:3602:3593:11540:3585:5124:4615:11540:5137:3866:3602:3602:3593:11540:5124:11540:5137:3866:3866:5156:5156:5156:5156:11540:5161:5161:5161:5161:5165:5165:3887:11540:5169:11540:3379:4404:4404:5165:3887:11540:2873:5169:3643:11540:3379:4404:3887:2624:2873:3906:5169:2884:3643:2630:3379:2624:2873:3906:3906:5169:2884:2884:3643:2630:
Solution:
+-------+-------+-------+
| 1 6 3 | 5 2 9 | 4 7 8 |
| 8 9 5 | 7 4 1 | 3 2 6 |
| 4 2 7 | 3 6 8 | 1 9 5 |
+-------+-------+-------+
| 3 7 4 | 2 8 6 | 5 1 9 |
| 2 1 8 | 9 5 7 | 6 4 3 |
| 9 5 6 | 4 1 3 | 7 8 2 |
+-------+-------+-------+
| 6 4 9 | 1 3 2 | 8 5 7 |
| 5 3 2 | 8 7 4 | 9 6 1 |
| 7 8 1 | 6 9 5 | 2 3 4 |
+-------+-------+-------+[/pre]
Quote:
Para, lead-in: Sometimes uniqueness might not be the only good constraint for a puzzle.This puzzle for example is also unique (even without the diagonal constraint). It was the result of an attempt to create a proper 5-cage diagonal Killer-X puzzle (based on Ruud's Killer-X special No4). But i don't think there is any proper way of solving this puzzle
Andrew (in 2012): I've just had a look at Para's SKX-4 V2 for the first time. An interesting cage pattern and starting steps. Then I got stuck and had the feeling that this puzzle will remain unsolvable by humans, unless there are masochists out there who want to try brute force T&E.
Andrew's start and comment on the cage pattern:
This is a Killer-X.

Prelims

a) 10(2) cage at R1C1 = {19/28/37/46}, no 5
b) 10(2) cage at R1C9 = {19/28/37/46}, no 5
c) 10(2) cage at R8C2 = {19/28/37/46}, no 5
d) 10(2) cage at R8C8 = {19/28/37/46}, no 5
e) 20(3) cage at R2C9 = {389/479/569/578}, no 1,2
f) 20(3) cage at R6C1 = {389/479/569/578}, no 1,2
g) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
h) 11(3) cage at R8C6 = {128/137/146/236/245}, no 9
i) And, of course, 45(9) cage at R3C3 = {123456789}

1. 45 rule on R5 1 innie R5C5 = 5, placed for both diagonals
[It’s interesting that this placement is also available from 45 rule on C5 and from 45 rule on both diagonals, with R5C5 on both diagonals.]

2. Writing (R3C3 + R4C4) = A, (R3C7 + R4C6) = B, (R6C4 + R7C3) = C, (R6C6 + R7C7) = D
2a. 45 rule on R1234 4 innies A + B = 16
2b. 45 rule on R6789 4 innies C + D = 24
2c. 45 rule on C1234 4 innies A + C = 22
2d. 45 rule on C6789 4 innies B + D = 18
2e. 45 rule on D\ 4 innies A + D = 20
2f. 45 rule on D/ 4 innies B + C = 20
2g. Taking the difference between steps 2a and 2c, C = B + 6, B + C = 20 (step 2f) -> B = 7, C = 13 -> R3C7 + R4C6 = {16/34}, R6C4 + R7C3 = {49/67}, killer pair 4,6 on D/ and for 45(9) cage at R3C3, no 4,6
2h. Taking the difference between steps 2b and 2c, D = A + 2, A + D = 20 (step 2e) -> D = 11, A = 9 -> R3C3 + R4C4 = 9 = {18/27}, R6C6 + R7C7 = {29/38}, killer pair 2,8 on D\

[Don’t know whether this is any help
(R1C1 + R2C2) + (R8C8 + R9C9) are “cloned” in (R3C7 + R4C6) + (R6C4 + R7C3)
and (R1C9 + R2C8) + (R8C2 + R9C1) are “cloned” in (R3C3 + R4C4) + (R6C6 + R7C7) although not “cloned” as the same pairs since A, B, C and D don’t total 10.]

This is also the stage reached by SudokuSolver v3.3.1, which gave the statement "No more eliminations". Clearly a human solver could try bifurcation, since each of the cell pairs A, B, C and D only have two possible pairs of values, but these don't immediately lead to any progress with the other cages so any human solving attempt would probably require a much deeper level of T&E.

While looking at this puzzle, I had the impression that the cage pattern has some similarity to the UK's national flag, so I re-coloured the diagram appropriately.

Image

9) Aug 1st 2007: Mhparker's Assassin61XV3
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4096:4096:4354:4354:2820:5125:5125:5127:5127:4096:5130:4354:4354:2820:5125:5125:3088:5127:6674:5130:5130:4629:2820:4631:3088:3088:4378:6674:6674:4629:4629:4631:4631:4631:4378:4378:4644:6674:6674:3111:3111:3111:7722:7722:4378:4644:4644:5423:5423:5423:3378:3378:7722:7722:4644:3383:2872:5423:3642:3378:2620:3901:7722:2623:3383:2872:4162:3642:4164:2620:3901:2887:2623:3383:4162:4162:3642:4164:4164:3901:2887:
Solution: 
+-------+-------+-------+
| 8 6 1 | 5 3 9 | 2 4 7 |
| 2 7 3 | 8 6 4 | 5 1 9 |
| 5 9 4 | 7 2 1 | 8 3 6 |
+-------+-------+-------+
| 7 1 2 | 9 8 3 | 6 5 4 |
| 3 4 9 | 1 5 6 | 7 8 2 |
| 6 5 8 | 4 7 2 | 3 9 1 |
+-------+-------+-------+
| 4 3 6 | 2 9 8 | 1 7 5 |
| 1 2 5 | 3 4 7 | 9 6 8 |
| 9 8 7 | 6 1 5 | 4 2 3 |
+-------+-------+-------+[/size]
Quote:
mhparker on solving ruudiculous killers: In my spare time, I do quite a lot of cycling. Like most keen cyclists, I have a watch with a heart rate monitor, capable of recording one's maximum pulse. The problem is, just how does one measure that (without risking collapsing with a heart attack afterwards!)? Now, if you ask most people how to do it, they will maybe suggest finding a good steep hill and battling up it. However, this is the wrong answer. The correct answer (surprisingly enough) is to take a flat(-ish) piece of road and do a prolonged sprint.

Get the analogy again? Make a parcours too difficult, and what happens is that instead of the contestants' performance increasing as expected, they just end up pacing themselves and losing form. This is clearly visible on this forum. Up to a point, the moves we make get more and more ingenious as the difficulty of the puzzle increases. But go beyond that threshold, and the quality of the moves tends to rapidly decrease again.

IMO (and I'm saying this with my puzzle setter hat on), the best puzzles are the ones that hit the above-mentioned threshold, without exceeding it. In other words, the ones that stretch the solvers, yet without over-strectching them.

We should not IMO fall into the trap of thinking we can and should solve every puzzle. Even if we could do the A60RP, it's not the end of the road, by any means. There are other - bigger - hills to climb beyond. Puzzles that are even more intractable than this one. For example, the A60RP looks like child's play in comparison to this one, which I came across when looking for a suitable A61X.
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Ed
PostPosted: Fri Jun 27, 2008 11:42 pm 
Offline
Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 61 by Ruud (July 07)
Puzzle pic:
Image
Solution: 
+-------+-------+-------+
| 2 3 8 | 6 1 4 | 5 7 9 |
| 6 5 9 | 7 8 2 | 1 3 4 |
| 1 7 4 | 3 9 5 | 8 6 2 |
+-------+-------+-------+
| 3 8 5 | 4 2 1 | 7 9 6 |
| 4 9 6 | 8 3 7 | 2 1 5 |
| 7 1 2 | 9 5 6 | 3 4 8 |
+-------+-------+-------+
| 5 4 3 | 1 6 8 | 9 2 7 |
| 9 2 7 | 5 4 3 | 6 8 1 |
| 8 6 1 | 2 7 9 | 4 5 3 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:2816:2816:7682:7682:4612:3077:3077:5127:5127:2816:4106:7682:7682:4612:3077:3077:4368:5127:6930:4106:4106:5141:4612:4631:4368:4368:5658:6930:6930:5141:5141:4631:4631:4631:5658:5658:4388:6930:6930:5141:4631:6185:5674:5674:5658:4388:4388:4399:4399:4399:6185:6185:5674:5674:4388:3127:2616:4399:4410:6185:3900:3901:5674:4415:3127:2616:2114:4410:4164:3900:3901:1095:4415:3127:2114:2114:4410:4164:4164:3901:1095:
Quote:
CathyW: Tougher than it looked from the preliminaries but got there eventually
Andrew: I found this one fairly straightforward and didn't need any wizard moves
Walkthrough by CathyW:
Tougher than it looked from the preliminaries but got there eventually - hopefully with no errors this time (or at least only minor typos)! First time I've knowingly used grouped x-wings to make eliminations.

Preliminaries:

a) 30(4) r12c34 = {6789}
b) 17(2) r89c1 = {89} not elsewhere in c1/N7 -> 10(2) r78c3 = {37/46}
c) 4(2) r89c9 = {13} not elsewhere in c9/N9
d) 8(3) r8c4+r9c34 = {125/134} Must have 1 -> r9c56 <> 1
e) 12(4) r12c67 = {1236/1245} Must have 1 and 2
f) 20(3) N3, no 1,2
g) 11(3) N1, no 9
h) 15(2) N9 = {69/78}
i) 18(5) at r3c6 = {12348/12357/12456} Must have 1 and 2

1. Outies c1234: r6c5 = 5

2. Innies c5: r45c5 = 5 = {14/23} -> r3c6+r4c67 = 13 (must have 1 or 2, can’t have both 3,4) = {148/157/238/256}

3. Innies N3: r12c7+r3c9 = 8 = {125/134} Must have 1, not elsewhere in N3
-> 1 locked to r12c7, not elsewhere in c7; r12c6 <> 1
No 3 in r3c9 -> r12c7 <> 4

4. Innies N9: r7c9+r9c7 = 11 = {29/47/56}

5. Outies c12: r35c3 = 10 = {19/28/37/46}

6. Outies c89: r35c7 = 10 = {28/37/46}

7. Outies – Innies r12: r3c5 – r2c28 = 1
-> r2c28 max 8, min 3 -> r3c5 = (46789), r2c2 = (1…6), r2c8 = (2…7)
Options: [412]/[614]/6{23}/[715]/7{24}/[816]/8{25/34}/[917]/9{26/35}

8. Innies r123: r3c1469 = 11 = {1235} not elsewhere in r3
-> r5c3 <> 7,8,9; r5c7 <> 7,8
-> split 8(3) N3 = {125}
-> 20(3) N3 = {389/479} Must have 9; If {389}, r1c8 = 3 -> r1c3 <> 8
-> 17(3) N3 = {368/467} Must have 6
-> 3 locked to r12c8, not elsewhere in c8 -> 3 locked to r456c7
-> r3c5 <> 4 (step 7)

9. Innies N7: r7c1+r9c3 = 6 = {15/24}

10. Grouped x-wing (1): 8(3) and 4(2) both in r89, 1 not elsewhere in r89

11. 12(3) N7 = {156/246/237} {147} blocked by 10(2) and {345} blocked by r7c1+r9c3 -> r7c2 <> 5

12. 15(3) N9 = {249/258/456} {267} blocked by 15(2)

13. 17(4) r6c345+r7c4 = 5{129/138/147/237/246} Must have at least one of 1,2

14. Innies N1: r12c3+r3c1 = 18 = {189/279/369/378/567}

15. Outies r6789: r5c1678 = 14 (no 9) = {1238/1247/1256/1346/2345}

16. 22(4) r349c9+r4c8 = 2{479/569/578}, 5{179/269/278/467}

17. O-I N8: r7c46 – r9c37 = 4
-> r7c46 min 7, max 17; r9c37 min 3, max 13

18. Grouped x-wing (5): in c34 5 locked within 20(4) and 8(3)
-> 8(3) = {125} -> r7c1 <> 2
-> r9c56 <>2, r9c6 <> 5 (cells see all of 8(3))

19. Grouped x-wing (9): in r12 both 30(4) and 20(3) must have 9
-> r12c5 <> 9

20. 16(3) N1, min from r3c23 = {46} = 10 but can’t repeat so r2c2 <> 6, r3c23 not {46}
Options: {169/178/268/349/367/457}
If {349/457} r2c2 = 3/5 -> r2c2 <> 4

21. Pointing cells: 6 locked to r4567c9 -> r5c78, r6c8 <> 6 -> r3c7 <> 4

22. Pointing cells: 1 locked to r45c5+r34c6 of 18(5) -> r56c6 <> 1

23. r3c9 = (25)
a) r3c9 = 2 -> r12c7 = {15} -> r12c6 = {24}
-> in N2 5 locked to r3c46 -> r3c1 <> 5
b) r3c9 = 5 -> r3c1 <> 5
Either case r3c1 <> 5
Further, if r3c9 = 5, r12c7 = {12}, then 5 locked to r12c6, so in either case 12(4) = {1245}
-> 4 locked to r12c6 not elsewhere in N2/c6
-> 18(3) N2 options: {189/279/369/378} -> r3c5 <> 6

24. 16(3) r8c6+r9c67 = {259/268/349/358/367/457}
If {259} r9c6 = 9; if {349} r9c7 = 4 -> r9c7 <> 9 -> r7c9 <> 2

25. r3c9 = (25)
a) r3c9 = 2 -> r4c8 <> 2
b) r3c9 = 5 -> r12c7 = {12} -> 2 locked to r789c8 -> r4c8 <> 2
Either case, r4c8 <> 2

26. 30(4) N12 = {6789}
a) if 9 within r12c3 -> r6c3 <> 9
b) if 9 within r12c4 -> r6c6 = 9 -> r6c3 <> 9
Either case r6c3 <> 9

27. r3c9 = (25)
a) r3c9 = 2 -> r12c7 = {15} -> r4c7 <> 5
b) r3c9 = 5 -> r3c4 <> 5 -> r89c4 must have 5 -> r9c3 <> 5 -> r4c3 = 5 -> r4c7 <> 5
Either case r4c7 <> 5

28. 18(5) = {12348/12357/12456}
Because r45c5 must be {14} or {23}, if 18(5) = {12357}, r45c5 = {23}, r3c6 = 5, r4c6 = 1, r4c7 = 7 -> r4c6 <> 7

29. 18(3) N2 = {189/279/369/378} Must have one of 7,9
-> 17(3) N8 not {179} -> r7c5 <> 1
-> 17(3) N8 = {269/278/368/467}

30. Split 18(3) N1 = {189/279/369/378} Must have one of 3,9
-> 16(3) N1 not {349}
-> 16(3) N1 = {169/178/268/367/457}

31. 16(3) N1 must have at least one of 6,7 within r3c23
17(3) N3 must have at least one of 6,7
If r2c8 = 6, r3c78 = [74]; else 6 in r3c78
-> r3c5 <> 7
-> 18(3) N2 = {18/27/36}9/{37}8

32. 1 locked to r45c5+r34c6 of 18(5)
a) if r3c6 = 1 -> 1 locked to r789c4 -> r456c4 <> 1
b) if r45c5 or r4c6 = 1 -> r456c4 <> 1
Either case r456c4 <> 1
-> 1 locked to r45c5+r4c6 of 18(5) -> r3c6 <> 1

33. In r3, 1 in r3c1 or r3c4
a) r3c1 = 1 -> r7c1 <> 1
b) r3c4 = 1 -> r7c6 = 1 -> r7c1 <> 1
Either case r7c1 <> 1
-> r9c3 <> 5
-> 5 locked to r89c4, not elsewhere in c4/N8
-> HS r4c3 = 5
-> r345c4 = 15 = {168/249/267/348} -> r45c4 <> 2,3

34. 9 locked tor123c3 -> r3c2 <> 9

35. Pointing pair: 5 locked to r5c89 -> r7c9 <> 5 -> r9c7 <> 6

36. 16(3) r8c6+r9c67 = {259/268/349/358/367}
If {367} r9c7 = 7 -> r89c6 <> 7

37. 7 locked to r567c6 -> 24(4) = {7…}-> r6c7 <> 7
24(4) = {2679/3678} Must also have 6

38. 1 locked to r789c4 -> r3c4 <> 1
-> r3c1 = 1 -> r12c3 = {89} not elsewhere in c3/N1 -> r12c4 = {67} not elsewhere in c4/N2
-> 11(3) N1 = {236/245}
-> 16(3) N1 = {367/457}, 7 not elsewhere in r3
-> HS r3c5 = 9 -> r12c5 = {18} not elsewhere in c5
-> r45c5 = {23} not elsewhere in c5/N5/18(5) -> r3c6 = 5
-> r3c9 = 2, r3c4 = 3
-> r45c4 = {48} not elsewhere in c4/N5 -> r6c4 = 9 -> r789c4 = {125} not elsewhere in N8
-> 8 locked to r3c78 -> 17(3) = 3{68} -> 20(3) = {479}

… Straightforward cage combinations and singles from here
Walkthrough by Andrew:
I liked Cathy's grouped x-wings, especially the one she used to fix the 8(3) cell as {125}, the pointing cells moves and the chains. Maybe the most impressive step was 27b. When I was adding comments while going through her walkthrough I typed "Wow! That’s not an obvious chain."
Ruud wrote:
Put aside your Deathly Hallows, for they will not help you beat this Assassin.
I found this one fairly straightforward and didn't need any wizard moves, unlike most recent Assassins which have required lots of them. This was the first Assassin that I've solved in one session for many weeks.

Thanks to Para for his comments.

Here is my walkthrough.

1. R78C3 = {19/28/37/46}, no 5

2. R78C7 = {69/78}

3. R89C1 = {89}, locked for C1 and N7, clean-up: no 1,2 in R78C3

4. R89C9 = {13}, locked for C9 and N9

5. 11(3) cage in N1 = {128/137/146/236/245}, no 9

6. 20(3) cage in N3 = {389/479/569/578}, no 1,2

7. 8(3) cage at R8C4 = 1{25/34}, no 1 in R9C56

8. 30(4) cage at R1C3 = {6789}

9. 12(4) cage at R1C6 = 12{36/45}, no 7,8,9

10. 18(5) cage at R3C6 = {12348/12357/12456} = 12{348/357/456}, no 9

11. 15(3) cage in N9 = {249/258/456} (cannot be {267} which clashes with R78C7), no 7

12. 12(3) cage in N7 = {156/237/246} (cannot be {147/345} which clash with R78C3)

13. Killer pair 6,7 in 12(3) cage (step 12) and R78C3, locked for N7

14. 45 rule on N7 2 innies R7C1 + R9C3 = 6 = {15/24}, no 3

15. 45 rule on N9 2 innies R7C9 + R9C7 = 11 = {29/47/56}, no 8

16. 45 rule on N1 3 innies R12C3 + R3C1 = 18, min R12C3 = 13 -> max R3C1 = 5
[Para commented. Actually minimum R12C3 = 14 already as {67} is blocked by R78C3. Doesn't matter much in the solving path though.]

17. 45 rule on N3 3 innies R12C7 + R3C9 = 8 = 1{25/34}, no 6,7,8,9, 1 locked in R12C7 for C7, N3 and 12(4) cage -> no 1 in R12C6
17a. {134} must have 4 in R3C9 -> no 4 in R12C7

18. 45 rule on R123 4 innies R3C1469 = 11 = {1235}, locked for R3

19. R12C7 + R3C9 (step 16) = {125} (only remaining combination), locked for N3

20. 45 rule on C12 2 outies R35C3 = 10 = {46}/[73/82/91] -> R5C3 = {12346}

21. 45 rule on C1234 1 outie R6C5 = 5
[I ought to have seen this much earlier! If I’d been looking at the original puzzle grid rather than my elimination grid I probably would have done so.]

22. 45 rule on C6789 3 innies R34C6 + R4C7 = 13 -> R34C5 = 5 = {14/23}
[Alternatively 45 rule on C5 2 innies R34C5 = 5 -> R34C6 + R4C7 = 13. In retrospect this is more obvious but after step 21 I was looking for something similar.]

23. 45 rule on C89 2 outies R35C7 = 10 = {46}/[73/82] -> R5C7 = {2346}, no 9 in R3C7

24. 45 rule on R6789 4 outies R5C1678 = 14 = {1238/1247/1256/1346/2345}, no 9

25. 45 rule on R1234 5 outies R5C23459 = 31 = {16789/25789/34789/35689/45679}, R5C5 = {1234} -> no 1,2 in R5C2349, clean-up: no 8,9 in R3C3
[Alternatively, using the result from step 24, 45 rule on R5 5 innies R5C23459 = 31]
25a. Cannot be {25789} because R5C3 only contains 3,4,6 -> R5C23459 = {16789/34789/35689/45679}, no 2, clean-up: no 3 in R4C5
25b. 2 in R5 locked in R5C1678 = {1238/1247/1256/2345}

26. Naked quad {3467} in R3578C3, locked for C3, clean-up: no 8,9 in R12C4, no 2 in R7C1 (step 14)

27. Naked pair {89} in R12C3, locked for C3 and N1
[Para commented. This of course sets R3C1 to 1 from innies N1 (step 16), which you eventually get to through combination analysis of 2 cages. But it doesn't matter much really as the puzzle is now broken down to the easier bits.

Agreed. Interesting that I missed two things about step 16. The combinations are still needed for the other eliminations although those steps would have been simpler if I'd fixed R3C1 first.]

28. Naked pair {67} in R12C4, locked for C4 and N2
[With hindsight, this reduced the 12(4) cage at R1C6 to one combination. I did that in step 33 so missing that here didn’t affect the solving path.]

29. 16(3) cage in N1 = {367/457} = 7{36/45}, no 1,2, 7 locked for N1
29a. No 3,5 in R3C23 -> R2C2 = {35}, R3C23 = {47/67}
29b. 7 locked in R3C23, locked for R3

30. 11(3) cage in N1 = {236/245} (cannot be {146} which clashes with R3C23), no 1, 2 locked for N1

31. R3C1 = 1 (hidden single in N1), clean-up: no 5 in R9C3 (step 14)

32. Naked pair {12} in R69C3, locked for C3 -> R4C3 = 5

33. 3 in R3 locked in R3C46, locked for N2

34. Hidden triple {189} in R123C5, locked for C5, clean-up: no 4 in R45C5 (step 22) -> R45C5 = [23] -> R3C6 = 5, R3C9 = 2, R3C4 = 3

35. Naked pair {24} in R12C6, locked for C6

36. Naked pair {15} in R12C7, locked for C7

37. Naked triple {467} in R789C5, locked for N8

38. R3C6 + R45C5 = [523] = 10 -> R4C67 = 8 = [17] (only remaining permutation), clean-up: no 8 in R78C7
[At this stage I saw grouped X-wing (9) with 20(3) cage in N3 and 22(4) cage at R3C9, 9 locked for C89, but then decided to use the naked pair in the next step.]

39. Naked pair {69} in R78C7, locked for C7 and N9

40. Naked pair {24} in R59C7, locked for C7 -> R3C7 = 8, R6C7 = 3, R3C5 = 9
[Alternatively R6C7 was hidden single in C7.]

41. R9C7 = {24} -> total of R89C6 must be even = {39} -> R9C7 = 4, R5C7 = 2
[I’ll admit that 16(3) cage at R8C6 = {39}4 (only remaining combination) is more direct but I like using the property of even/odd numbers when I can.]

42. Naked pair {39} in R89C6, locked for C6 and N8 -> R7C6 = 8

43. R7C9 = 7 (hidden single in N9), clean-up: no 3 in R8C3

44. R3C4 + R4C3 = [35] -> R45C4 = {48} (only remaining combination), locked for C4 -> R6C4 = 9

45. R3C7 = 8 -> R23C8 = 9 = [36] (only remaining permutation), R2C2 = 5, R12C7 = [51], R12C5 = [18], R12C3 = [89], R12C9 = [94], R1C8 = 7, R12C6 = [42], R2C1 = 6, R12C4 = [67]

46. R7C1 = 5 (hidden single in C1) -> R9C3 = 1 (step 14), R6C3 = 2, R7C4 = 1, R89C9 = [13], R89C6 = [39], R89C1 = [98], R78C7 = [96], R789C8 = [285], R89C4 = [52], clean-up: no 4 in R7C3

47. R6C2 = 1 (hidden single in C2)
[There is also a hidden single in R8C2 but I didn’t notice it originally. It’s possible that it wasn’t a hidden single when I first solved the puzzle. I had to edit some steps because I hadn’t made all the eliminations after fixing a couple of earlier cells.]

48. Naked pair {47} in R56C1, locked for C1 and N4 -> R4C1 = 3, R5C3 = 6

and the rest is naked singles
Walkthrough by Para:
Hi all

Just finished this one. Don't know why i couldn't break it the first time. But i found the breakthrough move (a hidden killer Quad in C4) at the second look.

Walk-Through Assassin 61

1. 11(3) at R1C1 = {128/137/146/236/245}: no 9

2. 30(4) at R1C3 = {6789}

3. 12(3) at R1C6 = {1236/1245}: no 7,8,9

4. 20(3) at R1C8 = {389/479/569/578}: no 1,2

5. 18(5) at R3C6 = {12348/12357/12456}: no 9

6. R78C3 = {19/28/37/46}: no 5

7. R78C7 = {69/78}: no 1,2,3,4,5

8. R89C1 = {89} -->> locked for C1 and N7
8a. Clean up: R78C3: no 1,2

9. 8(3) at R8C4 = {125/134}: no 6,7,8,9; 1 locked in 8(3) cage: R9C56: no 1

10. R89C9 = {13} -->> locked for C9 and N9

11. 45 on C1234: 1 outie: R6C5 = 5

12. 45 on N7: 2 innies: R7C1 + R9C3 = 6 = {15/24}: no 3,6,7

13. 45 on N9: 2 innies: R7C9 + R9C7 = 11 = {29/47/56}: no 8

14. 45 on C5: 2 innies: R45C5 = 5 = {14/23}: no 6,7,8

15. 45 on R123: 4 innies: R3C1469 = 11 = {1235} -->> locked for R3

16. 45 on N3: R12C7 + R3C9 = 8 = {125}({134} blocked by R3C9) -->> locked for N3
16a. 1 in N3 locked for C7 and 12(4) cage at R1C6
16b. 3 in N3 locked for C8
16c. 17(3) at R2C8 = {368/467}: no 9; 6 locked for N3

17. 45 on C89: 2 outies: R35C7 = 10 = [46/64/73/82]: R5C7 = {2346}

18. 45 on C12: 2 outies: R35C3 = 10 = [46/64/73/82/91]: R5C3 = {12346}

19. Killer Quad {6789} in R12C3 + R35C3 + R78C3 -->> locked for C3
19a. 8 in N4 locked for C2
19b. 9 in N4 locked for C2

20. 45 on N1: 3 innies: R12C3 + R3C1 = 18 = {89}[1]/{79}[2]/{69/78}[3]: {67}[5] blocked by R78C3: R3C1: no 5

21. 16(3) at R2C2 = {169/178/268/349/367/457}: {259/358 blocked by R3C2)

22. 5 in N1 either in 11(3) at R1C1 or 16(3) at R2C2; 11(3) = {245} or 16(3) = {457}
22a. 11(3) at R1C1: {137/146} blocked(clash with 16(3) = {457}; 16(3): {268/349} blocked(clash with 11(3) = {245})
22b. 11(3) at R1C1 = {236/245}: no 1,7; 2 locked for N1

23. 7 in C1 locked for N4

24. 17(4) at R6C3 = 5{129/138/147/237/246} = {6|7|8|9..} -->> R67C4 needs one of {6789}

25. Hidden Killer Quad {6789} in R12C4 + R67C4 + R45C4 -->> R45C4 needs one of {6789}
25a. 20(4) at R3C4 needs one of {6789} in R45C4 and no room for {6789} in any of ther other cells of that cage -->> 20(4) contains max. one of {6789} -->> 20(4) = {2459/3458}: no 1,6,7

26. CPE: R45C5: no 1, see all 1’s in N2
26a. R45C5 = {23}(step 14) -->> locked for C5, N5 and 18(5) cage at R3C6

27. 18(3) at R1C5 = {189/468}: no 7; 8 locked for N2 and C5
27a. 7 in N2 locked for C4 and 30(4) cage at R1C3
27b. 7 in N5 locked for C6

28. 18(5) in R3C6 = {12348/12357}: ({12456} blocked by R45C5): no 6; 1 locked within R34C6 -->> locked for C6

29. 45 on N1: R12C3 + R3C1 = 18 = {69}[3]/{89}[1] -->> 9 locked in R12C3 for N1 and 30(4) cage at R1C3
29a. Naked Pair {67} in R12C4 -->> locked for N2, C4 and 20(4) at R1C3
29b. Naked Pair {89} in R12C3 -->> locked for C3
29c. R12C3 = {89} -->> R3C1 = 1(step 29)
29d. R3C6 = 5; R3C9 = 2; R3C4 = 3; R4C6 = 1(hidden)

30. 18(5) at R3C6 = 15{237} -->> R4C7 = 7
30a. 20(4) at R3C4 = 3{458}: no 2,9 -->> R4C3 = 5; R45C4 = {48} -->> locked for N5 and C4
30b. R6C4 = 9

31. R78C7 = {69} -->> locked for C7 and N9
31a. Clean up: R35C7 = [82](step 17); R7C9 = 7 and R9C7 = 4(step 13); R8C3: no 3
31b. R6C7 = 3; R45C5 = [23]; R5C8 = 1(hidden); R5C9 = 5(hidden)
31c. R5C2 = 9(hidden); R7C3 = 3(hidden); R8C4 = 7; R9C5 = 7(hidden)
31d. R5C4 = 8(hidden); R4C4 = 4

32. 17(3) at R2C8 = 8{36} -->> R23C8 = [36]
And the rest is all singles

greetings

Para


Last edited by Ed on Wed Jul 16, 2008 10:38 am, edited 1 time in total.

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Ed
PostPosted: Fri Jun 27, 2008 11:57 pm 
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Location: Sydney, Australia
Assassin 61X by mhparker (July 07)
Puzzle pic: 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4608:4608:4610:4610:3076:5381:5381:4615:4615:4608:3338:4610:4610:3076:5381:5381:3088:4615:5394:3338:3338:4885:3076:5399:3088:3088:3354:5394:5394:4885:4885:5399:5399:5399:3354:3354:5412:5394:5394:6183:6183:6183:6698:6698:3354:5412:5412:3631:3631:3631:3634:3634:6698:6698:5412:5431:3384:3631:3898:3634:2364:3901:6698:831:5431:3384:4162:3898:4164:2364:3901:3143:831:5431:4162:4162:3898:4164:4164:3901:3143:
Solution: 
+-------+-------+-------+
| 8 1 7 | 4 3 2 | 9 6 5 |
| 9 3 2 | 5 1 6 | 4 8 7 |
| 5 4 6 | 9 8 7 | 1 3 2 |
+-------+-------+-------+
| 4 9 8 | 2 5 3 | 6 7 1 |
| 6 2 1 | 7 9 8 | 5 4 3 |
| 7 5 3 | 6 4 1 | 8 2 9 |
+-------+-------+-------+
| 3 8 4 | 1 2 5 | 7 9 6 |
| 1 7 9 | 3 6 4 | 2 5 8 |
| 2 6 5 | 8 7 9 | 3 1 4 |
+-------+-------+-------+
Quote:
mhparker, lead-in: (Est. difficulty: r1.75 level
CathyW: I found it easier than the A61
Andrew: I found Mike's A61X a real struggle.. 1.75 ... 1.5... ratings look right
Forum 2021 Revisit here
Ratings discussion by mhparker & Para:
About A61X mhparker wrote:
(Est. difficulty: V1.75 level)
Para: When did the number behind the V become an estimation of difficulty instead of just a second harder version? I mean V2 was just second version right?

mhparker: It was Ruud, wasn't it, who first released a "V1.5" instead of calling it a "V2" (A47 version 1.5). The obvious implication at the time was that the puzzle wasn't quite up to the difficulty level of a traditional V2. This was backed up recently by Andrew on the Assassin 58 thread:
Andrew wrote:
It took me several hours to solve this one so I feel it was hard enough for a V1. In my opinion V1 Assassins shouldn't become so hard that they drive away solvers like myself. Real evil is fine for V2 and V2+. V1.5 should be somewhere between those levels.
I intend to publish an estimated difficulty level with all of my own puzzles in the future. I wanted to make a post on this, but haven't got round to it yet. The idea was to use a numeric rating scheme based around the traditional "V1" and "V2" difficulty levels, which would have the numeric ratings 1.0 and 2.0, respectively. I would then just use the numeric rating "1.75", instead of the term "V1.75" (which is maybe confusing).

Looks like I'd better hurry up and make that other post!

mhparker: Hi folks,

As mentioned above, here's my suggestion for an informal numeric rating scheme. I'll be using this for any future puzzles of mine, unless we agree to use something different. All this shouldn't become "cast in stone", but it's useful to have some way (as a puzzle setter) of communicating the estimated difficulty of a puzzle.

Any feedback and suggestions welcome.

Quote:
0.5: Typical newspaper "Deadly". Intended to be done using only limited pencilmarks.

0.75: Easy Assassin, like some of the very early ones, such as A1. Rarely seen now: all recent Assassins would have at least a 1.0 rating on this scale.

1.0: "Average" V1 Assassin (looking back over a longer period of time). Something like A57, perhaps.

1.25: Harder Assassin. Actually, most recent Assassins seem to have become more difficult than they traditionally used to be. So a rating of "1.25" would be considered the norm now. Typical example: A59.

1.5: Hard Assassin, having a significantly longer and/or narrower solution path, and/or requiring more advanced techniques. The A60 was definitely one of these.

1.75: Very hard Assassin, but still not hard enough to require a team effort to solve. Does not require any hypotheticals. The A60RP-Lite could maybe deserve such a rating.

2.0: Traditional "V2" standard, typically requiring a team effort and maybe (but not necessarily) involving limited use of hypotheticals. Example: A55V2.

2.5: Requires a team effort and several short to medium length hypotheticals. The TJK18 and A48-Hevvie would probably fall into this category.

3.0: "Ruudiculous", requiring a team effort and massive hypotheticals to solve, if it can be solved at all. The A50V2 and (possibly) A60RP could be considered examples of this.

4.0: Puzzles with unique solutions, but which can only realistically be solved by computer programs using backtracking (brute force).
P.S. The puzzle I posted above (A61X) could possibly be rated 1.5 instead of 1.75 on this scale. Difficult to tell. Better to just try it and let me know! :wink:

Para: Wasn't Ruud working on some difficulty estimation for Killers as well? I remember him mentioning it once.

mhparker: True, but I suspect it's extremely difficult (if not impossible) to come up with a computer-generated rating that accurately reflects how easy a puzzle is for a human to solve. Such an approach would probably work well up to a point (i.e., up to a certain difficulty level), but rapidly break down when the program approaches its (solving) limits. This "breakdown point" would no doubt occur too early to reliably rate Assassins, which would often (or even usually) end up being "off the scale".

Also, computer programs don't tend to combine logical steps into compound moves like humans do, and don't usually contain the kind of Artificial Intelligence (AI) needed to distinguish the hypotheticals that humans can relatively easily detect from brute force T&E. Furthermore, computer-generated ratings tend to ignore an important difficulty factor: namely how narrow the solution path is.

In the end, I expect we would be no better off with computer-generated ratings than with a relative, subjective rating scheme as proposed above, where puzzles are assigned to relatively small number of difficulty "bands".
Discussion on hypotheticals, contradiction and T&E by Andrew, mhparker & CathyW:
Andrew: Since both hypotheticals and T&E have been mentioned in the above discussion, plus there are also contradiction moves, I was wondering what the difference is between these three categories.

Let me have a try at answering that myself.

A contradiction move is normally of the type If r.c. = . followed by a short chain to show that r.c. can't have that value. That is what I did for R3C7 + R7C3 in Assassin 60 to eliminate [22] and leave the remaining options {13} which I then solved by direct steps.

Hypotheticals are when one looks at the range of values for a cell, or possibly combos for a cage, works through a short chain and eliminates some candidates from other cells in the chain and possibly also from the original cell.

T&E is more like bifurcation where one tries the values for a cell and finds that all except one lead to an impossible position. That was what I did by accident in Assassin 24 where I started as a hypothetical and it just happened that one candidate led to an impossible position; BTW it did also work as a hypothetical for a different cell.

mhparker: It turned out to be exactly thís move (CathyW step 22) that shortcut the puzzle (A61X). The problem is that current solvers (even the latest JSudoku!) don't detect such moves, no doubt considering them to be hypotheticals (which, in turn, are assumed to be synonymous with limited trial and error (T&E)).

However, it's clear in this case that whilst this neat move is indeed strictly speaking a hypothetical, it is definitely not T&E, but (instead) perfectly sound human logic.

Clearly hypotheticals therefore need to be formalized and the boundaries of T&E precisely defined. This has opened up a whole new avenue of research I had not been considering up to now. For that reason alone, this puzzle has already proved to be more than worthwhile.

Many thanks for providing this insight. :D

CathyW: imho:

Hypothetical = logical "if ... then ... so ..." statements leading to elimination of candidate(s) and/or placement
Bifurcation/Trifurcation included in this provided each option is pursued to logical conclusion to prove conflict or validity.

So where does one draw the line for 'unacceptable' T&E?

mhparker: I'll take your points in reverse order.
Cathy wrote:
So where does one draw the line for 'unacceptable' T&E?
Speaking for myself, T&E is only acceptable if there is no other obvious way to proceed with a puzzle. That happens a lot around here! :)

However, what I was trying to say is not that T&E is acceptable, but rather that some very simple hypotheticals like this one should not be considered to be T&E at all. My reasoning is that humans are very quickly able to spot a promising-looking pattern. If the chains are very short and the options at this position are so few that they can be enumerated on the fingers of one hand, then I don't see how this has anything more to do with T&E than, say, Aligned Pair Exclusion (APE) does.
Cathy wrote:
Hypothetical = logical "if ... then ... so ..." statements leading to elimination of candidate(s) and/or placement
Bifurcation/Trifurcation included in this provided each option is pursued to logical conclusion to prove conflict or validity.
Yes, you're spot on there! Many people immediately raise the question as to why XY-Chains, Nice Loops, and so on, aren't also considered to be hypotheticals. After all, they're also a case of "if ... then ... so ...", aren't they? However, it's exactly the bifurcative aspect of hypotheticals you mention that is their major distinguishing feature.

The basic rule for a single chain or loop is: if you can identify the various (strong, weak, etc.) links in advance, construct the chain or loop from them, and come to some productive conclusion (candidate/combo elimination, etc.) without touching (i.e., altering) the state of the grid, then it's not a hypothetical.

If, on the other hand, one or more of the links in the chain depends on some side-effect (e.g., candidate elimination) of an earlier link in the chain, then one is dealing with a hypothetical. For example, consider the following x-cycle on the digit 'x' (in verbose notation, where "=>" and "->" imply strong and weak links, respectively):

A<>x => B=x -> ... -> E<>x => F=x

If (in this example) the strong link between E and F didn't exist at the start of the chain, but only arose dynamically due to the eliminations on x caused by the premise "B=x" at the end of the first link, then it's a hypothetical. The implication here is that one is building up the chain as one goes along, re-evaluating the situation at each step, creating new links dynamically as they become available. This "try first, evaluate later" approach explains why hypotheticals are generally considered to be a form of trial and error. The bifurcation you mention comes into play because a computer program has to bifurcate (i.e., fork a copy of the grid) in order to maintain the side-effects, such that they can be undone (e.g., by restoring the original grid) at the end of the sequence.

Bifurcation is also required in order to record and compare the effects of assuming each of a set of initial "possibilities" (candidates in a cell, combos in a cage, etc.) in turn. This is the case that Andrew mentioned above, which corresponds closely to forcing chains.

The "rule of thumb" I use is:

If bifurcation is required, then it's a hypothetical.
A61X Walkthrough by CathyW:
Here's my WT for the A61X. Do let me know about any typos!

Preliminaries

a) 3(2) r89c1 = {12} not elsewhere in N9/c1

b) 24(3) r5c456 = {789} not elsewhere in N5/r5

c) 13(4) r345c9+r4c8 must have 1 -> r6c9 <> 1


1. Innies N7: r7c1 + r9c3 = 8 = {35} not elsewhere in N7
-> r78c3 <> 8 -> 8 locked to r789c2, not elsewhere in c2

2. Innies N9: r7c9 + r9c7 = 9 = {18/27/36/45}

3. Outies c12: r35c3 = 7 = {16/25/34}

4. Outies c89: r35c7 = 6 = {15/24}

5. Innies r123: r3c1469 = 23

6. Innies r789: r7c1469 = 15 = 3{129/147/156/246}; 5{127/136} (no 8)
-> r9c7 <> 1

7. Outies r1234: r5c239 = 6 = {123} -> r3c3 <> 1,2,3
-> r5c18 = (456), r5c7 = (45) -> r3c7 = (12)
-> NP {12} not elsewhere on diagonal /
-> r9c7 <> 2 -> r7c9 <> 7
-> 9(2) r78c7 <> {45}

8. 16(3) r9c3+r89c4 = 3{49/58/67}; 5{29/38/47} -> r89c4 <> 1

9. Innies N1: r12c3+r3c1 = 14

10. Innies N3: r12c7+r3c9 = 15

11. Innies c5: r456c5 = 18 (only one of 7,8,9) = {567/468/369/459}
-> HS r6c6 = 1 -> r6c7, r7c6 <> 2,3; no 8 in r7c6 -> r6c7 <> 5
-> HS r4c4 = 2 -> r3c4, r4c3 = {89}
Clean up r8c7 <> 7,8

12. 14(4) r6c345+r7c4: min from r6c45 is 7 -> r6c3+r7c4 is max 7
-> r6c3 <> 7,8; r7c4 <> 7
14(4) options: {1256/1346/2345} -> r7c4 <> 6

13. Outies - Innies r12: r2c28 - r3c5 = 3
Min from r2c28 is 7 -> r3c5 = (4…9)
-> r12c5 <> 8,9

14. O-I N8: r9c37 - r7c46 = 2

15. Outies c1: r1456c2+r5c3 = 18 (no 8); 1,2,3 can be repeated between r1c2 and r5c3

16. Outies c9: r1456c8+r5c7 = 24; 4,5 can be repeated between r1c8 and r5c7

17. 13(3) N1: r3c3 min 4 -> r23c2 max 9 -> r23c2 <> 9
Options: 4[72]/{36}; 5[71/62]; 6[52]/{34} -> r3c2 <> 7

18. Pointing cells: r7c9 sees all with 2 in N6 -> r7c9 <> 2 -> r9c7 <> 7
-> split 15(4) r7c1469 now 3{147/156}; 5{136} -> r7c6 <> 9 -> r6c7 <> 4
Must have 1 -> r7c58 <> 1
Must have 3 -> r7c578 <> 3 -> r8c7 <> 6
If 3{147}, r7c6 = 7 -> r7c6 <> 4 -> r6c7 <> 9

19. 26(5) N69: {14579/14678/24569/34568}
Must have 4 -> r4c9 <> 4 since sees all cells of 26(5)

20. Elimination chain:
a) r3c4 = 9 -> r12c6, r3c8 <> 9
b) r3c4 = 8 -> r4c3 = 9 -> (r4c7 <> 9) 9 locked to r12c7 -> r12c6, r3c8 <> 9
Either case r12c6, r3c8 <> 9

21. 16(3) r9c3+r89c4 = 3{49/58/67}; 5{38/47}
Must have one of 7,8,9 within r89c4 -> Killer triple with r35c4 -> r12c4 <> 7,8,9
-> 9 locked to r3c456 -> r3c1 <> 9

22. a) if r7c6 = 5, r6c7 = 8
b) if r7c6 = 6, r6c7 = 7 -> r7c7 = 8
c) if r7c6 = 7, r6c7 = 6 -> r7c7 = 8
-> 8 locked to r67c7 not elsewhere in c7 -> r7c9 <> 1
-> 8 locked to r6c789 -> r6c1 <> 8

23. 21(4) r567c1+r5c2 = {3459/3567} -> r6c2 <> 2
Must have 3 and 5, r4c1 sees all cells of 21(4) -> r4c1 <> 3,5

24. HS r7c4 = 1
-> r6c345 = {256/346} 6 not elsewhere in r6 -> r7c6 <> 7
-> split 15(4) r7c1469 = {1356} -> r8c7 <> 3, r9c7 <> 5
-> NP {12} r38c7, not elsewhere in c7
-> NP {12} r8c17, not elsewhere in r8 -> r8c3 <> 7
-> NP {78} r67c7, r124c7 <> 7

25. 1 locked to 13(4) within N6 -> r3c9 <> 1

26. 4 locked to 26(5) within N6 -> r4c78 <> 4

27. 26(5) N69 = {24569/34568} Must have 4,5,6
6 in either r5c8 or r7c9 -> r4c9, r89c8 see both therefore <> 6
split 9(2) N9 = {36} -> r89c89 <> 3 -> r89c9 <> 9
-> 15(3) r789c8 = {159/249}, 9 not elsewhere in c8
-> r9c8 <> 5
(Andrew commented that perhaps additional explanation required for this step: Elimination chain r7c9 = 5 -> r5c7 = 4 clashes with r9c7 = 4.)

28. 3 locked to r8c456 -> r9c456 <> 3

29. 26(5) must have 4,5,6; 5 now within 26(5) within N6 -> r4c789 <> 5

30. 13(4) r345c9+r4c8 = {1237/1246} -> r3c9 <> 5
-> 2 locked to r35c9 -> r26c9 <> 2

31. 3 locked to r1c1, r2c2 on diagonal \, not elsewhere in N1

32. 6 locked to r1c1, r2c2, r3c3 on diagonal \, not elsewhere in N1

33. Pointing cells: 7 locked to r1234c8 -> r3c9 <> 7

34. Split 15(3) r12c7+r3c9 = {249/456} 4 not elsewhere in N3

35. 13(4) r345c9+r4c8:
a) if {1237} -> r3c9 = 2
b) if {1246} -> r3c9 = 4
-> r3c9 <> 6

36. 12(3) N3 = {138/156/237} -> r3c8 <> 1,2

37. 13(3) r23c2+r3c3 = [724/715/625/526/346]
-> r2c2 <> 4, r3c2 <> 5
-> NT {124} r3c279, not elsewhere in r3 -> r5c3 <> 3; r12c5 <> 7
-> 13(3) N1 = [715/625/526/346]

38. 12(3) r123c5 = {12}9/{13}8/{14/23}7/15/24}6/{16}5 ({345} blocked by r46c5}

39. Split 14(3) N1: {158/248/257}
-> r12c3 <> 9
-> 18(3) N1 = 9{18/27}; {936} not possible as 3,6 only in r1c1, {945} blocked by split 14(3)
-> r12c1 = (789); r1c2 = (12)
-> 9 locked to r12c1, not elsewhere in c1
-> {248} blocked for split 14(3) -> r12c3 <> 4
-> HS r3c2 = 4 -> r3c9 = 2, r3c7 = 1
-> r8c7 = 2, r7c7 = 7, r6c7 = 8, r9c1 = 2, r8c1 = 1, r7c6 = 5, r7c1 = 3, r7c9 = 6 …

Straightforward combos and singles from here

:D
Alternate ending to CathyW Walkthrough by mhparker:
Hi folks,

Here is an alternative partial WT for the A61X starting from Cathy's step 21:

After cathy's step 21:

22. Innies n9 cannot be {45} due to r5c7 (common peer)
22a. -> no 4,5 in r7c9+r9c7

23. Hidden killer pair on {45} in n9 ({45} locked within 15(3)n9 and 12(2)n9)
23a. 15(3) cannot contain both of {45} (i.e., cannot be {456}) due to r5c8
23b. -> 15(3)n9 and 12(2)n9 must each contain exactly 1 of {45}
23c. -> 12(2)n9 = {48/57} (no 3,9)

24. 9 in n9 locked in 15(3) = {159/249} (no 3,6,7,8)
24a. 1 only in r9c8 -> no 5 in r9c8
24b. 9 locked for c8
24c. Cleanup: no 1 in r3c8

25. 3 in d\ locked in n1 -> not elsewhere in n1

26. Innie/outie diff. n1: r12c4 = r3c1 + 4
26a. min. r3c1 = 4 -> min. r12c4 = 8
26b. -> no 1 in r12c4
26c. max. r12c4 = 11 -> max. r3c1 = 7
26d. -> no 8 in r3c1

27. Hidden single (HS) in c4 at r7c4 = 1
27a. Cleanup: no 8 in r9c7 (innies n9)

28. Naked pair (NP) on {36} at r7c9+r9c7 -> no 3,6 elsewhere in n9

29. NP on {12} at r38c7 -> no 1,2 elsewhere in c7

30. NP on {12} at r8c17 -> no 2 elsewhere in r8

31. 3 in r8 locked in n8 -> not elsewhere in n8

32. 6 in d\ locked in n1 -> not elsewhere in n1

33. Split 13(3) at r6c345 = {256/346}
33a. 2 only in r6c3 -> no 5 in r6c3
33b. 6 locked in r6c345 for r6
33c. Cleanup: no 7 in r7c6

34. NP on {78} at r67c7 -> no 7,8 elsewhere in c7

35. Naked triple (NT) on {356} at r7c169 -> no 5,6 elsewhere in r7
35a. Cleanup: no 7 in r8c3

36. Hidden killer triple on {789} in c3 at r12478c3
36a. -> r12c3 must contain 1 of {789}
36b. -> min. r12c3 = 8 -> max. r12c4 = 10 -> max. r3c1 = 6 (step 26)
36c. -> no 7 in r3c1

37. 13(3)n1 = {157/247/256/346} = {(4/5)..}
37a. -> r3c1 and 13(3)n1 form killer pair on {45} -> no 4,5 elsewhere in n1

38. 18(3)n1 = {189/279/378} (no 6)
38a. (Note: {369} unplaceable, since {36} only in r1c1)
38b. {12} only in r1c2
38c. -> no 9 in r1c2

39. 6 in n1 and d\ locked in 13(3)n1 = {256/346) (no 1,7)
39a. 2 only in r3c2
39b. -> no 5 in r3c2
39c. 3 only in r2c2
39d. -> no 4 in r2c2

40. 4 in n1 locked in r3 -> not elsewhere in r3

41. 1 in n6 locked in 13(4)n36 -> no 1 in r3c9

42. HS in r3 at r3c7 = 1

This is the first of a run of naked/hidden singles that continues until the end of the puzzle.
Walkthrough by Andrew:
I found Mike's A61X a real struggle. I was stuck and left this one for a few days while I worked on other puzzles and walkthroughs. When I returned my next move, step 57, proved to be my breakthrough. Now that I've finished it and worked through Cathy's walkthough I can see where I missed key moves.

Cathy and I reached almost similar positions before her step 22 (my step 41) but using slightly different moves to get there. I also had a couple of extra 45s which locked 3 to two cells on D/ but that doesn't seem to be significant for the solving path.

Soon after that I started struggling. I missed some points about the 26(5) cage, see comments below, which probably weren't significant at the time. I think the key things I missed were the eliminations in N9 that allowed the locked 3 and 6 on D\ in N1. Cathy also used the split cage in N1 which I didn't use; I don't know why because I did use the similar split cage in N3. N9 and N1 were clearly the key areas for the final stages of this puzzle.

Thanks Mike for posting your partial walkthrough showing the intended solving path if Cathy's hypothetical hadn't provided a shortcut.

Mike's steps 22 and 23 appear to be the key ones; the equivalent of step 22 was also in Cathy's walkthrough although expressed differently. I must admit I'm still finding it hard to spot common peer eliminations although I did find a couple in this puzzle. The one in Mike's step 22 is definitely a harder one to find. Then there's the hidden killer pair in step 23. We've been getting those moves more recently but this one is harder to see because it also depends on one combination in R789C8 not being available.

Mike estimated this puzzle as 1.75 when he posted the puzzle and Ed has it currently rated as 1.5. Having seen both Cathy's and Mike's walkthroughs those ratings look right.

In case anyone is interested, here is how I solved A61X.

1. R78C3 = {49/58/67}, no 1,2,3

2. R78C7 = {18/27/36/45}, no 9

3. R89C1 = {12}, locked for C1 and N7

4. R89C9 = {39/48/57}, no 1,2,6

5. 19(3) at R3C4 = {289/379/469/478/568}, no 1

6. R5C456 = {789}, locked for R5 and N5
6a. Max R4C4 = 6 -> min R3C4 + R4C3 = 13, no 2,3

7. R789C2 = {489/579/678}, no 3

8. 13(4) at R3C9 = {1237/1246/1345}, no 8,9

9. 14(4) at R6C3 = {1238/1247/1256/1346/2345}, no 9

10. 3 in N7 only in R7C1 + R9C3
10a. 45 rule on N7 2 innies R7C1 + R9C3 = 8 = {35}, locked for N7, clean-up: no 8 in R78C3
10b. R789C2 = {489/678} = 8{49/67}, 8 locked for C2
10c. Max R9C3 = 5 -> min R89C4 = 11, no 1

11. 45 rule on N9 2 innies R7C9 + R9C7 = 9 = {18/27/36/45}, no 9

12. 45 rule on C1234 1 innie R5C4 – 3 = 1 outie R6C5 -> R6C5 = {456}

13. 45 rule on C6789 1 innie R5C6 – 3 = 1 outie R4C5 -> R4C5 = {456}

14. 45 rule on C12 2 outies R35C3 = 7 = {16/25/34}, no 7,8,9

15. 45 rule on C89 2 outies R35C7 = 6 = {15/24}
15a. R78C7 (step 2) = {18/27/36} (cannot be {45} which clashes with R35C7)

16. 45 rule on R1234 3 outies R5C239 = 6 = {123}, locked for R5, clean-up: no 1,2,3 in R3C3 (step 14), no 4,5 in R3C7 (step 15)
16a. Min R3C3 = 4 -> max R2C23 = 9, no 9

17. 14(4) cage at R6C3 = {1247/1256/1346/2345} (cannot be {1238} because R6C5 only contains 4,5,6), no 8

18. 21(4) cage at R5C1 has R5C1 = {456}, R7C1 = {35}, valid combinations {1569/2568/3459/3468/3567}

19. 45 rule on R789 4 innies R7C1469 = 15 with R7C1 = {35}, valid combinations {1239/1257/1347/1356/2346}, no 8, clean-up: no 1 in R9C7 (step 11)

20. Naked pair {12} in R3C7 + R9C1, locked for D/ -> no 2 in R9C7, clean-up: no 7 in R7C9 (step 11)
[This has been there since step 16. Only just spotted it.]

21. 12(3) cage in N3 with R3C7 = {12} = {129/138/147/156/237/246}
21a. 12(3) cage at R2C8, 1,2 only in R3C78 -> no 9 in R3C8

22. R6C6 = 1 (hidden single in N5), locked for D\, clean-up: no 8 in R8C7
22a. R6C7 + R7C6 = {49}/{67}/[85], no 2,3, no 5 in R6C7

23. R4C4 = 2 (hidden single in N5), locked for D\, clean-up: no 7 in R8C7
23a. R3C4 + R4C3 = 17 = {89}

24. 3 in N5 locked in R4C6 + R6C4, locked for D/
24a. 12(3) cage in N3 (step 21) = 12(3) cage in N3 with R3C7 = {12} = {129/138/147/156/237/246}
24b. 8 of {128} must be in R2C8 -> no 8 in R3C8

25. 16(3) cage at R8C4 = {349/358/367/457} -> R89C4 must contain 7/8/9
25a. Killer triple 7,8,9 in R3C4, R5C4 and R89C4, locked for C4

26. 14(4) at R6C3 (step 17) = {1256/1346/2345} (cannot be {1247} because 2,7 only in R6C3), no 7
26a. {1256/1346} must have 1 in R7C4 -> no 6 in R7C4
26b. {1256/2345} must have 2 in R6C3 -> no 5 in R6C3

27. 45 rule on C5 3 innies R456C5 = 18 = {459/468/567}

28. R123C5 = {129/138/147/237} (cannot be {156/246/345} which clash with R456C5), no 5,6

29. R789C5 = {159/168/249/267/348/357} (cannot be {258/456} which clash with R456C5)

30. R123C5 (step 27) = {129/138/237} (cannot be {147} which clashes with all combinations for R789C5 in step 29), no 4

31. 13(3) cage at R2C2 min R2C2 + R3C3 = 7 -> max R3C2 = 6

32. R789C8 = {159/168/249/258/267/348/357} (cannot be {456} which clashes with R5C8)

33. R35C7 = 6 (step 15), 9 in C7 locked in R1246C7
33a. 45 rule on C7 5 innies R12469C7 = 30 = 9{1578/2478/3468/3567} (cannot be {25689} which clashes with R35C7)

34. CPE 2 in R7C9 can see every 2 in N6 -> no 2 in R7C9, clean-up: no 7 in R9C7 (step 11)

35. 26(5) cage at R5C7 must have R5C7 = {45}, R5C8 = {456} and R7C9 = {13456}, valid combinations {14579/14678/24569/34568}
[I missed the fact that 4 is now locked in the 26(5) cage leading to a CPE elimination.]

36. R4C3 = {89}, R78C3 must contain 7/9 -> 18(4) cage at R1C3 must contain 7/8/9 in R12C3 with no other 7/8/9 in the cage, valid combinations {1269/1359/1368/1458/1467/2349/2358/2367/2457}

37. 45 rule on N1 2 outies R12C4 – 4 = 1 innie R3C1, max R12C4 = 11 -> max R3C1 = 7

38. 8,9 in R3 locked in R3C456, locked for N2

39. 45 rule on N89 3 innies R7C469 – 7 = 1 outie R9C3, R9C3 = {35} -> R7C469 = 10,12 = {136/145/147/156/345}, no 9, clean-up: no 4 in R6C7 (step 22a)

40. R7C1469 (step 19) = {1347/1356} = 13{47/56}, 1,3 locked for R7, clean-up: no 6 in R8C7
40a. 7 only in R7C6 -> no 4 in R7C6, clean-up: no 9 in R6C7

41. R7C6 = {567}
41a. If R7C6 = 5 -> R6C7 = 8
41b. If R7C6 = 6 -> R6C7 = 7 -> R7C7 = 8
41c. If R7C6 = 7 -> R6C7 = 6 -> R7C7 = 8
41d. 8 in C7 locked in R67C7, locked for C7, clean-up: no 1 in R7C9 (step 11)
41e. 8 in N6 locked in R6C789, locked for R6

42. R7C4 = 1 (hidden single in R7)
42a. 14(4) at R6C3 (step 26) = 1{256/346} = 16{25/34}, 6 locked for R6, clean-up: no 7 in R7C6 (step 22a)
42b. R789C5 (step 29) = {249/267/348/357}

43. 1 in C5 locked in R123C5 (step 30) = 1{29/38}, no 7
43a. 8,9 only in R3C5 -> no 1,2,3 in R3C5
43b. Naked pair {89} in R3C45, locked for R3
43c. 7 in N2 locked in R123C6, locked for C6, clean-up: no 4 in R4C5 (step 13)

44. 21(4) cage at R5C1 (step 18) = {3459/3567} = 35{49/67}, no 2
44a. CPE 3,5 in R4C1 can see every 3,5 in 21(4) cage -> no 3,5 in R4C1

45. 26(5) cage at R5C7 (step 35) = {24569/34568} = 456{29/38}, no 7
[Now we have 4,5,6 locked but I still missed the CPE elimination.]

46. R789C8 (step 32) = {159/168/249/267} (cannot be {348/357} which clash with R7C9 + R9C7, cannot be {258} -> R7C9 + R9C7 = {36} and there are no remaining combinations in R78C7 and R89C9), no 3
46a. 1 only in R9C8 -> no 5,8 in R9C8

47. 45 rule on N78 2 remaining innies R7C16 – 5 = 1 outie R9C7, R7C16 = [35/36/56] = 8,9,11 -> R9C7 = {346}, clean-up: no 4 in R7C9 (step 11)

48. Naked triple {356} in R7C169, locked for R7, clean-up: no 7 in R8C3, no 3 in R8C7

49. Naked pair {12} in R8C17, locked for R8

50. Naked quad (actually two naked pairs) {1278} in R3678C7, locked for C7
50a. R12469C7 (step 33a) = 9{3468/3567} = 369{48/57}

51. 1 in N6 locked in R4C89 + R5C9 -> no 1 in R3C9

52. 45 rule on N3 3 innies R12C7 + R3C9 = 15 = {249/357/456} (cannot be {267} because 2,7 only in R3C9)
52a. 7 only in R3C9 -> no 3 in R3C9

53. 16(3) cage at R8C6 = {268/349/358} (cannot be {259} because R9C7 only contains 3,4,6)
53a. 6 of {268} must be in R9C7 -> no 6 in R89C6
53b. 2 only in R9C6 -> no 8 in R9C6
53c. 8 only in R8C6 -> no 5 in R8C6

54. 21(4) cage at R3C6 = {2469/3459/3567} (cannot be {2379} because R4C5 only contains 5,6)
54a. Only combination with both 5,6 is {3567} when 7 must be in R3C6 -> no 5,6 in R3C6
54b. 9 in {2469/3459} must be in R4C7 -> no 4 in R4C7
54c. 21(4) cage at R3C6 must contain either 7 in R3C6 or 9 in R4C7 -> 21(4) cage at R1C6 must contain 7/9, valid combinations {2469/3459/3567}

55. 18(4) cage at R1C3 (step 36) = {1359/1368/1458/1467/2349/2358/2367/2457} (cannot be {1269} because 1,2,9 only in R12C4)
55a. R12C4 = {3456} -> R12C3 = {17/18/19/27/28/29}, no 3,4,5,6
55b. R12C3 contains 1/2 -> R13C2 must contain 1/2, R45C2 must contain 1/2 and R56C3 must contain 1/2

56. 13(3) cage at R2C2 = {157/247/256/346}
56a. 1,2 only in R3C2 -> no 5 in R3C2

57. R12C4 cannot be {56} (no valid combination for 18(4) cage at R1C3) -> R12C6 must contain 5/6 (cannot be both as that would clash with R7C6)
57a. Killer pair 5,6 in R12C6 and R7C6, locked for C6
57b. R12C6 contains 5/6 -> R12C4 must contain 5/6 -> 18(4) cage at R1C3 cannot be {2349}

58. 16(3) cage at R8C6 (step 53) = {268/349}
58a. R89C6 = {28/39/49}
58b. R789C5 (step 42b) = {267/357} (cannot be {249/348} which clash with R89C6) = 7{26/35}, no 4,8,9, 7 locked for C5

59. R5C4 = 7 (hidden single in C4)

60. Killer pair 5,6 in R7C6 and R89C5, locked for N8

61. Killer pair 5,6 in R4C5 and R89C5, locked for C5 -> R6C5 = 4, R4C6 = 3
[Alternatively R6C5 = 4 (hidden single in C5)]
61a. 14(4) at R6C3 (step 42a) = {1346} (only remaining combination) -> R6C34 = [36], 6 locked for D/
61b. R4C5 = 5, R9C3 = 5, R7C1 = 3, clean-up: no 7 in R8C9, no 6 in R9C7 (step 11)

62. 16(3) cage at R8C6 (step 58) = {349} (only remaining combination) -> R9C7 = 3, R7C9 = 6 (step 11), R7C6 = 5, R6C7 = 8 (cage sum), R78C7 = [72], 7 locked for D\, R3C7 = 1, locked for D/ -> R89C1 = [12], R7C5 = 2 (hidden single in R7), clean-up: no 5 in R8C9, no 9 in R89C9

63. Naked pair {49} in R89C6, locked for C6 and N8

and the rest is straightforward, remembering to make eliminations along the diagonals
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Ed
PostPosted: Fri Jun 27, 2008 11:59 pm 
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Posts: 1044
Location: Sydney, Australia
Assassin 62 by Ruud (Aug 07)
Puzzle pic:
Attachment:
a62.JPG
a62.JPG [ 96.47 KiB | Viewed 18301 times ]
Code: Select, Copy & Paste into solver:
3x3::k:4096:4096:4610:3587:3587:3589:3589:3847:3847:4096:4096:4610:4610:3587:2830:3589:6672:3847:5394:5394:5394:4610:5910:2830:6672:6672:1562:5394:4380:5910:5910:5910:4640:4640:6672:1562:4900:4380:4380:2087:4640:4640:5162:5162:1562:4900:4654:2087:2087:4401:4401:4401:5162:4917:4900:4654:4654:4409:4401:4667:4917:4917:4917:4671:4654:3393:4409:3907:4667:4667:6214:6214:4671:4671:3393:3393:3907:3907:4667:6214:6214:
Solution: 
+-------+-------+-------+
| 4 3 5 | 6 7 8 | 1 2 9 |
| 2 7 6 | 3 1 9 | 5 8 4 |
| 1 8 9 | 4 5 2 | 7 6 3 |
+-------+-------+-------+
| 3 4 2 | 7 9 6 | 8 5 1 |
| 8 6 7 | 5 3 1 | 9 4 2 |
| 5 9 1 | 2 8 4 | 3 7 6 |
+-------+-------+-------+
| 6 5 3 | 9 2 7 | 4 1 8 |
| 7 1 4 | 8 6 3 | 2 9 5 |
| 9 2 8 | 1 4 5 | 6 3 7 |
+-------+-------+-------+
Quote:
Para: Somewhere between 0.5-0.75 probably
Walkthrough by Para:
Hi all

This one was very basic. Nothing special to solve it. It was a relaxing puzzle.

Walk-through Assassin 62

1. R23C6 = {29/38/47/56}: no 1

2. 26(4) at R2C8 = {2789/3689/4589/4679/5678}: no 1

3. 6(3) at R3C9 = {123} -->> locked for C9

4. 8(3) at R5C4 = {125/134}: no 6,7,8,9; 1 locked in 8(3) cage -->> R6C56: no 1

5. 20(3) at R5C7 = {389/479/569/578}: no 1,2

6. R78C4 = {89} -->> locked for C4 and N8

7. 45 on R89: 2 innies and 1 outie: R8C24 = R7C6 + 2: Min R8C24 = 9 -->> Min R7C6 = 7 -->> R7C6 = 7; R8C24 = [18]
7a. R7C4 = 9
7b. Clean up: R23C6: no 4

8. 15(3) at R8C5 = {456} -->> locked for N8

9. 45 on N9: 2 outies: R6C9 + R8C6 = 9 = [63/72] -->> R6C9 = {67}

10. 18(4) at R7C6 = 7{236/245} ({128} blocked because {18} only in R9C7, {146} blocked by R8C6): no 1,8,9

11. 7 and 9 in N9 locked within 24(4) cage at R8C8 -->> 24(4) = {2679/3579}: no 1,4,8
11a. R9C4 = 1(hidden)
11b. R6C3 = 1(only place within 8(3) cage(step 4))

12. 1 and 8 within N9 locked in 19(4) cage at R6C9 -->> 19(4) = {1378/1468}: no 2,5; 8 locked for R7
12a. 19(4) needs one of {67} which has to go in R6C9 -->> R7C789: no 6
12b. 5 and 6 in R7 locked for N7

13. 45 on N7: 3 innies: R7C123 = 14 = {356}(5 and 6 locked in these cells) -->> locked for N7 and R7
13a. R7C5 = 2; R8C6 = 3; R6C9 = 6(step 9); R9C8 = 3(hidden)
13b. Clean up: R23C6: no 8; R89C7 = {26}(step 10) -->> locked for C7 and N9
13c. R89C3 = [48](last possible combination within 13(3) at R8C3)

14. 45 on C12: 3 outies: R357C3 = 19 = {379} -->> R7C3 = 3; R35C3 = {79} -->> locked for R3

15. 45 on N1: 3 outies: R23C4 + R4C1 = 10 = {26}[2]/{35}[2]/{25}[3]/{34}[3]/{24}[4]/{23}[5] -->> R23C4 = {23456}; R4C1 = {2345}

16. 18(4) at R1C3 = {3456}(last possible combination): no 2 -->> R12C3 = {56} -->> locked for C3 and N1; R23C4 = {34} -->> locked for C4 and N2
16a. R4C3 = 2; R4C1 = 3(step 15); R4C9 = 1; R5C9 = 2(hidden); R56C4 = [52]
16b. R3C9 = 3; R23C4 = [34]; R1C2 = 3(hidden)

17. 17(3) at R4C2 = {467}: ({458} blocked by R5C3) -->> R5C3 = 7; R45C2 = {46} -->> locked for C2 and N4
17a. R3C3 = 9; R7C12 = [65]; R6C2 = 9; R56C1 = [85]

18. 14(3) at R1C4 = {167}: ({158} blocked by R1C4) -->> locked for N2
18a. R5C6 = 1(hidden); R3C1 = 1(hidden); R3C2 = 8

And the rest is all naked singles

greetings

Para
Walkthrough by Andrew:
Para wrote:
This one was very basic. Nothing special to solve it. It was a relaxing puzzle.
Somewhat surprising considering the complicated cage pattern with so many 4-cell cages. Still we know by now that the choice of cage totals tends to contribute much more than the cage pattern to the level of difficulty. I haven't (yet?) looked at V2 but that could well be a monster since nothing has been posted so far.
Cathy wrote:
I didn't keep a complete WT this time but early placements made with outies - innies of r89. Thereafter some combo crunching did the job.
It was interesting that there were two ways to look at outies - innies of r89 depending whether one included r78c4. It just happened that Para did it one way and I used the other way; both lead to the same early placements.

Para's steps 14, 15 and 16 made his more direct than my solving path. I've made a comment after my step 18.

Here is my walkthrough for A62.

1. R23C6 = {29/38/47/56}, no 1

2. R78C4 = {89}, locked for C4 and N8

3. R345C9 = {123}, locked for C9
3a. Min R12C9 = 9 -> max R1C8 = 6

4. R567C1 = {289/379/469/478}, no 1

5. 8(3) cage at R5C4 = 1{25/34}, no 1 in R6C56

6. 20(3) cage in N6 = {389/479/569/578}, no 1,2

7. 26(4) cage at R2C8 = {2789/3689/4589/4679/5678}, no 1

8. 45 rule on N1 3 outies R23C4 + R4C1 = 10, min R23C4 = 3 -> max R4C1 = 7

9. 45 rule on R89 2 outies R7C46 – 15 = 1 innie R8C2 -> R7C46 = [97] (only possible permutation greater than 15), R8C2 = 1, R8C4 = 8, clean-up: no 4 in R23C6

10. 15(3) cage in N8 = {456} (only remaining combination), locked for N8

11. 18(4) cage at R7C6 = 7{236/245} (cannot be {1278} because 1,8 only in R9C7, cannot be {1467} because only 2,3 in R8C6) = 27{36/45}, no 1,8,9, no 2 in R8C8

12. 45 rule on N9 2 remaining outies R6C9 + R8C6 = 9 = [63/72]

13. 18(4) cage at R6C2 max R7C23 + R8C2 = 15 -> min R6C2 = 3

14. 24(4) cage in N9 must contain 7,9 = 79{26/35}, no 1,4,8
14a. 2 only in R9C8 -> no 6 in R9C8

15. 1,8 in N9 locked in R7C789, locked for R7
15a. 19(4) cage at R6C9 = 18{37/46}, no 2,5
15b. 6 in {1468} must be in R6C9 -> no 6 in R7C789
15c. 5,6 in R7 locked in R7C123, locked for N7
15d. 18(4) cage at R6C2 max R7C23 + R8C2 (step 13) = 12 -> min R6C2 = 6

16. R9C4 = 1 (hidden single in R9)
16a. R89C3 = 12 = {39}/[48], no 2,7, no 4 in R9C3

17. R6C3 = 1 (only remaining cell for 1 in 8(3) cage, step 5)
17a. R56C4 = {25/34}

18. 45 rule on C89 3 outies R357C7 = 20 = {389/479/569/578}, no 1,2
[I didn’t do the corresponding 3 outies for C12 at this stage because it didn’t do anything useful and had forgotten about it by the time that R7C3 had been reduced to 3 candidates, at which stage Para’s step 14 showed that it was very useful.]

19. R7C8 = 1 (hidden single in R7)

20. R23C4 + R4C1 = 10 (step 8)
20a. Min R4C1 + either R2C4 or R3C4 = 4 -> max R2C4 or R3C4 = 6
20b. Min R23C4 = 5 -> max R4C1 = 5

21. 7 in N7 locked in 18(3) cage = 7{29/38}, no 4

22. R89C123 = 18 + 1 + 12 (step 16a) = 31 -> R7C123 = 14 and must contain 5,6 = {356}, locked for R7 and N7 -> R7C5 = 2, R8C6 = 3, clean-up: no 8 in R23C6, no 9 in R89C3 (step 16a) = [48]

23. Naked pair {48} in R7C79, locked for N9 -> R6C9 = 6 (step 15a)

24. R9C8 = 3 (hidden single in N9)
24a. 24(4) cage in N9 = {3579} (step 14), locked for N9

25. Naked pair {26} in R78C6, locked for C7

26. 18(4) cage at R6C2 = {1359/1368} = 13{59/68}, no 7
26a. 3 locked in R7C23, locked for R7
26b. Max R7C1 = 6 -> min R56C1 = 13, no 2,3

Ruud wrote:
... but there are some nice quadruple outies in here.

Now for a couple of quadruple outies, one of which was very useful.

27. 45 rule on N3 4 outies R1C6 + R4C89 + R5C9 = 16, max R4C89 + R5C9 = 14 -> min R1C6 = 2

28. 45 rule on N7 4 outies R5C1 + R6C12 + R9C4 = 23, R9C4 = 1 -> R5C1 + R6C12 = 22 = 9{58/67}, 9 locked for N4
28a. R567C1 = {568} (only remaining combination), locked for C1
28b. R5C1 + R6C12 = {589} (only remaining combination)
28c. Naked pair {58} in R56C1, locked for N4 -> R7C1 = 6, R6C2 = 9
28d. 9 in N7 locked in R89C1, locked for C1

29. 17(3) cage in N4 = {467} (only remaining combination), locked for N4
29a. 4 in N4 locked in R45C2, locked for C2

30. Naked pair {23} in R4C13, locked for R4 -> R4C9 = 1
30a. R5C9 = 2 (hidden single in N6) -> R3C9 = 3, clean-up: no 5 in R6C4 (step 17a)

31. R5C6 = 1 (hidden single in C6)

32. 1 in N3 locked in R12C7, 14(3) cage at R1C6 = 1{49/58/67}, no 2 in R1C6

33. 2 in C6 locked in R23C6 = {29}, locked for C6 and N2
33a. No 9 in R1C6 -> no 4 in R12C7 (step 32)

34. R6C4 = 2 (hidden single in C4) -> R5C4 = 5 (step 17a), R56C1 = [85]

35. R7C5 = 2 -> R6C567 = 15 = {348} (only remaining combination), locked for R6 -> R6C8 = 7
35a. R5C78 = 13 = {49} (only remaining combination), locked for R5 and N6

36. R6C7 = 3 (hidden single in N6) -> R6C56 = {48}, locked for N5 -> R4C6 = 6, R4C45 = [79], R5C5 = 3, R4C2 = 4, R4C7 = 8 (cage sum), R4C8 = 5, R7C79 = [48], R5C78 = [94], R8C8 = 9

37. Naked pair {57} in R89C9, locked for C9
37a. R12C9 = {49} -> R1C8 = 2

38. Naked triple {346} in R123C4, locked for N2

39. R4C45 = [79] -> R3C5 + R4C3 = 7 = [52] (only remaining permutation) -> R4C1 = 3

40. R8C5 = 6 -> R9C56 = [45], R6C56 = [84], R89C7 = [26], R8C1 = 7, R9C12 = [92], R89C9 = [57], R1C6 = 8, R3C7 = 7

41. Naked pair {68} in R3C28, locked for R3 -> R3C3 = 9, R3C4 = 4, R23C6 = [92], R3C1 = 1, R1C12 = [42], R12C9 = [94], R3C2 = 8 (cage sum), R23C8 = [86]

41. R12C5 = {17} -> R1C4 = 6 (cage sum), R2C4 = 3

42. R1C2 = 3 (hidden single in N1), R2C2 = 7 (cage sum)

and the rest is naked singles
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Ed
PostPosted: Sat Jun 28, 2008 12:03 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 62v2 by Ruud (Aug 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:7168:7168:3074:4099:4099:4869:4869:1799:1799:7168:7168:3074:3074:4099:2830:4869:5136:1799:4626:4626:4626:3074:4886:2830:5136:5136:4122:4626:5660:4886:4886:4886:5920:5920:5136:4122:2084:5660:5660:4647:5920:5920:3626:3626:4122:2084:4910:4647:4647:3889:3889:3889:3626:6197:2084:4910:4910:2873:3889:4667:6197:6197:6197:2367:4910:5697:2873:4419:4667:4667:4934:4934:2367:2367:5697:5697:4419:4419:4667:4934:4934:
Solution: 
+-------+-------+-------+
| 9 8 2 | 7 3 5 | 6 4 1 |
| 7 4 5 | 1 6 9 | 8 3 2 |
| 6 3 1 | 4 8 2 | 7 9 5 |
+-------+-------+-------+
| 8 6 4 | 2 5 7 | 9 1 3 |
| 1 9 7 | 6 4 3 | 2 5 8 |
| 2 5 3 | 9 1 8 | 4 7 6 |
+-------+-------+-------+
| 5 7 6 | 3 2 4 | 1 8 9 |
| 3 1 9 | 8 7 6 | 5 2 4 |
| 4 2 8 | 5 9 1 | 3 6 7 |
+-------+-------+-------+
Quote:
mhparker: not so much of an Assassin, this one. More like a Terminator!
mhparker: I could eventually solve without hypotheticals, but only by using several grouped AICs..I rated the A62V2 at 2.0

Andrew (in 2014): Another puzzle I've just tried to the first time. Since I don't know how to do AICs I used forcing chains for my key moves; three fairly heavy ones and several short ones. It's hard to know how to rate the heavy ones; they must be close to Mike's rating.
Walkthrough by mhparker using several creative moves:
"Here's one I made earlier..."

Just noticed the Magic Roundabout topic on the DJApe Killer forum, so thought I'd "join in" by starting this thread with a now-famous quote from another cult British children's TV program. (Hint: do the words "sticky-back plastic" mean anything to you? If not, see here).

The walkthrough below really is one I made earlier. More than 3 months ago to be exact. It's very long, even in fairly optimized form, and uses several AICs, and some other creative moves. I note that SudokuSolver (SS) can now complete this puzzle, without using any chains. However, some of the "45" moves it uses are quite involved and very difficult to "spot" (read: work out using pen and paper), even if they're not classified as "extreme" or "insane". I personally prefer to use inference chains when the going gets tough. However, the SS log also makes interesting reading, and I may follow up this WT by elaborating on one or two of the steps that SS makes.

For now, here's my WT as I left it on August 20, in unmodified form:


Assassin 62V2 Walkthrough

----
Note: New shorthand notation (example): "CPE(R7): no 3 in R6C8"
=> "Common Peer Elimination(CPE): R6C8 sees all 3's in R7 -> no 3 in R6C8".
----

1. 28/4 at R1C1 = {(47/56)89}: no 1,2,3
1a. {89} locked for N1

2. 12/4 at R1C3 = {12(36/45)}: no 7,8,9

3. 19/3 at R1C6: no 1

4. 7/3 at R1C8 = {124}, locked for N3

5. 11/2 at R2C6: no 1

6. 22/3 at R4C2 = {(58/67)9}: no 1,2,3,4
6a. 9 locked for N4

7. 8/3 at R5C1 = {1(25/34)}: no 6,7,8,9
7a. 1 locked for C1

8. 11/2 at R7C4: no 1

9. 22/3 at R8C3 = {(58/67)9}: no 1,2,3,4
9a. no 9 in R9C12

10. 9/3 at R8C1 = {126/234}: no 5,7,8,9
10a. (Note: {135} blocked by 8/3 at R5C1 (step 7))
10b. 2 locked for N7
10c. 1 only in R9C2 -> no 6 in R9C2
10d. no 3 in R9C2 (requires {24} in R89C1 - blocked by 8/3 at R5C1)
10e. no 4 in R9C2 (requires {23} in R89C1 - blocked by 8/3 at R5C1)
10f. Summary: 9/3 at R8C1 = {26}[1]/{34}[2]

11. 9 in C1 locked in R12C1 -> not elsewhere in N1

12. Outies C12: R357C3 = 14/3 = {149/158/167/239/248/257/347/356}
12a. min. R5C3 = 5
12b. -> max. R37C3 = 9
12c. -> no 9 in R7C3

13. Outies C89: R357C7 = 10/3 = {127/136/145/235} (no 8,9)
13a. {12} only in R57C7
13b. -> no 7 in R57C7

14. Outies R6789: R5C1478 = 14/4 = {1238/1247/1256/1346/2345} (no 9)
14a. Cleanup: no 1,2 in R6C4

15. CPE(C1): no 7 in R3C23

16. CPE(N1): no 1,2,3 in R3C4

17. {123} in 12/4 at R1C3 (step 2) only in R1C3+R2C34
17a. -> no 6 in R1C3+R2C34

18. 12/4 at R1C3 and 7/3 at R1C8 form grouped X-Wing on {12} in R12
18a. -> no 1,2 elsewhere in R12
18b. Cleanup: no 9 in R3C6
18c. CPE(N2): no 1 in R4C4

19. 20/4 at R2C8: {124} only available in R4C8
19a. -> {1289},{1469},{1478},{2459},{2468} all blocked
19b. -> 20/4 at R2C8 = {1379/1568/2369/2378/2567/3458/3467} = {(1/2/4)..}
19c. -> R4C8 = {124}

20. from steps 7 and 10f: 8/3 at R5C1 and 9/3 at R8C1 together lock {234} in C1
20a. -> no 2,3,4 in R1234C1

21. Outies N7: R5C1+R6C12+R9C4 = 13/4
21a. min. R5C1+R6C12 = 6
21b. -> max. R9C4 = 7 (no 8,9)
21c. min. R9C4 = 5
21d. -> max. R5C1+R6C12 = 8
21e. -> R5C1+R6C12 = {123/124/125/134} (no 6,7,8)
21f. 1 locked in R5C1+R6C12 for N4

22. 9 in 22/3 at R8C3 locked in R89C3 for C3 and N7
22a. {89} in 22/3 only in R89C3
22b. -> no 5 in R89C3

23. 18/4 at R3C1 = {1368/1458/1467/2358/2367/2457/3456}
23a. (Note: {1278} blocked by 12/4 at R1C3)
23b. -> must contain 2 of {1234}
23c. {1234} only in R3C23
23d. -> R3C23 = {1234} (no 5,6)
23e. Cleanup: no 1 in R7C3 (step 12)

24. 1 in C3 locked in N1 -> not elsewhere in N1

25. Innies N1: R123C3+R3C12 = 17/5, w/ {123} locked = {123(47/56)}
25a. 6 only in R3C1
25b. -> no 5 in R3C1

26. Outies N1: R23C4+R4C1 = 13/3
26a. min. R4C1 = 5
26b. -> max. R23C4 = 8
26c. -> max. R2C4 = 3 (no 4,5)

27. R3C23 cannot contain {34}. Here's how:
27a. R3C123 = [6]{34} blocked by 28/4 at R1C1 (step 1), and...
27b. ...[7]{34} would require 4 in R4C1 for 18/4 cage sum - unavailable
27c. -> R3C23 must contain exactly 1 of {12} (step 23)
27d. -> R12C3 must contain exactly 1 of {12} (only other place in N1)
27e. -> other of {12} for 12/4 at R1C3 (step 2) must go in R2C4
27f. -> no 3 in R2C4

28. Only other place for {12} in R3/N2 is R3C56
28a. -> either R3C5 = {12} -> R2C4+R3C5 = {12} -> R1C6+R3C4 = 15 (innies N2) = [96], or...
28b. ...R3C6 = 2 -> R2C6 = 9
28c. -> 9 locked in R12C6 for C6 and N2
28d. from steps 28a/28b: either way, no 6 in 11/2 at R2C6
28e. -> R23C6 <> {56}

29. 9 in R3 locked in N3 -> not elsewhere in N3

30. 19/3 at R1C6 = {379/478/568}
30a. (Note: {469} now blocked because {49} only available in R1C6)
30b. 9 only in R1C6
30c. -> no 3,7 in R1C6

31. I/O difference R12: R2C68 = R3C4 + 8 = 12, 13 or 14
31a. if 12 (R3C4 = 4): R2C68 = [93] ([75] blocked because it would require 4 in R3C6)
31b. if 13 (R3C4 = 5): R2C68 = [76/85]
31c. if 14 (R3C4 = 6): R2C68 = [86/95]
31d. Summary: R2C6 = {789} (no 3,4), R2C8 = {356} (no 7,8)
31e. Cleanup: no 7,8 in R3C6

32. Innies C1234: R1C4+R4C34 = 13/3
32a. Min. R1C4+R4C3 = 5
32b. -> Max. R4C4 = 8 (no 9)

33. Hidden killer pair on {78} in C1 as follows:
33a. {78} in C1 locked in R1234C1
33b. R34C1 cannot contain both of {78} (step 23a)
33c. 9 in C1 already locked in R12C1 (step 11)
33d. -> R12C1 = {79/89} (no 5,6), 18/4 at R3C1 = {(7/8)..}

34. I/O difference R89: R7C46 = R8C2 + 6
34a. R7C46 cannot sum to 11, otherwise R7C6 would clash w/ R8C4
34b. -> no 5 in R8C2

35. 5 in N7 locked in R7 -> not elsewhere in R7

36. no 7 in R4C1. Here's how:
36a. 1 in R3 locked in N12 innies (= 23/5) within R3C35
36b. -> if R3C3 <> 1, then R3C5 must be 1 -> max. R3C5+R1C6 = 10 -> min. R3C123 = 13
36c. -> if R4C1 = 7, then R3C123 <> [6]{23}, as this only sums to 11
36d. furthermore, if R4C1 = 7, then R3C123 cannot be [6]{14} as this blocked by 28/4 at R1C1 (step 1)
36e. but these are the only 2 options available with 7 in R4C1
36f. -> no 7 in R4C1

37. 7 in C1 locked in N1 -> not elsewhere in N1

38. from step 6: if 22/3 at R4C2 <> {679}, then it must be {589} -> R4C1 = 6
38a. -> 6 in N4 locked in R4C1 or 22/3 at R4C2
38b. -> no 6 elsewhere in N4 (R46C3)

39. 18/4 at R3C1 = {1368/1467/2367/2457}
39a. {2457} blocked by grouped AIC. Here's how ('=>' = strong link, '->' = weak link):
39b. if R34C1 <> {6..} => R89C1 = {6..} -> R789C3 <> {6..} => R5C3 = 6, R3C3 <> {24} (step 12)
39c. In other words, if R34C1 doesn't contain a 6, R3C3 cannot contain a 2 or 4
39d. -> {2457} combo blocked
39e. -> 18/4 at R3C1 = {1368/1467/2367}
39f. -> no 5 in R4C1

40. 5 in C1 now locked in 8/3 at R5C1 = {125} (no 3,4)
40a. 2 locked in R56C1 for C1 and N4
40b. Cleanup: no 8 in R4C4 (step 32)

41. Hidden single (HS) in N7 at R9C2 = 2
41a. Cleanup: R89C1 = {34}, locked for N7

42. Recall step 27c
42a. {12} now not available in R3C2
42b. -> R3C3 = {12} (no 3,4)

43. R4C1 and 22/3 at R4C2 (step 9) form killer pair on {68} within N4
43a. -> no 8 in R46C3

44. Outies C12 revisited: R357C3 = 14/3 = {158/167/257}
44a. -> R57C3 can only contain at most 1 of {689}
44b. Only other places for {689} in C3 are R89C3
44c. -> R57C3 and R89C3 form hidden killer triple on {689} in C3
44d. -> R57C3 must contain exactly 1 of {689} -> R357C3 = {1(58/67)} (no 2) and...
44e. ...R89C3 must contain exactly 2 of {689} -> no 7 in R89C3
44f. Cleanup: no 6 in R9C4

45. Naked single (NS) at R3C3 = 1
45a. Note: R57C3 now = {58/67} (step 44)

46. HS in N2 at R2C4 = 1

47. 1 in C2 locked in 19/4 at R6C2
47a. 7 in N7 locked in 19/4 at R6C2
47b. -> 19/4 at R6C2 = {17..} = {17(38/56)} (no 4) = {(3/5)..}

48. 4 in C2 locked in N1 -> not elsewhere in N1

49. 12/4 at R1C3: 4 now only available in R3C4
49a. -> no 5 in R3C4 (step 2)

50. Split 17/3 at R3C12+R4C1 = [638/746]
50a. -> R3C12 = [63/74] = {(4/6)..}
50b. -> R3C12 and R3C4 form killer pair on {46} in R3
50c. -> no 4,6 elsewhere in R3
50d. Cleanup: no 7 in R2C6

51. Innies N2: R1C6+R3C45 = 17/3 = {269/368/458/467}
51a. R3C45 cannot be [67] due to R3C1
51b. -> if {467}: no 4 in R1C6
51c. if {458}: 4 must go in R3C4
51d. Conclusion: no 4 in R1C6

52. 4 now unavailable to 19/3 at R1C6 = {379/568} = {(6/7)..} (no eliminations yet)

53. Another grouped AIC:
53a. Either R3C2 = 3, or...
53b. R3C2 <> 3 => R3C2 = 4
53c. -> R3C4 <> 4 => R3C4 = 6
53d. -> R3C1 <> 6 => R3C1 = 7
53e. -> R3C789 <> 7 => R12C7 = {7..} = {37} (step 52)
53f. Thus, either or both of R3C2 and R12C7 contain(s) a 3
53g. -> no 3 in R3C789
53h. Cleanup: no 5,6 in R57C7 (step 13)

54. Hidden killer pair on {36} in N3, as follows:
54a. Only places for {36} in N3 are R2C8 and within 19/3 (R12C7)
54b. 19/3 cannot contain both of {36} (step 52)
54c. -> R2C8 and R12C7 must each have one of {36}
54d. -> R2C8 = {36} (no 5); no 6 in R1C6

55. {467} combo now blocked for N2 innies (step 51), as none of these digits present in R1C6
55a. -> R1C6+R3C45 = {269/368/458} (no 7)

56. 7 in N2 now locked in 16/3 at R1C4 = {367/457} (no 8)

57. Another hidden killer pair, this time in C4:
57a. Only places for {89} in C4 are within 18/3 (R56C4) and within 11/2 (R78C4)
57b. Neither can contain both of {89} (due to no 1 in R6C3), so both must contain exactly one of {89}
57c. -> 11/2 at R7C4 = {29/38} (no 4,5,6,7); {567} combo blocked for 18/3 at R5C4

58. Back to grouped AIC's:
58a. Either R7C1 = 5, or...
58b. R7C1 <> 5 => R7C1 = 1
58c. -> R56C1 <> 1 => R6C2 = 1 (strong link N4)
58d. -> R6C2 <> 3 => R3C2 = 3 (strong link C2)
58e. -> R12C3 <> 3 => R12C3 = {25} (ALS node)
58f. Thus, either R7C1 = 5 or R12C3 = {25}
58g. -> no 5 in R7C3 (common peer)
58h. Cleanup: no 8 in R5C3 (step 45a)

59. 8 in C3 locked in N7 -> not elsewhere in N7

60. Outies C1: R123C2 = 15/3, w/ 4 locked = {348/456} = {(3/5)..}
60a. {35} not available in R7C3+R8C2
60b. -> R123C2 and R67C2 (step 47b) form killer pair on {35} within C2
60c. -> no 5 in R45C2

61. I/O difference R89: R8C24 = R7C6 + 5. Analysis follows:
61a. if R8C2 = 1, R7C6+R8C4 = [48]
61b. if R8C2 = 6, R7C6+R8C4 = [32/43]
61c. if R8C2 = 7, R7C6+R8C4 = [42]
61d. Summary: R7C6 = {34} (no 1,2,6,7,8}; R8C4 = {238} (no 9)
61e. Cleanup: no 2 in R7C4

62. 17/3 at R8C5 = {179/269/368/458/467} = {(3/4/9)..}
62a. (Note: {278} and {359} blocked by 11/2 at R7C4 (step 57c))
62b. 11/2 at R7C4 = {(3/9)..} (step 57c)
62c. -> 17/3 at R8C5, 11/2 at R7C4 and R7C6 form killer triple on {349} in N8
62d. -> no 3,4,9 elsewhere in N8 (R7C5+R8C6)

63. 24/4 at R6C9:
63a. Max. R7C7 = 4 -> Min. R6C9+R7C89 = 20
63b. -> no 1,2 in R6C9+R7C89

64. CPE(R7): no 4 in R89C7

65. CPE(C7): no 4 in R6C9

66. Outies N9: R6C9+R78C6 = 16
66a. R7C6 = {34} -> R6C9+R8C6 = 12 or 13
66b. -> no 3,9 in R6C9; no 1,2 in R8C6

67. 17/3 at R8C5 must contain exactly one digit in range {1..4} (step 62)
67a. 11/2 at R7C4 must contain exactly one digit in range {1..4} (step 57c)
67b. Only other places for remaining 2 of {1..4} in N8 are R7C56
67c. -> 17/3 at R8C5, 11/2 at R7C4 and R7C56 form hidden killer quad on {1234} in N8
67d. -> no 6,7,8 in R7C5

68. I/O difference R789: R6C29 = R7C15 + 4
68a. R7C15 = [12/51/52] -> sum to 3, 6 or 7
68b. -> R6C29 = 7, 10 or 11. Analysis follows:
68c. if 7, R6C29 = [16]
68d. if 10, R6C29 = [37]
68e. if 11, R6C29 = [38/56]
68f. -> R6C9 = {678} (no 5)

69. 24/4 at R6C9 (revisited): 5 no longer available
69a. -> valid combos are {1689/2679/3678} (no 4)
69b. (Note: {3489} blocked by R7C6)
69c. -> 24/4 at R6C9 must have exactly one of {123}, which must be in R7C7
69d. -> no 3 in R7C89
69e. 6 locked in R6C9+R7C89
69f. -> no 6 in R89C9

70. HS in R7 at R7C6 = 4

71. 4 in C7 locked in N6 -> not elsewhere in N6

72. Split 14/3 at R8C67+R9C7 = {158/167/257/356} (no 9)
72a. (Note: {239} combo blocked, as none of these digits present in R8C6)

73. 9 in C7 locked in N6 -> not elsewhere in N6

74. Innies N8: R7C5+R8C6+R9C4 = 13/3, w/ {349} unavailable
74a. -> valid combos are {157/256} (no 8)
74b. 5 locked in R8C6+R9C4 for N8
74c. -> no 5 in R9C7 (CPE)

75. Permutations 20/4 at R2C8:
75a. 20/4 cannot have both of {67} due to 19/3 at R1C6 (step 52)
75b. -> 20/4 at R2C8 = {1379/1568/2378} = {(6/7)..}
75c. -> must have exactly one of {57}, which must go in R3C7
75d. -> no 5,7 in R3C8

76. 19/3 at R1C6 (step 52) and 20/4 at R2C8 (step 75b) form killer pair on {67} within N3
76a. -> no 7 in R3C9

77. 7 in N3 locked in C7 -> not elsewhere in C7

78. {257} combo now blocked for split 14/3 at R8C67+R9C7 (step 72), as none of these digits present in R9C7
78a. -> 14/3 at R8C67+R9C7 = {158/167/356} (no 2) = {(1/3)..}
78b. {36} in R89C7 blocked by R12C7
78c. -> no 3 in R8C7

79. 10/3 at R357C7 (step 13) = [7]{12}/[541]/[5]{23} = {(1/3)..}
79a. R8C6 has neither of {13}
79b. -> R57C7 and R89C7 (step 78a) form killer pair on {13} within C7
79c. -> no 1,3 elsewhere in C7

80. HS in N3 at R2C8 = 3

81. 19/3 at R1C6 = {568} (no 7,9)
81a. 6 only available in R12C7, locked for C7

82. HS in C6/N2 at R2C6 = 9
82a. -> R3C6 = 2

83. HS in C1/N1 at R1C1 = 9

84. HS in C7/N3 at R3C7 = 7

85. NS at R3C1 = 6
85a. -> R4C1 = 8, R3C4 = 4
85b. -> R2C1 = 7, R3C2 = 3

86. Naked pair (NP) on {25} at R12C3 -> no 2,5 elsewhere in C3 and N1

87. Hidden pair (HP) on {34} in C3 at R46C3
87a. -> R46C3 = {34} (no 7)

88. HP on {37} in R1/N2 at R1C45
88a. -> R1C45 = {37} (no 5,6)
88b. -> R2C5 = 6 (cage-split)

89. HS in R1/C7/N3 at R1C7 = 6

90. 6 in C4 locked in N5 -> not elsewhere in N5

91. 3 no longer available to 19/4 at R6C2
91. -> (from step 47b) 19/4 at R6C2 = {1567} (no 8)
91a. 6 locked for N7

92. Cage-split of 22/3 at R8C3: R89C3 = {89} = 17
92a. -> R9C4 = 5

93. Revisit split 14/3 at R8C67+R9C7 (step 78a):
93a. {158} blocked because R8C6 has none of these digits
93b. {167} blocked because {67} only in R8C6
93c. -> 14/3 at R8C67+R9C7 = {356} = [653] (only permutation possible)

Now it's just naked and hidden singles to end.
Walkthrough by Andrew:
In order to make it easier to follow, I've listed all the cages and hidden cages which I've used for each of my forcing chains. Therefore my walkthrough seems somewhat longer than it really is. Also first time through I overlooked a permutation in step 33b and another one in step 44a, so I had to insert steps 34a,b and extra moves in step 44a to eliminate those permutations.

Prelims

a) R23C6 = {29/38/47/56}, no 1
b) R78C4 = {29/38/47/56}, no 1
c) 19(3) cage at R1C6 = {289/379/469/478/568}, no 1
d) 7(3) cage at R1C8 = {124}
e) 22(3) cage at R4C2 = {589/679}
f) 8(3) cage at R5C1 = {125/134}
g) 9(3) cage at R8C1 = {126/135/234}, no 7,8,9
h) 22(3) cage at R8C3 = {589/679}
i) 28(4) cage at R1C1 = {4789/5689}, no 1,2,3
j) 12(4) cage at R1C3 = {1236/1245}, no 7,8,9

Steps resulting from Prelims
1a. 7(3) cage at R1C8 = {124}, locked for N3
1b. 22(3) cage at R4C2 = {589/679}, 9 locked for N4
1c. 8(3) cage at R5C1 = {125/134}, 1 locked for C1
1d. 28(4) cage at R1C1 = {4789/5689}, 8,9 locked for N1
1e. 9 in C1 only in R12C1, locked for N1

2. 9(3) cage at R8C1 = {126/234} (cannot be {135} which clashes with 8(3) cage at R5C1), no 5, 2 locked for N7
2a. Min R89C1 = {34} = 7 (R89C1 cannot be {23/24} which clash with 8(3) cage) -> max R9C2 = 2
[I noticed later that an alternative way for step 2a is
8(3) cage at R5C1 and 9(3) cage at R8C1 form combined 17(6,5+1) cage. Minimum total for 5 cells in the column is 15 -> max R9C2 = 2.]

3. 45 rule on C89 3 outies R357C7 = 10 = {127/136/145/235}, no 8,9
3a. 7 of {127} must be in R3C7 -> no 7 in R57C7

4. 45 rule on R6789 4 outies R5C1478 = 14 = {1238/1247/1256/1346/2345}, no 9
4a. 45 rule on R6789 2 outies R5C14 = 1 innie R6C8, min R5C14 = 3 -> min R6C8 = 3

5. 45 rule on N7 4 outies R5C1 + R6C12 + R9C4 = 13
5a. Min R5C1 + R6C12 = 6 -> max R9C4 = 7
5b. Min R9C4 = 5 -> max R5C1 + R6C12 = 8, no 6,7,8, 1 locked for N4
5c. 22(3) cage at R8C3 = {589/679}, 9 locked for C3 and N7
5d. 5 of {589} must be in R9C4 -> no 5 in R89C3

6. 1,2,3 in N1 only in R12C3 + R3C123, CPE no 1,2,3 in R3C4
6a. 12(4) cage at R1C3 = {1236/1245}
6b. 6 of {1236} must be in R3C4 -> no 6 in R12C3 + R2C4
[I missed caged X-Wing for 1,2 in 12(4) cage and 7(3) cage at R1C8, no other 1,2 in R12, which would have simplified things slightly.]

7. 45 rule on R12 2 innies R2C68 = 1 outie R3C4 + 8, IOU no 8 in R2C8
7a. Min R3C4 = 4 -> min R2C68 = 12, no 2 in R2C6, clean-up: no 9 in R3C6

8. 20(4) cage at R2C8 = {1379/1568/2369/2378/2567/3458/3467} (cannot be {1289/1469/1478/2459/2468} because 1,2,4 only in R4C8)
8a. 1,2,4 only in R4C8 -> R4C8 = {124}

9. 45 rule on R89 2 innies R8C24 = 1 outie R7C6 + 5, IOU no 5 in R8C2
9a. 5 in N7 only in R7C123, locked for R7, clean-up: no 6 in R8C4

10. 9(3) cage at R8C1 (step 2) = {126/234} = {26}1/{34}2, 8(3) cage at R5C1 = {125/134}
10a. Combined cage 8(3) + R89C1 = {125}{34}/{134}{26}, 2,3,4 locked for C1

11. 45 rule on C1234 2 outies R34C5 = 1 innie R1C4 + 6, IOU no 6 in R4C5

12. 18(4) cage at R3C1 = {1278/1368/1467/2358/2367/2457} (cannot be {1458} which clashes with 28(4) cage at R1C1, cannot be {3456} which clashes with combined cage 8(3) at R5C1 + R89C1)
12a. 1,2,3,4 only in R3C23 -> no 5,6,7 in R3C23
12b. Hidden killer pair 1,2 in 12(4) cage at R1C3 and 18(4) cage for N1, 12(4) cage must contain at least one of 1,2 in N1 -> 18(4) cage cannot contain both of 1,2 -> 18(4) cage = {1368/1467/2358/2367/2457} (cannot be {1278} which contains both of 1,2)
12c. Hidden killer pair 1,2 in 12(4) cage and 18(4) cage for N1, 18(4) cage contains one of 1,2 -> 12(4) cage must contain one of 1,2 in N1 -> R2C4 = {12}
12d. Hidden killer pair 6,7 in 28(4) cage and R3C1 for N1, 28(4) cage contains one of 6,7 -> R3C1 = {67}
12e. 18(4) cage = {1368/1467/2367/2457} (cannot be {2358} which doesn’t contain 6 or 7)

13. 45 rule on C12 3 outies R357C3 = 14, max R35C3 = 12 -> no 1 in R7C3
13a. 1 in C3 only in R123C3, locked for N1

14. 45 rule on N89 2 innies R7C5 + R9C4 = 1 outie R6C9 + 1
14a. Min R7C5 + R9C4 = 6 -> min R6C9 = 5
14b. Max R7C5 + R9C4 = 10 must contain one of 1,2,3,4 -> R7C5 = {1234}

15. 18(4) cage at R3C1 (step 12e) = {1368/1467/2367/2457}
15a. 1 of {1368} must be in R3C3, 3 of {2367} must be in R3C2 (cannot be [6237/7236] which clashes with 9(3) cage at R8C1 = {34}2) -> no 3 in R3C3

16. 18(4) cage at R3C1 (step 12e) = {1368/1467/2367/2457}
16a. Consider combinations for 9(3) cage at R8C1 (step 2) = {126/234}
9(3) cage = {126}, 6 locked for C1 => 18(4) cage = {2457} => 3 in N1 only in R12C3, locked for C3
or 9(3) cage = {234}, locked for N7
-> no 3 in R7C3

17. R357C3 (step 13) = 14 = {158/167/248/257}
17a. 1,2 only in R3C3 -> R3C3 = {12}
17b. 18(4) cage at R3C1 (step 12e) = {1368/1467/2367/2457}
17c. 3,4 only in R3C2 -> R3C2 = {34}
17d. 2 in N1 only in R123C3, locked for C4

18. 19(4) cage at R6C2 = {1378/1468/1567/2458/3457} (cannot be {2368/2467} which clash with 9(3) cage at R8C1)
18a. Hidden killer pair 1,2 in 19(4) cage and R9C2 for C2, R9C2 = {12} -> 19(4) cage must contain one of 1,2 -> 19(4) cage at R6C2 = {1378/1468/1567/2458} (cannot be {3457} which doesn’t contain 1 or 2)
[Rest of step 18 deleted, I’d used a forcing chain for placement of 2 in C2 but it was unnecessary after the more powerful step 19.]

19. 19(4) cage at R6C2 (step 18a) = {1378/1468/1567/2458}
19a. 18(4) cage at R3C1 (step 12e) = {1368/1467/2367/2457}
19b. 9(3) cage at R8C1 (step 2) = {126/234}
19c. Consider placement for 3 in C2
R3C2 = 3 => 18(4) cage = {1368/2367}, 6 locked for C1 = 9(3) cage = {234} => R9C2 = 2
or 3 in 19(4) cage = {1378}, 1 locked for C2 => R9C2 = 2
-> R9C2 = 2 -> R89C1 = {34}, locked for C1 and N7
19d. Naked triple {125} in 8(3) cage at R5C1, locked for C1
19e. 18(4) cage = {1368/1467/2367}, 6 locked for C1

20. 12(4) cage at R1C3 = {1236/1245}
20a. Consider placement for 1 in R3
R3C3 = 1 => 1 in 12(4) cage must be in R2C4
or R3C5 = 1
-> 1 in R2C4 + R3C5, locked for N2
20b. 1 in N2 only in R2C4 + R3C5, CPE no 1 in R4C4
20c. 45 rule on N2 4 innies R1C6 + R2C4 + R3C45 = 18 contains 1 = {1269/1359/1368/1458/1467} (cannot be {1278} because R3C4 only contains 4,5,6)
20d. R23C6 = {38/47}/[92] (cannot be {56} which clashes with R1C6 + R2C4 + R3C45), no 5,6 in R23C6

21. R4C1 = {678}, 22(3) cage at R4C2 = {589/679} -> combined half cage R4C1 + 22(3) cage = {67}{589}/8{679}, 8 locked for N4

[Now a heavier forcing chain]
22. R5C1 + R6C12 + R9C4 = 13 (step 5), R357C3 (step 13) = 14, 28(4) cage at R1C1 = {4789/5689}, 22(3) cage at R4C2 = {589/679}
22a. 18(4) cage at R3C1 (step 19e) = {1368/1467/2367}
22b. Consider placements for R9C4 = {567}
R9C4 = 5 => R5C1 + R6C12 = 8 = {125} (because must contain 1,2 for N4), locked for N4 => 22(3) cage = {679}, locked for N4 => R4C1 = 8 => R3C123 = [631]
or R9C4 = 6 => R5C1 + R6C12 = 7 = {12}4 => 28(4) cage = {5689}, 5,6 locked for C2, R34C1 = [76] => 22(3) cage = {589} => R5C3 = 5, R9C4 = 6 => R89C3 = 16 = {79} => R7C3 = 8 (hidden single in N7), R57C3 = [58] = 13 => R3C3 = 1 (cage sum)
or R9C4 = 7 => R5C1 + R6C12 = 6 = {12}3 => R3C2 = 4 => 18(4) cage = {1467} => R3C3 = 1
-> R3C3 = 1
22c. R2C4 = 1 (hidden single in N2)
[Note that the longest part of this forcing chain, starting with R9C4 = 6, can be extended slightly to reach a contradiction but I won’t take it that far.]
22d. 2 in R3 only in R3C56, locked for N2

[I stepped through the earlier steps with SudokuSolver, without studying the details closely and saw that it reduced R3C6 to {23}. That looks like a more satisfying way to eliminate 18(4) cage at R3C1 = {2367} leading to R3C3 = 1.]

23. 28(4) cage at R1C1 = {4789/5689}
23a. R3C3 = 1 -> 18(4) cage at R3C1 (step 19e) = {1368/1467}
Considering combinations for 18(4) cage
18(4) cage = {1368} = [6318] => 28(4) cage = {4789} => R12C1 = {79}, R12C2 = {48} => R12C3 = {25} => R3C4 = 4 (cage sum)
or 18(4) cage = {1467} => R3C2 = 4 => 28(4) cage = {5689} => R12C2 = {56} => R12C3 = {23} => R3C4 = 6 (cage sum)
-> 4 in R3C24, locked for R3, 6 in R3C14, locked for R3, R3C4 = {46}, R12C2 = {48/56}, no 7
23b. 7 in N1 only in R123C1, locked for C1
23c. Killer pair 6,8 in R4C1 and 22(3) cage at R4C2, locked for N4
23d. 4 in R3C24, CPE no 4 in R12C3
23e. 4 in N1 only in R123C2, locked for C2
Clean-up: no 7 in R2C6

24. 19(4) cage at R6C2 (step 18a) = {1378/1567}, 7 locked for N7
24a. 22(3) cage at R8C3 = {589/679}
24b. 5,7 only in R9C4 -> R9C4 = {57}

25. R2C68 = R3C4 + 8 (step 7)
25a. R3C4 = {46} -> R2C68 = 12,14 = {39}/[86/95] (cannot be {48} because no 4,8 in R2C8, cannot be {57} because no 5,7 in R2C6), no 4 in R2C6, no 7 in R2C8, clean-up: no 7 in R3C8

26. R1C6 + R2C4 + R3C45 (step 20c) = {1269/1368/1458/1467} (cannot be {1359} because R3C4 only contains 4,6)
26a. 2 of {1269} must be in R3C5 -> no 9 in R3C5
26b. 7 of {1467} must be in R3C5 -> no 7 in R1C6
26c. 9 in R3 only in R3C89, locked for N3
26d. Max R12C7 = 15 -> min R1C6 = 4

27. R357C3 (step 13) = 14, R3C3 = 1 -> R57C3 = 13 = {58/67}
27a. 18(4) cage at R3C1 (step 23a) = {1368/1467}, 22(3) cage at R4C2 = {589/679}, 22(3) cage at R8C3 = {589/679}
27b. Consider placements for R3C4 = {46}
R3C4 = 4 => 18(4) cage = {1368} => R4C1 = 8 => 22(3) cage at R4C2 = {679} => R57C3 = {67}, locked for C3 => R89C3 = {89} => R9C4 = 5
or R3C4 = 6 => R4C1 = 6 => 22(3) cage at R4C2 = {589} => R57C3 = {58}, locked for C3 => R89C3 = {69} => R9C4 = 7
-> R39C4 = [45/67]
27c. R78C4 = {29/38} (cannot be {47/56} which clash with R39C4), no 4,5,6,7 in R78C4

28. 45 rule on C1234 3(1+2) innies R1C4 + R4C34 = 13
28a. Min R4C3 = 3 -> max R14C4 = 10, no 1,2 in R1C4, no 1 in R4C4 -> no 9 in R1C4, no 8,9 in R4C4
28b. Max R14C4 = 10 cannot be [82] which clashes with R78C4 -> no 8 in R1C4

29. 18(4) cage at R3C1 (step 23a) = {1368/1467}
29a. Consider combinations for 22(3) cage at R4C2 = {589/679}
22(3) cage = {589}, locked for N4
or 22(3) cage = {679}, locked for N4 => R4C1 = 8 => R4C123 = [631] => naked triple {125} in R56C1 + R6C2, locked for N4
-> no 5 in R46C3

30. Hidden killer pair 8,9 in R56C4 and R78C4 for C4, R78C4 contains one of 8,9 -> R56C4 must contain one of 8,9
30a. 18(3) cage at R5C4 = {279/369/378/459/468} (cannot be {567} which doesn’t contain 8 or 9)
30b. 4 of {459/468} must be in R6C3 -> no 4 in R56C4
30c. 9 of {279/459} must be in R6C4 -> no 2,5 in R6C4

31. R1C4 + R4C34 = 13 (step 28)
31a. R4C3 = {347} -> R4C3 + R14C4 = 3[37]/4{36}/4[45]/4[72]/7[42] (cannot be 3{46} which clashes with R3C4), no 5 in R1C4, no 4 in R4C4
31b. 4 in C4 only in R13C4, locked for N2

32. 16(3) cage at R1C4 = {358/367/457} (cannot be {349} which clashes with R23C6), no 9
32a. 9 in N2 only in R12C6, locked for C6

33. R8C24 = R7C6 + 5 (step 9)
33a. R8C24 cannot total 6,7,12,13 -> no 1,2,7,8 in R7C6
33b. R7C6 = {346} -> R8C24 = 8,9,11 = [62/18/63/72/83], no 9 in R8C4, clean-up: no 2 in R7C4

34. 45 rule on N8 4 innies R7C56 + R8C6 + R9C4 = 17 = {1358/1367/1457/2456} (cannot be {1268/2348} because R9C4 only contains 5,7, cannot be {2357} which clashes with R78C4)
34a. R7C6 + R8C24 (step 33b) cannot be [683] which clashes with R7C56 + R8C6 + R9C4 = {1367} and clashes with {2456} because 22(3) cage at R8C3 = {89}5 clashes with R8C2 = 8) -> no 6 in R7C6
34b. R7C6 = {34} -> R8C24 (step 33b) = [62/18/63/72], no 8 in R8C2
34c. 1,2,3,4 of R7C56 + R8C6 + R9C4 must be in R7C56 -> no 1,2,3,4 in R8C6
34d. 1,2 only in R7C5 -> R7C5 = {12}
34e. 4 in R7 only in R7C6789, CPE no 4 in R89C7
34f. R7C5 + R9C4 = R6C9 + 1 (step 14)
34g. Max R7C5 + R9C4 = 9 -> max R6C9 = 8

35. R2C68 = R3C4 + 8 (step 7), min R3C4 = 4 -> min R2C68 = 12, max R2C8 = 6 -> no 3 in R2C6, clean-up: no 8 in R3C6

36. 45 rule on N36 3 innies R4C7 + R6C79 = 1 outie R1C6 + 14
36a. Min R1C6 = 5 -> min R4C7 + R6C79 = 19, no 1 in R46C7

37. R7C6 = {34} -> R8C24 = 8,9 = [62/18/63/72] (step 34b) -> R7C6 + R8C24 = 3[62]/4[18]/4[63]/4/72
37a. R7C56 + R8C6 + R9C4 (step 34) = {1358/1457/2456} (cannot be {1367} which clashes with R7C6 + R8C24 = 3[62], killer combo clash), 5 locked for N8
37b. 5 in N8 only in R8C6 + R9C4, CPE no 5 in R9C7

38. R357C7 (step 3) = {127/136/145/235}
38a. Consider combinations for R7C56 + R8C6 + R9C4 (step 37a) = {1358/1457/2456}
R7C56 + R8C6 + R9C4 = {1358} => R78C6 = [38] = 11 => R89C7 = 7 = {16} (cannot be {25} because no 2,5 in R9C7), locked for C7 => R357C7 = {235} => R7C7 = {23}
or R7C56 + R8C6 + R9C4 = {1457/2456} => R7C6 = 4
-> no 4 in R7C7
38b. 24(4) cage at R6C9 = {1689/2589/2679/3579/3678} (cannot be {3489} which clashes with R7C6, cannot be {4569} which clashes with 18(4) cage at R7C6 = [38]{16} or contains 4 in R7C6, cannot be {4578} because no 4,5,7,8 in R7C7) -> no 4 in 24(4) cage

39. R7C6 = 4 (hidden single in R7)
39a. 45 rule on N9 2(1+1) remaining outies R6C9 + R8C6 = 12 = [57/66/75], no 8 in R6C9, no 8 in R8C6
39b. Min R78C6 = 9 -> max R89C7 = 9, no 9 in R89C7
39c. 4,9 in C7 only in R456C7, locked for N6
39d. R5C14 = R6C8 (step 4a), max R6C8 = 8 -> no 8 in R5C4

40. 19(3) cage at R1C6 = {379/568}
40a. 20(4) cage at R2C8 (step 8) = {1379/1568/2369/2378} (cannot be {2567} which clashes with 19(3) cage)
40b. 8,9 only in R3C8 -> R3C8 = {89}
40c. 6 of {1568} must be in R2C8 -> no 5 in R2C8

41. R2C68 (step 25a) = [86/93] -> R2C68 + R3C6 = [863/932], CPE no 3 in R3C79
41a. 20(4) cage (step 40a) = {1379/1568/2378} (cannot be {2369} because 3,6 only in R2C8)

42. R357C7 (step 3) = {127/145/235} (cannot be {136} because R3C7 only contains 5,7), no 6 in R57C7
42a. R3C7 = {57} -> no 5 in R5C7
42b. Max R5C7 = 4 -> min R56C8 = 10, no 1 in R5C8

43. 45 rule on R123 2 innies R3C59 = 2 outies R4C18 + 4
43a. R4C1 = {68}, R4C8 = {12} -> R4C18 = 7,8,9,10 -> R3C59 = 11,12,13,14 = [29/38/39/58/85/59] (cannot be {57} which clashes with R3C7), no 7 in R3C59
43b. 7 in N3 only in R123C7, locked for C7

44. 20(4) cage at R2C8 (step 41a) = {1379/1568/2378}, R357C7 (step 42) = {127/145/235}, R3C59 = 11,12,13,14 = [29/38/39/58/85/59] (step 43a)
44a. Consider placements for R3C4 = {46}
R3C4 = 4 => R3C2 = 3 => R3C6 = 2 => R3C59 = [58/85/59], 5 locked for R3 => R3C7 = 7 => R57C7 = {12}, naked pair {12} in R4C8 + R5C7, locked for N4, min R45C9 = {35} = 8 => max R3C9 = 8, R3C8 = 9 (hidden single in R3) => 20(4) cage = {1379} => R4C8 = 1
or R3C4 = 6 => R3C1 = 7 => R3C7 = 5 => 20(4) cage = {1568} => R4C8 = 1
-> R4C8 = 1
44b. R1C9 = 1 (hidden single in N3)

45. 24(4) cage at R6C9 (step 38b) = {1689/2589/2679/3579/3678}
45a. R7C7 = {123} -> no 2,3 in R7C89

46. R7C6 = 4 -> 18(4) cage at R7C6 = {1458/1467/3456} (cannot be {2457} because no 2,5,7 in R9C7), no 2 in R8C7

47. 45 rule on N3 1 remaining outie R1C6 = 1 innie R3C9, R3C9 = {589} -> R1C6 = {589}

48. 16(3) cage at R3C9 = {259/268/358} (cannot be {367} because R3C9 only contains 5,8,9), no 7
48a. 14(3) cage at R5C7 = {248/257/347} (cannot be {356} which clashes with 16(3) cage), no 6
48b. 4 of {347} must be in R5C7 -> no 3 in R5C7
48c. 8 of {248} must be in R6C8 -> no 8 in R5C8
48d. R5C14 = R6C8 (step 4a)
48e. {257} must be [257] (cannot be [275] which clashes with R5C12 = [23], killer combo clash) -> no 5 in R6C8
[Cracked, at long last. The rest is fairly straightforward.]

49. R357C7 (step 3) = {127/145/235}
49a. 1,3 only in R7C7 -> R7C7 = {13}

50. R7C5 = 2 (hidden single in R7), clean-up: no 9 in R7C4
50a. Naked pair {38} in R78C4, locked for C4 and N8
50b. 17(3) cage at R8C5 = {179} (only remaining combination), locked for N8, 9 also locked for C5 -> R9C4 = 5, R8C6 = 6
50c. R78C6 = [46] = 10 -> R89C7 = 8 = [53], R89C1 = [34], R78C4 = [38], R8C3 = 9, R9C3 = 8 (cage sum)
50d. R9C5 = 9 (hidden single in N8)
50e. R7C7 = 1, R7C1 = 5
50f. Naked pair {67} in R7C23, locked for R7 and N7 -> R8C2 = 1, R6C2 = 5 (cage sum), R8C5 = 7, R9C6 = 1

51. 22(3) cage at R4C2 = {679} (only remaining combination), locked for N4 -> R4C1 = 8

52. R3C6 = 2 (hidden single in R3) -> R2C6 = 9, R12C1 = [97], R3C1 = 6, R3C24 = [34]

53. R3C7 = 7, R12C7 = {68}, locked for C7, N3 and 19(3) cage at R1C6 -> R1C6 = 5, R3C5 = 8, R3C8 = 9

54. R3C9 = 5 -> R45C9 = 11 = [38]
54a. R6C8 = 7, R5C8 = 5 (hidden single in N6) -> R5C7 = 2 (cage sum)

and the rest is naked singles.
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