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PostPosted: Sun Jul 06, 2008 10:38 pm 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Old SSv3.2.1 scores:
Killer rating table      
Rounded Score from SSv3.2.1
! = 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.66v1.5 1.75 1.65|TransX 1.75 2.45|A.68v3 3 !(t&E)5.45|
|A.67 1.00 1.05|TransX-L 1.25 0.95|A.68v1.5 1.25 1.10|
|Vortex 1.75 |A.68 1.25 1.05|A.69 1.25 1.20|
|Vort-Lite 1.25 |A.68v2 1.75 1.30|A.69v1.5 E1.50 !1.60|
|====================================================================|
page #5
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.66v1.5 1.75 !1.45|TransX H1.5 !2.80|A.68v3 3 !(t&E)7.30|
|A.67 1.00 !1.20|TransX-L 1.25 1.00|A.68v1.5 1.25 !1.25|
|Vortex 1.75 1.70|A.68 1.25 1.05|A.69 1.25 1.15|
|Vort-Lite 1.25 0.95|A.68v2 1.75 !1.45|A.69v1.5 E1.50 !1.90|
|====================================================================|
page #5
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|
|-----------------------------+-----------------------------+-----------------------------|
|A66v1.5 mhp 1.75 1.35|TransX Para H1.50 2.20|A.68v3 Ruud 3.00 5.10|
|A.67 Ruud 1.00 1.15|TranX-L Para 1.25 1.15|A68v1.5 mhp 1.25 1.20|
|Vortex mhp 1.75 1.45|A.68 Ruud 1.25 1.15|A.69 Ruud 1.25 1.35|
|V-Lite mhp 1.25 1.10|A.68v2 Ruud 1.75 1.35|A69v1.5 mhp E1.50 1.70|
|=========================================================================================|
page #5


Assassin 66 v1.5
by mhparker (Sep 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3072:3072:3072:4355:4355:3589:2822:2822:2822:2825:2825:5131:4355:3589:3589:2063:5904:5904:7698:2825:5131:5131:2070:2063:2063:5904:4890:7698:4636:4636:4636:2070:4384:4384:4384:4890:7698:2853:2853:5159:5159:5159:2858:2858:4890:7698:2094:2094:2094:2865:4658:4658:4658:4890:7698:3383:2872:2872:2865:3643:3643:5693:4890:3383:3383:2872:3906:3906:3652:3643:5693:5693:4680:4680:4680:3906:3652:3652:2894:2894:2894:
Solution:
+-------+-------+-------+
| 6 2 4 | 9 5 8 | 7 1 3 |
| 1 7 9 | 3 2 4 | 5 8 6 |
| 8 3 5 | 6 7 1 | 2 9 4 |
+-------+-------+-------+
| 7 4 6 | 8 1 5 | 9 3 2 |
| 2 8 3 | 7 4 9 | 6 5 1 |
| 9 5 1 | 2 3 6 | 8 4 7 |
+-------+-------+-------+
| 4 9 2 | 1 8 7 | 3 6 5 |
| 3 1 8 | 5 6 2 | 4 7 9 |
| 5 6 7 | 4 9 3 | 1 2 8 |
+-------+-------+-------+
Quote:
CathyW: Edit: Definitely much harder - at least a 1.5. ..Edit 2: ... scratch Rating up to 2 probably. Either needs hypotheticals or I'm missing something....Edit 3: Give up ..just checked through Para's WT. Nothing short of brilliant imho!
Para: That was a good puzzle somewhere between 1.50 and 1.75. Some fancy combination work and/with 45-tests did the trick.
Andrew: I would rate A66V1.5 as a solid 1.75; possibly a bit higher for my solving path with 1.75 for Para's slightly more direct route.
Forum 2021 revisit to this puzzle here
Walkthrough by Para:
Hi all

That was a good puzzle somewhere between 1.50 and 1.75. Some fancy combination work and/with 45-tests did the trick.
Enjoy :lol:

Walk-Through Assassin 66V1.5

1. 11(3) at R1C7, R2C1, R7C3 and R9C7 = {128/137/146/236/245}: no 9

2. 20(3) at R2C3 and R5C4= {389/479/569/578}: no 1,2

3. 8(3) at R2C7 = {125/134}: no 6,7,8,9

4. 23(3) at R2C8 = {689} -->> locked for N3

5. R34C5 = {17/26/35}: no 4,8,9

6. 11(2) at R5C2, R5C7 and R6C5 = {29/38/47/56}: no 1

7. 8(3) at R6C2 = {125/134}: no 6,7,8,9; 1 locked for R6

8. 22(3) at R7C8 = {589/679}: no 1,2,3,4; 9 locked for N9
8a. 9 in C7 locked for N6

9. 45 on R1: 3 innies: R1C456 = 22 = {589/679}: no 1,2,3,4; 9 locked for R1 and N2
9a. 45 on R1: 3 outies: R2C456 = 9 = {126/135/234}: no 7,8

10. 45 on N1: 3 innies: R23C3 + R3C1 = 22 = {589/679}: no 1,2,3,4
10a. 45 on N1: 1 innie and 1 outie: R3C1 = R3C4 + 2: R3C4: no 8
10b. 8 in N2 locked within R1C456: R1C456 = {589} -->> locked for R1 and N2
10c. 11(3) at R1C7 = {137}(last combo) -->> locked for R1 and N3
10d. R1C123 = {246} -->> locked for N1
10e. 11(3) at R2C1 = {137}(last combo) -->> locked for N1
10f. Clean up: R2C456 = {126/234}: 2 locked for R2 and N2; R3C4: no 4

11. 45 on N3: 1 innie and 1 outie: R3C6 + 3 = R3C9 -->> R3C6 = 1; R3C9 = 4(only possible combo)
11a. R23C7 = [52]
11b. Clean up: R2C456 = {234} -->> locked for R2 and N2
11c. R3C2 = 3(hidden)
11d. Clean up: R3C1: no 5
11e. R3C3 = 5(hidden)
11f. 6 in R2 locked for N3
11g. More Clean up: R4C5 = {12}; R5C2: no 6; R5C3: no 8; R5C8: no 6

12. 45 on R1: 1 innie and 1 outie: R2C4 + 5 = R1C6 -->> R1C6: no 5; R2C4: no 2

13. 45 on R5: 2 innies: R5C19 = 3 = {12} -->> locked for R5
13a. Clean up: R5C237: no 9
13b. 9 in R5 locked for N5
13c. Clean up: R7C5: no 2

14. 19(5) at R3C9 = 4{1257/1356}: no 8; 1,5 locked for C9
14a. Killer Pair {37} in R1C9 + 19(5) at R3C9 -->> locked for C9

15. 30(5) at R3C1 = {24789/25689}: {15789} blocked by R2C1, {34689/35679/45678} blocked by R5C1 -->> no 1,3; R5C1 = 2; 8,9 locked for C1
15a. R5C9 = 1
15b. 3 in C1 locked for N7
15c. Killer Pair {46} in R1C1 + 30(5) at R3C1 -->> locked for C1

16. 45 on N7: 1 innie and 1 outie: R7C1 = R7C4 + 3 -->> R7C4: no 7,8

17. 45 on N9: 1 innie and 1 outie: R7C6 = R7C9 + 2 -->> R7C6: no 2,3,6

18. 45 on R9: 1 innie and 1 outie: R8C6 + 2 = R9C4 -->> R9C4: no 1,2,3; R8C6: no 8,9

19. 45 on R4: 3 innies: R4C159 = 10 = [415/523/613/712] -->> R4C1: no 8,9; R4C9: no 6,7

20. 14(3) at R7C6 = [9]{14}/[5]{18}/[7]{16}/{347}: [5]{36} blocked by step 17 -->> R7C6: no 8
20a. Clean up: R7C9: no 6

21. 45 on N78: 3 innies: R7C156 = 19 = {7[3]9/{469/478/68[5]} -->> R7C15: no 5
21a. Clean up: R7C4: no 2

22. 45 on N8: 3 innies: R7C456 = 16 = [169]/[187]/{349}/[385]/{36}[7]/{457}

23. Combining R7C456 with I/O N7 and N9
23a. R7C14569 = [41697/41875/96375/85742/63497/63942] -->> R7C1: no 7; R7C4: no 4; R7C6: no 5; R7C9: no 3

24. 45 on N7: 3 innies: R7C1 + R78C3 = 14 = [4]{28}/[6]{17}/[8]{24}/[9]{14} -->> R78C3: no 6

25. 18(3) at R9C1 = {369/378/459/567}: {189/279/468} blocked by innies N7: no 1,2

26. 13(3) at R7C2 = {139/157/238/256/346}: {148/247} blocked by innies N7

27. Combining steps 23 and 24: R7C14569 = [85742] clashes with R7C1 + R78C3 = [8]{24}
27a. R7C1 + R78C3 = [4]{28}/[6]{17}/[9]{14} = {4|6..}: R7C1: no 8; R7C14569 = [41697/41875/96375/63497/63942]: R7C4: no 5; R7C5: no 7
27b. Clean up: R6C5: no 4,6

28. 14(3) at R7C6 = [9]{14}/[7]{16}/[4]{37}/[7]{34} -->> R78C7: no 8

29. R7C9 = {257}; 22(3) at R7C8 = {5|7..}/{6|8..}
29a. 11(3) at R9C7 = {128/146/236}: no 7 = {6|8..}: {245} blocked by R7C9 + 22(3) at R7C8
: no 5
29b. Killer Pair {68} in 22(3) at R7C8 + 11(3) at R9C7 -->> locked for N9

30. 45 on R6: 3 innies: R6C159 = 19 = [982]/[9]{37}/[487]/{568} -->> R6C1: no 7; R6C5: no 2
30a. Clean up: R7C5: no 9
30b. R7C14569 = [41697/41875/96375/63497]: R7C6: no 4; R7C9: no 2

31. 14(3) at R7C6 = [9]{14}/[7]{34} -->> R78C7 = {14/34}: no 7, 4 locked for C7 and N9
31a. Clean up: R5C8: no 7

32. 11(3) at R9C7 = {128/236} = {3|8..} -->> 2 locked for R9
32a. 18(3) at R9C1 = {369/459/567} = {4|6..}: {378} blocked by 11(3) at R9C7
32b. Killer Pair {46} in 18(3) at R9C1 + Innies N7(step 27a) -->> locked for N7

33. 2 in N8 locked in R8C456 for R8
33a. 45 on R9: 3 outies: R8C456 = {238/247/256}: no 1,9

34. 30(5) at R3C1 = 2{4789/5689}: 7 only in R4C1 -->> R4C1: no 4
34a. R4C159 = 10 = [523/613/712]: R4C9: no 5

35. 19(5) at R3C9 = 14(257/356}: One of {23} goes in R4C9 -->> R6C9: no 2,3

36. R6C159 = 19 = [937/487]/{568} -->> R6C5: no 7
36a. Clean up: R7C5: no 4

37. R7C14569 = [41697/41875/96375] -->> R7C1: no 6; R7C4: no 3

38. R7C1 + R78C3 = [4]{28}/[9]{14}: no 7 -->> 4 locked for N7

39. 18(3) at R9C1 = {369/567}: no 8 -->> 6 locked for R9

40. 11(3) at R9C7 = {128}(last combo): no 3 -->> locked for R9 and N9

41. 22(3) at R7C8 = {679}(last combo): no 5 -->> locked for N9
41a. R7C9 = 5; R7C6 = 7
41b. R8C8 = 7(hidden); R7C4 = 1(hidden); R4C5 = 1(hidden); R3C45 = [67]
41c. R7C15 = [48](step 37); R6C5 = 3

42. 8(3) at R6C2 = {125}(last combo): no 4

And the rest is all naked and hidden singles

Andrew a welcome to the Masters from the Grandmaster :wink: :lol:

greetings

Para
Brief notes solution by gary w & extra clarification by mhparker:
gary w:
i haven't got time to post a complete wt but my solution to a66 v1.5 was along the following line...hope you can follow it.

various easy prelims include showing r2c12=1/7 therefore r5c1=2 and r5c9=1 because of combos in 30/5 cage

r7c1 is not a 9 otherwise both r3c4 and r7c4 would be a 6

r67c1 + r67c9 = 25 r7c159 = 17

in c9 2 is either in the 19/5 cage (at r4c9) or at r9c9 but if the latter then r67c9 = 65 (only combo that works for the 19/5 cage) which then >> r67c1 = 86

but then there are two 6s in r7c159

thus the 19/5 cage in c9 is (r3-r7) 421 5/7... this breaks the puzzle



couldn't solve 65v3 without a hypothetical but it wasn't TOO hard to deduce the number that has to go in r7c9 after which no more hypos were needed.Has anyone solved it logically yet?

both a66 and a67 took me about 1.5-2 hrs to do.A "deadly" killer in The Times normally takes me about 0.5 hrs so i'ld rate both of these as significantly harder

...I appreciate my brief notes are difficult to follow.Essentially if the 19/2 cage doesn't contain a 2 it must be (r3-r7) 4 (356) 1 (356)(356).But all these can be eliminated using very short chains based on

r7c1 is not 9
r67c1+ r67c9=25
r7c159=17
r7c6=r7c9+2
and the numbers in the 11/2 cage at r67c5 which fix r3c5 as either a 6 or 7 and thus r3c1 aseither an 8 or a 9.

No guesswork or pencil marks needed.Does this,then,constitute an AIC ???

mhparker:
Hi Gary,

I've been doing some more thinking about your moves in the light of your last post, which I'd like to clarify a bit more here so that I (and other forum members) can better understand them. To make things easier, I'll refer to the following grid state, which corresponds to how far SumoCue v1.3 gets if one manually sets r5c19 to [21] (due to logic you mentioned in your first post, corresponding to step 15 in Para's WT):

Code:
.-----------------------------------.-----------------------.-----------.-----------------------------------.
| 46          246         246       | 589         589       | 89        | 137         137         37        |
:-----------------------.-----------:           .-----------'           :-----------.-----------------------:
| 17          17        | 89        | 34        | 234         234       | 5         | 689         689       |
:-----------.           |           '-----------+-----------.-----------'           |           .-----------:
| 89        | 3         | 5           67        | 67        | 1           2         | 89        | 4         |
|           :-----------'-----------------------:           :-----------------------'-----------:           |
| 4567      | 1456789     1346789     12345678  | 12        | 2345678     346789      2345678   | 235       |
|           :-----------------------.-----------'-----------'-----------.-----------------------:           |
| 2         | 4578        3467      | 3456789     3456789     3456789   | 34678       34578     | 1         |
|           :-----------------------'-----------.-----------.-----------'-----------------------:           |
| 456789    | 145         134         1234      | 234578    | 2345678     346789      2345678   | 2567      |
|           :-----------.-----------------------:           :-----------------------.-----------:           |
| 46789     | 12456789  | 124678      13456     | 346789    | 4579        134678    | 56789     | 2357      |
:-----------'           |           .-----------'-----------+-----------.           |           '-----------:
| 1357        12456789  | 124678    | 123456789   12345678  | 234567    | 134678    | 56789       5689      |
:-----------------------'-----------:           .-----------'           :-----------'-----------------------:
| 1357        2456789     246789    | 456789    | 123456789   23456789  | 13468       1234568     2568      |
'-----------------------------------'-----------'-----------------------'-----------------------------------'


This corresponds roughly (but not exactly) to the position reached by Para after his step 21.

Note that {67} have been eliminated from r4c9 due to permutations on r4 innies (r4c159 = 10(3)) (Para's step 19). The 6 in r7c9 has been eliminated due to fact that r7c6 cannot be 8, otherwise cannot make up 14(3) cage total. Hence, via innie/outie difference n9, r7c9 can't be 6 (Para's step 20). Lastly, the 3 has gone from r6c9, because otherwise there would be no way of reaching the 19(5) cage total.

As you say, the next step is to remove the 9 from r7c1. Since my last post, I've realized that there is another way of doing this, namely by using innie/outie difference on n245689 (yes, 6 nonets!!), as follows:

Code:
2 outies (r37c1) = 2 innies (r37c4) + 5

Using this equation, r7c4 can't be a 6, because this would force r3c4 to be 7, resulting in the innies summing to 13. This would require that the outies sum to 18, which is clearly impossible, because they are peers of each other. Thus we can eliminate 6 from r7c4, which in turn eliminates 9 from n7 due to innie/outie difference on n7.

This results in the following grid state:

Code:
.-----------------------------------.-----------------------.-----------.-----------------------------------.
| 46          246         246       | 589         589       | 89        | 137         137         37        |
:-----------------------.-----------:           .-----------'           :-----------.-----------------------:
| 17          17        | 89        | 34        | 234         234       | 5         | 689         689       |
:-----------.           |           '-----------+-----------.-----------'           |           .-----------:
| 89        | 3         | 5           67        | 67        | 1           2         | 89        | 4         |
|           :-----------'-----------------------:           :-----------------------'-----------:           |
| 4567      | 1456789     1346789     12345678  | 12        | 2345678     346789      2345678   | 235       |
|           :-----------------------.-----------'-----------'-----------.-----------------------:           |
| 2         | 4578        3467      | 3456789     3456789     3456789   | 34678       34578     | 1         |
|           :-----------------------'-----------.-----------.-----------'-----------------------:           |
| 456789    | 145         134         1234      | 234578    | 2345678     346789      2345678   | 2567      |
|           :-----------.-----------------------:           :-----------------------.-----------:           |
| 4678      | 12456789  | 124678      1345      | 346789    | 4579        134678    | 56789     | 2357      |
:-----------'           |           .-----------'-----------+-----------.           |           '-----------:
| 1357        12456789  | 124678    | 123456789   12345678  | 234567    | 134678    | 56789       5689      |
:-----------------------'-----------:           .-----------'           :-----------'-----------------------:
| 1357        2456789     246789    | 456789    | 123456789   23456789  | 13468       1234568     2568      |
'-----------------------------------'-----------'-----------------------'-----------------------------------'


Now to apply your key move:

If 19(5) at r3c9 is {13456}, then r67c9 must be [63/65]. Since r67c19 = 25 (innies r6789), possible permutations for r67c19 (where "x" = r7c5) are:

Code:
9       6
7---x---3

8       6
6---x---5


However, both of these possibilities are blocked due to innies of r789 = r7c159 = 17(3), which would require x (i.e., r7c5) = 7 in the first case and x = 6 in the second. In both cases, r7c5 would clash with r7c1. Thus, we can reject the {13456} combo for 19(5) at r3c9, which must now be {12457}.

Nice creative move! Congratulations! :D

Apart from my alternative way of stripping the 9 from r7c1, does this correspond more or less to the logic you were using?

Thanks once again for providing this move sequence. Keep up the good work! BTW, do you do your puzzles by keeping close track of the remaining candidates using software, or do you do them all on paper, using only rough pencilmarks? In any case, 30 minutes for a Times Deadly is pretty good going, by any means! I used to take about 50 minutes for them on average, although it's a long time ago since I did the last one. (Cathy: if you're reading this, how long do you take to do them?).
Walkthrough by Andrew:
In the past few weeks I've been working on Assassin 66V1.5 and Vortex Killer as "background" jobs.

Vortex Killer was definitely the easier of these two puzzles.

I would rate A66V1.5 as a solid 1.75; possibly a bit higher for my solving path with 1.75 for Para's slightly more direct route. Ed currently has Vortex Killer as 1.75 and A66V.15 as 1.5. Maybe he should consider swapping those ratings?

I had been encouraged to keep going by Para's comment "Some fancy combination work".

We both used the same type of breakthrough in R7 with Para's solving path being more effective, possibly because he used R7C14569 together while I used R7C159 and R7C456 separately. That was clearly the key area which made it a narrow solving path that, together with the need to use steps together and "fancy combination work", made me assess it as a solid 1.75.

I missed two of his 45s, outies from R1 and innies for N78, which would have made it easier for me.


Here is my walkthrough for A66V1.5

1. R34C5 = {17/26/35}, no 4,8,9

2. R5C23 = {29/38/47/56}, no 1

3. R5C78 = {29/38/47/56}, no 1

4. R67C5 = {29/38/47/56}, no 1

5. 11(3) cage in N1 = {128/137/146/236/245}, no 9

6. 20(3) cage at R2C3 = {389/479/569/578}, no 1,2

7. 8(3) cage at R2C7 = 1{25/34}, CPE no 1 in R3C89

8. 23(3) cage in N3 = {689}, locked for N3

9. R5C456 = {389/479/569/578}, no 1,2

10. R6C234 = 1{25/34}, 1 locked for R6
10a. R6C678 = {279/369/378/468/567} (cannot be {459} which clashes with R6C234)

11. 11(3) cage at R7C3 = {128/137/146/236/245}, no 9

12. 22(3) cage in N9 = 9{58/67}, 9 locked for N9
12a. 9 in C7 locked in R456C7, locked for N6, clean-up: no 2 in R5C7

13. R34567C9 = {12358/12367/12457/13456}, 1 locked for C9

14. 45 rule on R1 1 innie R1C6 – 5 = 1 outie R2C4 -> R1C6 = {6789}, R2C4 = {1234}
[I missed 45 rule on R1 3 outies R2C456 = 9 which would have made the early stages a lot quicker.]

15. 45 rule on R9 1 innie R9C4 – 2 = 1 outie R8C6, no 1,2 in R9C4, no 8,9 in R8C6

16. R1C789 = {137/245}

17. R1C123 = {138/246} (cannot be {129/147/156/237/345} which clash with R1C789), no 5,7,9

18. 45 rule on R1 3 innies R1C456 = 22 = 9{58/67}
[Alternatively Killer quad 1,2,3,4 in R1C123 and R1C789, locked for R1 but the split 22(3) cage gives a bit more.]
18a. 9 in R1 locked in R1C456, locked for N2

19. 14(3) cage in N2 = {149/158/239/248/257/347/356} (cannot be {167} which clashes with R1C456)

20. 45 rule on R123 3 innies R3C159 = 19 = {289/379/469/478/568}, no 1, clean-up: no 7 in R4C5

21. 45 rule on R4 3 innies R4C159 = 10 = {127/136/145/235}, no 8,9

22. 45 rule on R5 2 innies R5C19 = 3 = {12}, locked for R5, clean-up: no 9 in R5C2378
22a. 9 in R5 locked in R5C456, locked for N5, clean-up: no 2 in R7C5
22b. R5C456 = 9{38/47/56}

23. 45 rule on N1 1 innie R3C1 – 2 = 1 outie R3C4, no 2,3,4 in R3C1, no 8 in R3C4

24. 45 rule on N1 3 innies R23C3 + R3C1 = 22 = 9{58/67}, no 3,4 in R23C3

25. Killer pair 6,8 in R1C123 and R23C3 + R3C1, locked for N1
25a. 11(3) cage in N1 = {137/245}

26. 45 rule on N3 1 innie R3C9 – 3 = 1 outie R3C6, no 3,5 in R3C6, no 2,3 in R3C9

27. 8(3) cage at R2C7 (step 7) = 1{25/34}
27a. No 2 in R3C6 because R23C7 = {15} would clash with R1C789, clean-up: no 5 in R3C9 (step 26)

28. 45 rule on N3 3 innies R23C7 + R3C9 = 11 = {137/245}
28a. 4 of {245} must be in R3C9 -> no 4 in R23C7

29. R3C159 (step 19) = {379/469/478} (cannot be {289/568} because R3C9 only contains 4,7), no 2,5, clean-up: no 3 in R3C4 (step 23), no 3,6 in R4C5

30. 45 rule on N2 3 innies R3C456 = 14 = {167/347} (cannot be {356} because R3C6 only contains 1,4) = 7{16/34}, no 5, 7 locked for R3 and N2 -> R3C9 = 4, R3C6 = 1, R3C45 = {67}, clean-up: no 6 in R1C6, no 2 in R2C4 (both step 14), no 2,5 in R1C789 (step 15), no 3 in R23C7 (step 7), no 5 in R4C5, no 3 in R9C4 (step 15)
30a. Naked pair {67} in R3C45, locked for R3 and N2

31. Naked triple {137} in R1C789, locked for R1, clean-up: no 8 in R1C123 (step 17)
31a. Naked triple {246} in R1C123, locked for N1, clean-up: no 7 in R2C3 (step 24), no 5 in 11(3) cage (step 25a) -> R3C2 = 3, clean-up: no 8 in R5C3
31b. Naked triple {589} in R1C456, locked for N2

32. R3C7 = 2 (hidden single in R3), R2C7 = 5, clean-up: no 6 in R5C8

33. R3C3 = 5 (hidden single in R3) , clean-up: no 6 in R5C2

34. 45 rule on N7 1 innie R7C1 – 3 = 1 outie R7C4, no 1,2,3 in R7C1, no 7,8 in R7C4

35. 45 rule on N7 3 innies R7C1 + R78C3 = 14

36. 45 rule on N9 1 outie R7C6 – 2 = 1 innie R7C9, no 2,6 in R7C6, no 8 in R7C9

37. 45 rule on N9 3 innies R78C7 + R7C9 = 12 = {138/147/237/246/345} (cannot be {156} which clashes with the 22(3) cage)
37a. 2 of {246} must be in R7C9 -> no 6 in R7C9, clean-up: no 8 in R7C6 (step 36)
37b. THEN R78C7 + R7C9 cannot be {246} because that would make 14(3) cage at R7C6 4{46} -> no 6 in R78C7
37c. R78C7 + R7C9 = {138/147/237/345}
37d. R9C789 = {128/146/236/245} (cannot be {137} which clashes with R78C7 + R7C9), no 7

38. 45 rule on R9 3 innies R9C456 = 16 = {169/178/259/349/358/367/457} (cannot be {268} which clashes with R9C789)

39. 45 rule on N8 3 innies R7C456 = 16 = {169/178/259/349/358/367/457} (cannot be {268} because R7C6 only contains 3,4,5,7,9)

40. R34567C1 must have 1,2 in R5C1 = {24789/25689} (cannot be {15789} which clashes with R2C1), no 1,3, 8,9 locked for C1 -> R5C1 = 2, R5C9 = 1, clean-up: no 3 in R7C6 (step 36)

41. Killer pair 4,6 in R1C1 and R34567C1, locked for C1

42. 3 in C1 locked in R89C1, locked for N7

43. R6C234 (step 10) = 1{25/34}
43a. 2 only in R6C4 -> no 5 in R6C4

44. R4C159 (step 21) = {127/136/145/235}
44a. 7 of {127} must be in R4C1 -> no 7 in R4C9
44b. 6 of {136} must be in R4C1 -> no 6 in R4C9

45. R34567C9 (step 13) = {12457/13456} = 145{27/36}, no 8, 5 locked for C9
45a. 6 of {13456} must be in R6C9 -> no 3 in R6C9

46. Killer pair 3,7 in R1C9 and R34567C9, locked for C9

47. 11(3) cage at R7C3 = {128/137/146/245} (cannot be {236} which must have 3 in R7C4; that would make R7C1 + R78C3 6{26})

48. R7C1 + R78C3 (step 35) = {149/158/167/248} (cannot be {257}; that would make 11(3) cage at R7C3 {27}2)

49. 13(3) cage in N7 = {139/157/238/247/256/346} (cannot be {148} which clashes with R7C1 + R78C3)

50. R9C123 = {279/369/378/459/567} (cannot be {189/468} which clash with R7C1 + R78C3), no 1

51. 14(3) cage at R8C6 = 2{39}/2{57}/3{29}/3{47}/4[19]/ 4{37}/5[18]/5{36}/ 6[17]/6{35}/7[16]/7{25}/7{34} (cannot be 2{48}/3{56}/5{27} because R9C4 is 2 more than R8C6 which would make R9C456 4{48}/5{56}/7{27}, cannot be 4{28} because R9C456 = 6{28} clashes with R9C9), no 8 in R9C5

52. 45 rule on R789 3 innies R7C159 = 17 = {269/278/359/368/458/467}
52a. Using steps 34 and 36 the following relationships are not allowed
R7C1 2 more than R7C9 which would make R7C1 the same as R7C6
R7C1 3 more than R7C9 which would make R7C4 the same as R7C9
R7C1 5 more than R7C9 which would make R7C4 the same as R7C6
R7C1 3 more than R7C5 which would make R7C4 the same as R7C5
R7C5 2 more than R7C9 which would make R7C5 the same as R7C6
52b. R7C159 = [467/485/647/692/872/935] (cannot be [593/683/782/845/863/953/962] which are all inconsistent with step 52a), no 5,7 in R7C1, no 5 in R7C5, no 3 in R7C9, clean-up: no 2,4 in R7C4 (step 34), no 5 in R7C6 (step 36), no 6 in R6C5
52c. R7C456 = [169/187/349/394/574/637], using steps 34 and 36
[The relationships in step 52a were meant to be indented but neither Tab nor a set of leading spaces did that. Very strange!]

53. 14(3) cage at R7C6 = {149/347}, no 8
53a. R78C7 contains 1/3 -> R9C789 (step 37d) must contain 1/3 = {128/146/236}, no 5
53b. R78C7 + R7C9 (step 37) = {147/237/345}

54. Looking at the interactions between R7C159, R7C456, the 11(3) cage at R7C3 and the 14(3) cage at R7C6
54a. R7C159 = [467], R7C456 = [169] clashes with R7C7 = {14}
54b. R7C159 = [485], R7C456 = [187], R78C7 = [34], R78C3 = [28]
54c. R7C159 = [647], R7C456 = [349], R78C7 = [14], R78C3 = [26]
54d. R7C159 = [692], R7C456 = [394], R78C7 = [73], R78C3 = [17]
54e. R7C159 = [872], R7C456 = [574], R78C7 = [37] clashes with R7C3 = {24}
54f. R7C159 = [935], R7C456 = [637], R78C7 = [43], R78C3 = [14]
54g. 4 locked in R7C1567, locked for R7
54h. 7 locked in R7C679, locked for R7, clean-up: no 4 in R6C5
54i. Other eliminations from steps 54a to 54f: no 8 in R7C1, no 6,8 in R7C3, no 5 in R7C4, no 6 in R7C5, no 1,2 in R8C3, no 1,7 in R8C7, clean-up: no 5 in R6C5
54j. R7C159 = [485/647/692/935], R7C456 = [187/349/394/637]

55. 22(3) cage in N9 (step 12) = 9{58/67}
55a. 7 of {679} must be in R8C8 -> no 6 in R8C8

56. R7C1 + R78C3 (step 48) = {149/167/248}
56a. 6 of {167} must be in R7C1 -> no 6 in R8C3

[At this stage I made a flawed step, which happened to lead to the correct solution. After a week away from this puzzle, I looked at this position again and found the following.]
57. 45 rule on R6 3 innies R6C159 = 19 = {289/379/478/568} (cannot be {469} because no 4,6,9 in R6C5)
57a. 2 of {289} must be in R6C9 -> no 2 in R6C5
57b. 9 of {379} and 4 of {478} must be in R6C1 -> no 7 in R6C1
57c. 5,6 of {568} must be in R6C19 and 4 of {478} must be in R6C1 -> no 8 in R6C1
57d. 3 of {379} must be in R6C5, 7 of {478} must be in R6C9 -> no 7 in R6C5
57e. Summary R6C1 = {4569}, R6C5 = {38}, R6C9 = {2567}, clean-up: no 4,9 in R7C5

58. R3C1 = 8 (hidden single in C1), R2C3 = 9, R3C4 = 6, R3C5 = 7, R3C8 = 9, R4C5 = 1, clean-up: no 4 in R8C6 (step 15)

59. R8C9 = 9 (hidden single in C9), R78C8 = {58/67}
59a. Killer pair 6,8 in R2C8 and R78C8, locked for C8, clean-up: no 3 in R5C7

60. Naked pair {38} in R67C5, locked for C5

61. 11(3) cage at R7C3 (step 47) = {128/137}, no 4
61a. 1 locked in R7C34, locked for R7, clean-up: no 9 in R7C6 (step 53), no 7 in R7C9 (step 36)

62. 14(3) cage at R7C6 = {347}
62a. 3 locked in R78C7, locked for C7 and N9

63. R9C789 (step 53a) = 1{28/46}
63a. 6 of {146} must be in R9C9 -> no 6 in R9C7

64. R7C456 (step 54j) = [187] (only remaining permutation) -> R7C19 = [45] (steps 34 and 36), R7C378 = [236], R7C2 = 9, R6C5 = 3, R8C7 = 4, R8C3 = 8, R8C8 = 7, R2C89 = [86], R1C123 = [624], clean-up: no 4 in R6C24 (step 10) -> R6C234 = [512], R46C1 = [79], R2C12 = [17], R6C8 = 4, R6C9 = 7, R1C9 = 3, R1C78 = [71], R49C9 = [28], R9C78 = [12], R9C2 = 6, R9C3 = 7, R9C1 = 5, R8C1 = 3, R8C2 = 1, R8C4 = 5

65. R5C4 = 7 (hidden single in R5) -> R5C56 = {49} (step 22b)
65a. Naked pair {49} in R5C56, locked for R5 and N5

and the rest is naked singles


Last edited by Ed on Sun Dec 28, 2008 9:06 am, edited 3 times in total.

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PostPosted: Sun Jul 06, 2008 10:42 pm 
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Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Assassin 67 by Ruud (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:3073:3073:3075:3075:5381:5381:4871:4871:2560:2570:3073:3340:3075:2318:5381:784:4871:3858:2570:3860:3340:4374:2318:3608:784:2842:3858:4380:3860:3340:4374:2318:3608:2594:2842:3858:4380:3860:3623:2600:3113:3608:2594:2842:2861:2094:4655:3623:2600:3113:3123:3892:4149:2861:2094:4655:3623:2874:3113:3123:3892:4149:2367:2624:4655:3394:2874:5700:3123:4166:4149:2367:2624:2624:3394:3394:5700:5700:4166:4166:
Solution:
+-------+-------+-------+
| 3 1 2 | 7 4 8 | 9 5 6 |
| 7 6 9 | 5 1 3 | 4 2 8 |
| 8 4 5 | 6 9 2 | 7 1 3 |
+-------+-------+-------+
| 1 9 3 | 2 8 4 | 5 6 7 |
| 6 8 7 | 9 3 5 | 2 4 1 |
| 2 5 4 | 1 7 6 | 3 8 9 |
+-------+-------+-------+
| 9 3 6 | 4 5 1 | 8 7 2 |
| 4 2 8 | 3 6 7 | 1 9 5 |
| 5 7 1 | 8 2 9 | 6 3 4 |
+-------+-------+-------+
Quote:
Howard S: This seemed straight forward - certainly easier than 66
Para: agree with Howard that this one was easier than last weeks. Probably a low 0.75 on the rating scale.
CathyW: I can't say I found it easier than A66 but would agree with the 0.75 rating
Andrew: I solved in one session, apart from a break for dinner, so that makes it easier than average Assassins over the last few months
Andrew: I'd put it between 0.75 and 1.0. Maybe when giving ratings we forget that Mike has defined typical recent Assassins as 1.25.
gary w: a67 took me about 1.5-2 hrs to do.A "deadly" killer in The Times normally takes me about 0.5 hrs so ....significantly harder
Walkthrough by Howard S:
This seemed straight forward - certainly easier than 66

1 Cage 17/2 C2 ={98}
2 => Cage 10/2 N1={64I73}
3 Rule 45 N1=> R3C13={94I85} (76 locked in Cage 10/2)
4 Rule 45 C6=> R189C6=24={987}
5 Cage 22/3 C67 = [{98I97}I{5I6}]
6 =>R1C6<>9 =>R12C7={94I86}
7 => Cage 19/3 N3={973I865}
8 Cage 3/2 N3 = {12}
9 => R3C79= 73 (54 not possible due to point 3 above)
10 => Cage 19/3 N3={865}
11 => R12C7={94} => R9C7=6 =>R89C6={97} =>R1C6=8
12 => Cage 10/2 N1={64} => R3C13={85}

Thereafter it is very easy with nothing special required



Let's hope no typos this week Rating 0.75 max
Walkthrough by Para:
Ruud wrote:
It will definitely score higher than last week's Assassin
I have to agree with Howard that this one was easier than last weeks. Probably a low 0.75 on the rating scale.


Walk-through Assassin 67

1. R12C1, R23C2, R45C8 and R56C5 = {19/28/37/46}: no 5

2. 21(3) at R1C6 = {489/579/678}: no 1,2,3

3. 19(3) at R1C8 = {289/379/469/578}: no 1

4. 9(3) at R2C6 = {126/135/234}: no 7,8,9

5. 11(3) at R3C9 = {128/137/146/236/245}: no 9

6. R67C1 and R78C5 = {29/38/47/56}: no 1

7. R67C2 = {17/26/35}: no 4,8,9

8. R67C8 = {69/78}: no 1,2,3,4,5

9. R89C1 = {18/27/36/45}: no 9

10. 10(3) at R8C2 = {127/136/145/235}: no 8,9

11. 22(3) at R8C6 = {589/679}: no 1,2,3,4

12. R23C8 = {12} -->> locked for C8 and N3
12a. R45C8 = {37/46}: no 8,9
12b. Killer Pair {67} in R45C8 + R67C8 -->> locked for C8

13. R34C5 = {89} -->> locked for C5
13a. R56C5 = {37/46} = {4|7..}: no 1,2
13b. R78C5 = {56}: {47} blocked by R56C6 -->> locked for C5 and N8
13c. R56C5 = {37} -->> locked for C5 and N5

14. R45C2 = {89} -->> locked for C2 and N4
14a. Naked Pair {89} in R4C25 -->> locked for R4
14b. Clean up: R23C2: no 1,2

15. 22(3) at R8C6 = {79}[6]/{89}[5] -->> R9C7 = {56}; R89C6 = {79/89} -->> locked for C6 and N8

16. 45 on C6: 1 innie and 1 outie: R9C7 + 2 = R1C6 -->> R1C6 = {78}
16a. Naked Triple {789} in R189C6 -->> locked for C6

17. 45 on N2: 2 innies and 2 outies: R1C6 + R3C5 = R4C46 + 11: R1C6 + R3C5 = 15/16/17 -->> R4C46 = 5/6 (4 not possible) = {14/15/24}: no 6; R1C6 + R3C5 = [79/89] -->> R3C5 = 9
17a. R4C5 = 8; R45C2 = [98]

18. 45 on N1: 2 innies: R3C13 = 13 = {58}: {67} blocked by R23C2 -->> locked for R3 and N1
18a. Clean up: R12C1: no 2

19. 12(3) at R1C4 needs 2 of {124} in R12C5 -->> 12(3) = [7]{14}/[6]{24}: R1C4 = {67}; R12C5 = {14/24} -->> 4 locked for C5 and N2

20. 45 on C4: 3 innies: R189C4 = 18 = [6]{48}/[7]{38} -->> R89C4 = {38/48}: no 1,2,7; 8 locked for C4 and N8
20a. R1C6 = 8(hidden); R9C7 = 6(step 16); R2C9 = 8(hidden)

21. 21(3) at R1C6 = 8{49}(last combo) -->> R12C7 = {49} -->> locked for C7 and N3

22. 19(3) at R1C8 = [56]8(last combo) -->> R1C89 = [56]
22a. R1C4 = 7; R3C2 = 4(hidden); R2C2 = 6
22b. Clean up: R2C1: no 3; R12C5 = {14} -->> locked for C5 and N2; R67C2: no 2; R8C1: no 3
22c. R9C5 = 2
22d. Clean up: R8C1: no 7; R89C4 = {38} -->> locked for C4 and N8

23. 14(3) at R5C4 = {149} (last combo) -->> locked for C4

24. 12(3) at R5C6 = {56}[1]/{26}[4] -->> R56C6 = {26/56} -->> 6 locked for C6 and N5
24a. R3C4 = 6(hidden); R3C8 = 1(hidden); R2C8 = 2; R2C46 = [53]; R3C6 = 2; R4C46 = [24]
24b. R7C46 = [41]

25. 45 on C9: 1 innies: R9C9 = 4
25a. 16(3) at R8C8 = 4{39}: R89C8 = {39} -->> locked for C8 and N9
25b. R45C8 = [64](last combo); R6C9 = 9(hidden); R56C4 = [91]

26. 16(3) at R6C9 = 9{25} (last combo) -->> R78C9 = {25} -->> locked for C9 and N9
26a. R8C7 = 1(hidden)

27. 14(3) at R3C7 = {257}(last combo) -->> R345C7 = [752]
27a. R78C7 = [38]; R78C8 = [87]; R56C5 = [37]; R67C2 = [53]; R56C6 = [56]; R3C9 = 3

28. R67C1 = [29](last combo)

And the rest is naked and hidden singles.

greetings

Para
Walkthrough by Andrew:
I agree with Howard and Para. Definitely easier than A66. I used less steps, slightly easier ones and had less notes about relationships below my working diagram for A67 than I did for A66. I'd put it between 0.75 and 1.0. Maybe when giving ratings we forget that Mike has defined typical recent Assassins as 1.25.

This was another one that I solved in one session, apart from a break for dinner, so that makes it easier than average Assassins over the last few months.

Here is my walkthrough.
Thanks Para for the comments and corrections.

1. R12C1 = {19/28/37/46}, no 5

2. R23C2 = {19/28/37/46} no 5

3. R23C8 = {12}, locked for C8 and N3

4. R34C5 = {89}, locked for C5

5. R45C2 = {89}, locked for C2 and N4, clean-up: no 1,2 in R23C2

6. R45C8 = {37/46}, no 5,8,9

7. R56C5 = {37/46}, no 1,2,5

8. R67C1 = [29/38]/{47/56}, no 1, no 2,3 in R7C1

9. R67C2 = {17/26/35}, no 4

10. R67C8 = {69/78}

11. R78C5 = {47/56}, no 1,2,3
[Para commented this should be R78C5 = {56} (only remaining combination, {47} clashes with R56C5), locked for C5. I ought to have spotted that. I saw it the other way round in step 20, only just after the preliminary steps, so this didn’t really affect the solution path.]

12. R89C1 = {18/27/36/45}, no 9

13. 21(3) cage at R1C6 = {489/579/678}, no 1,2,3

14. R234C6 = {126/135/234}, no 7,8,9

15. R345C9 = {128/137/146/236/245}, no 9

16. 10(3) cage at R8C2 = {127/136/145/235}, no 8,9

17. 22(3) cage at R8C6 = 9{58/67}
17a. CPE no 9 in R9C4

18. Naked pair {89} in R4C25, locked for R4

19. Killer pair 6,7 in R45C8 and R67C8, locked for C8

20. R56C5 (step 7) = {37} (cannot be {46} which clashes with R78C5), locked for C5 and N5, clean-up: no 4 in R78C5 = {56}, locked for C5 and N8
[Alternatively 45 rule on C5 3 innies R129C5 = 7 = {124}, locked for C5]

21. 22(3) cage at R8C6 (step 17) = 9{58/67}
21a. R89C6 = {79/89} -> R9C7 = {56}
21b. 9 locked in R89C6, locked for C6 and N8

22. 45 rule on C2 3 innies R189C2 = 10 = {127/145/235} (cannot be {136} which clashes with R67C2), no 6

23. 45 rule on C4 1 innie R1C4 – 5 = 1 outie R9C5 -> R1C4 = {679}
[Alternatively R1C4 comes from remaining combinations for 12(3) cage]

24. 45 rule on C7 1 outie R1C6 – 2 = 1 innie R9C7 -> R1C6 = {78}
24a. Naked triple {789} in R189C6, locked for C6

25. 45 rule on C7 3 innies R129C7 = 19 = {49}6/{68}5 (cannot be {58}6 which clashes with 21(3) cage at R1C6)
25a. R12C7 = {49/68}, no 5,7

26. 45 rule on C8 1 innie R1C8 – 1 = 1 outie R9C9 -> R9C9 = {23478}

27. 45 rule on N1 2 innies R3C13 = 13 = {49/58} (cannot be {67} which clashes with R23C2)
27a. Killer pair 8,9 in R3C13 and R3C5, locked for R3

28. 12(3) cage in N1 = {129/156/237/246} (cannot be {138} which clashes with all combinations for R23C2 together with R3C13, cannot be {147} which clashes with R23C2, cannot be {345} which clashes with R3C13), no 8

29. 45 rule on N3 2 innies R3C79 – 2 = 1 outie R1C6 -> R3C79 = 9 or 10 = {36/37/46} (cannot be {45} which clashes with R3C13), no 5
[Para pointed out that R3C79 cannot be {46} either because this clashes with R12C7. Well spotted! It’s so easy to forget which combinations remain from previous steps, particularly ones like this that don’t have a fixed total for these two cells. He added "Get's eliminated in 29a though" so it didn’t matter this time.]
29a. 5 in N3 locked in 19(3) cage = {568} (only remaining combination), locked for N3, 6 locked in R12C9, locked for C9, clean-up: R9C9 = {47} (step 26)

30. Naked pair {49} in R12C7, locked for C7 and N3 -> R1C6 = 8 (cage sum), R34C5 = [98], R45C2 = [98], R1C89 = [56], R2C9 = 8, R9C7 = 6 (step 24), R9C9 = 4 (step 26), clean-up: no 2 in R1C1, no 2,4 in R2C1, no 4 in R3C13 (step 26), no 9 in R6C8, no 3,5 in R8C1

31. R9C9 = 4 -> R89C8 = 12 = {39}, locked for C8 and N9, clean-up: no 7 in R45C8, no 6 in R6C8

32. R6C9 = 9 (hidden single in C9), R78C9 = 7 = {25} (only remaining combination), locked for C9 and N9
32a. R5C4 = 9 (hidden single in R5)

33. 1 in N9 locked in R78C7, locked for C7
33a. R678C7 = {138} (only remaining combination) -> R6C7 = 3, R78C7 = {18}, locked for N9 -> R7C8 = 7, R6C8 = 8, R3C79 = [73], R56C5 = [37], clean-up: no 3,7 in R2C2, no 4 in R6C1, no 1 in R6C2, no 4,8 in R7C1, no 1,5 in R7C2
33b. Naked pair {46} in R23C2, locked for C2 and N1, clean-up: no 2 in R67C2 = [53], no 6 in R7C1, no 6 in R8C1

34. 10(3) cage in N7 = {127} (only remaining combination), locked for N7, clean-up: no 8 in R89C1 = [45], R3C1 = 8, R3C3 = 5, R7C1 = 9, R6C1 = 2, clean-up: no 1 in R12C1
34a. Naked pair {37} in R12C1, locked for C1 and N1

35. Naked triple {127} in R9C235, locked for R9 -> R89C6 = [79], R89C8 = [93], R9C4 = 8
35a. R8C4 = 3 (hidden single in R8), R9C5 = 2 (cage sum), R1C4 = 7 (step 23), R12C1 = [37]
35b. R8C2 = 2 (hidden single in N7), R1C2 = 1, R9C23 = [71], R78C9 = [25], R78C5 = [56], R78C3 = [68], R78C7 = [81], R6C3 = 4, R45C3 = [37], R45C9 = [71], R45C1 = [16], R45C8 = [64], R12C5 = [41], R12C7 = [94], R12C3 = [29], R23C2 = [64], R23C8 = [21], R2C46 = [53]

36. R2C6 = 3 -> R34C6 = 6 = [24]

and the rest is naked singles

There were several hidden singles near the end and probably others that I missed which may have made the solution slightly more direct. No need to point them out to me unless I missed any before step 30


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PostPosted: Sun Jul 06, 2008 10:43 pm 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
[edit April 2011: my apologies to Mike for missing these next two puzzles from the archive. Thanks for noticing Andrew! Fortunately, the Google cache still shows page 1 of the original thread]

Vortex Killer by Mike (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver; 1-9 cannot repeat on the diagonals:
3x3:d:k:4096:3073:3073:4099:4099:4869:4869:4869:5128:7177:4096:3073:3073:2829:2829:4869:5128:4369:7177:7177:4096:2325:2325:2829:5128:4369:4369:7177:3868:3868:6942:2325:6942:3105:4369:3619:804:3868:3878:3878:6942:3105:3105:4907:3619:804:5678:3878:6942:3889:6942:4907:4907:5429:5678:5678:3128:3641:3889:3889:2876:5429:5429:5678:3128:5697:3641:3641:6212:6212:2876:5429:3128:5697:5697:5697:2892:2892:6212:6212:2876:
Solution:
+-------+-------+-------+
| 3 4 1 | 7 9 6 | 8 2 5 |
| 9 7 2 | 5 1 8 | 3 6 4 |
| 5 8 6 | 4 3 2 | 9 1 7 |
+-------+-------+-------+
| 6 3 7 | 9 2 1 | 4 5 8 |
| 2 5 8 | 3 4 7 | 1 9 6 |
| 1 9 4 | 8 6 5 | 7 3 2 |
+-------+-------+-------+
| 8 1 3 | 6 5 4 | 2 7 9 |
| 4 2 5 | 1 7 9 | 6 8 3 |
| 7 6 9 | 2 8 3 | 5 4 1 |
+-------+-------+-------+
Quote:
Mike, lead-in:(Est. rating: 1.75)...The 4-way rotational symmetry of this puzzle makes one feel dizzy just by looking at it. Miss a few critical tricks (especially a quick first placement), and it will suck you in. Mercilessly.
Cathy: (After finishing the Lite version) I'll leave Mike's original Vortex for the Masters!
Para: still busy tweaking, so will get to a Walk-through when i am done with that.
Andrew: I would rate Vortex Killer between 1.5 and 1.75
2021 forum Revisit here
Walkthrough by Para:
Hi all

I have to admit, this isn't the nicest walk-through i have ever written, but couldn't find the break-through move i used before. But here's a walk-through, i hope to improve on it though. Someone else have any thought, feel free to share.

Walk-through Vortex Killer

1. 12(4) at R1C2 = {1236/1245}: no 7,8,9; 1,2 locked within cage: R2C12: no 1,2

2. R1C45 = {79} -->> locked for R1 and N2

3. 20(3) at R1C9 = {389/479/569/578}: no 1,2

4. 28(4) at R2C1 = {4789/5689}: no 1,2,3; 8 locked within cage: R1C1: no 8

5. 11(3) at R2C5 = {128/146/236/245}

6. 9(3) at R3C4 = {126/135/234}: no 7,8,9

7. R45C9 = {59/68}: no 1,2,3,4,7

8. R56C1 = {12} -->> locked for C1 and N4

9. 15(3) at R4C2 = {348/357/456}: no 9

10. 19(3) at R5C8 = {379/478}: {289/469/568} blocked by R45C9: no 1,2,5,6; 7 locked for N6

11. 11(3) at R7C7 = {128/137/146/236/245}: no 9

12. R9C56 = {29/38/47/56}: no 1

13. 45-overlap on both diagonals: Total sum 5 cages = 86 -->> R5C5 = 90-86 = 4
13a. 11(3) at R7C7 = {128/137/236}: no 5
13b. Clean up: R9C6: no 7

14. 45 on D/: 2 outies: R4C4 + R6C6 = 14 = {59/68}: no 1,2,3,7

15. 45 on D\: 2 outies: R4C6 + R6C4 = 9 = {18/27/36}: no 5,9

16. 45 on N1: 2 outies: R2C4 + R4C1 = 11 = [29/38/47/56/65]: R2C4: no 1; R4C1: no 4
16a. 1 in 12(4) cage at R1C2 locked within R1C23 + R2C3 -->> locked for N1

17. 45 on N3: 2 outies: R1C6 + R4C8 = 11 = [29]/{38/56}: no 1,4 R4C8: no 2

18. 45 on N7: 2 outies: R6C2 + R9C4 = 11 = [92]/{38/47/56}: R9C4: no 1,9

19. 45 on N9: 2 outies: R6C9 + R8C6 = 11 = {29/38/56}/[47]: no 1; R8C6: no 4

20. 1 in D\ locked within 11(3) cage at R7C7: 11(3) = {128/137}: no 6; 1 locked for N9

21. 45 on R123: 3 outies: R4C158 = 13 = [526/625/715/823/913]: R4C5: no 3,5,6; R4C8: no 8,9
21a. Clean up: R1C6: no 2,3(step 17)

22. 45 on C123: R259C4 = 10 = [217/316/415/514/613]/{235}: R5C5: no 6,7,8,9; R9C4: no 8
22a. Clean up: R6C2: no 3(step 18)

Missed earlier
23. Killer Pair {89} in 14(2) at R4C9 and 19(3) at R5C8 -->> locked for N6
23a. Clean up: R8C6: no 2,3

24. 45 on C789: 3 outies: R158C6 = 22 = {589/679}: no 1,2,3,; 9 locked for C6: When {679}, R1C6 = 6: R58C6: no 6
24a. Clean up: R4C4: no 5(step 14), R6C9: no 5(step 19)

25. 1 locked in N6 for C7 and 12(3) cage at R4C7
25a. 12(3) = [9]{12}/[8]{13}/[7][41]/[5]{16}: R45C7: no 5

26. R158C6 = {589}/[6]{79} -->> R6C6 = [6]/{58} -->> 6 locked in R16C6 for C6; R1C1: no 6
26a. Clean up: R6C4: no 3; R9C5: no 2,5

27. 16(3) at R1C1 = [592]/[3]{67}: {358} blocked by R4C4 + R6C6(step 14): R2C2: no 3,5,8; R3C3: no 3,5,8,9

28. 12(4) at R1C2 = {1236/1245}, can’t have both {26} within N1 because of 16(3) cage at R1C1 -->> R2C4: no 3
28a. Clean up: R4C1: no 8

29. R259C4 = 10 = [217/613/415/514]/{235}: R9C4: no 6
29a. Clean up: R6C2: no 5

30. R259C4 = 10 = [217/613/415/514]/{235}: When R5C4 = 1 -->> R4C5 = 2, so either R4C5 = 2 or R259C4 = {235}: R3C4 + R6C4: no 2(sees all 4 cells)
30a. Clean up: R4C6: no 7

31. 7 in R4 locked for N4
31a. Clean up: R9C4: no 4

32. 7 in R4 locked within R4C123: R4C1 = 7 or 15(3) at R4C2 = {357} -->> R4C1: no 5
32a. Clean up: R4C2: no 6

33. 15(3) at R5C3 = [1]{59}/[1]{68}/[2][94]/[3][84]/[5][64]: [2]{58} blocked by 15(3) cage at R4C2: R56C3: no 3

34. 3 in N4 locked within 15(3) cage at R4C2: 15(3) = {348/357}: no 6

35. R4C158 = 13 = [625/715/913]: R4C8: no 6; Clean up: R1C6: no 5

36. Killer Triple {358} in R4C8 + R45C9 + 19(3) at R5C8 -->> locked for N6
36a. Clean up: R5C6: no 8; R8C6: no 8

37. 45 on N5: 4 innies: R46C5 + R5C46 = 18: Max R5C46 = 14: Min R46C5 = 4: R6C5: no 1,2

I know this is not nice, I will find something else later.
38. 45 on R6789: 2 outies and 3 innies: R5C18 + 6 = R6C346: Analysis: R6C4 = 7 -->> R6C2 = 4 -->> R9C4 = 7: R6C4: no 7
38a. Clean up: R4C6: no 2

39. 27(5) at R4C4 = 4{1589/3569} -->>R4C4 = 9; R6C6 = 5
39a. R1C45 = [79]; R1C1 = 3; R5C6 = 7; R8C6 = 9; R1C6 = 6(step 24)

40. R45C7 = [41](last combo)
40a. R56C1 = [21]; R5C4 = 3; R4C5 = 2(hidden); R4C6 = 1(hidden)
40b. R6C45 = [86]; R6C9 = 2; R1C9 = 5; R4C8 = 5(hidden)

41. R56C3 = [84](last combo in 15(3) at R5C3)

And the rest is all hidden and naked singles.

greetings

Para
Walkthrough by Andrew:
In the past few weeks I've been working on Assassin 66V1.5 and Vortex Killer as "background" jobs.

Vortex Killer was definitely the easier and more enjoyable of these two puzzles. Maybe it was easier for those of us who solved SKX4 earlier this year.

I would rate Vortex Killer between 1.5 and 1.75. Ed currently has Vortex Killer as 1.75 and A66V.15 as 1.5. Maybe he should consider swapping those ratings?

Thanks Mike for some excellent comments.

Here is my walkthrough for Vortex Killer

1. R1C45 = {79}, locked for R1 and N2

2. R45C9 = {59/68}

3. R56C1 = {12}, locked for C1 and N4

4. R9C56 = {29/38/47/56}, no 1

5. 9(3) cage at R3C4 = {126/135/234}, no 7,8,9

6. 20(3) cage in N3 = {389/479/569/578}, no 1,2

7. 15(3) cage at R4C2 = {348/357/456}, no 9

8. 19(3) cage in N6 = {379/469/478/568} (cannot be {289} which clashes with R45C9), no 1,2
[Para also had {469/568} clash with R45C9, giving 7 locked for N6. Missed them! Fortunately it didn’t have a significant effect on the solving path.]
8a. Killer pair 8,9 in R45C9 and 19(3) cage, locked for N6
[This shouldn’t really be here, because it’s not a preliminary step, but I’ve moved it after Mike pointed out that I had two step 15s.]


9. 11(3) cage in N9 = {128/137/146/236/245}, no 9

10. 12(4) cage at R1C2 = 12{36/45}
10a. CPE no 1,2 in R2C2

11. 28(4) cage at R2C1 = {4789/5689} = 89{47/56}
11a. CPE no 8 in R1C1

After the preliminary steps the first placement

12. 45 rule on D/ + D\, R5C5 counts toward both diagonals, total 86 -> R5C5 = 4, locked for D/ and D\, clean-up: no 7 in R9C6

13. 45 rule on D/ 2 outies R4C4 + R6C6 = 14 = {59/68}

14. 45 rule on D\ 2 outies R4C6 + R6C4 = 9 = {18/27/36}, no 5,9

15. 16(3) cage in N1 = {259/268/358/367} (cannot be {169} which clashes with R4C4 + R6C6, cannot be {178} because R1C1 only contains 3,5,6), no 1
15a. 2 of {259} must be in R3C3 -> no 9 in R3C3
15b. 1 in N1 locked in R1C23 + R2C3 of 12(4) cage -> no 1 in R2C4

16. 1 in D\ locked for 11(3) cage = 1{28/37}, no 5,6, 1 locked for N9

17. 16(3) cage in N1 (step 15) = {259/268/367} (cannot be {358} which clashes with 11(3) cage in N9)
17a. 2 of {268} must be in R3C3 -> no 8 in R3C3
17b. 5 of {259} must be in R1C1 -> no 5 in R2C2 + R3C3

18. 12(3) cage in N7 = {129/138/156/237}
18a. 9 in {129} must be in R9C1 -> no 9 in R8C2 + R7C3

19. 45 rule on N1 2 outies R2C4 + R4C1 = 11 = [29/38/47/56/65], no 4 in R4C1

20. 45 rule on N3 2 outies R1C6 + R4C8 = 11 = [47/56/65/83], no 1,2,3 in R1C6, no 1,2,4 in R4C8

21. 45 rule on N7 2 outies R6C2 + R9C4 = 11 = {38/47/56}/[92], no 1,9 in R9C4

22. 45 rule on N9 2 outies R6C9 + R8C6 = 11 = [29/38]/{47/56}, no 1 in R6C9, no 1,2,3 in R8C6

23. 1 in N6 locked in R45C7, locked for C7 and 12(3) cage at R4C7 -> no 1 in R5C6
23a. 12(3) cage at R4C7 = 1{29/38/47/56}
23b. 7 of {147} must be in R5C6 -> no 7 in R45C7
23c. 8,9 only in R5C6 -> no 2,3 in R5C6

24. 45 rule on R123 3 outies R4C158 = 13 = {139/157/238/256}
24a. 1,2 only in R4C5 -> R4C5 = {12}

25. 45 rule on C123 3 outies R259C4 = 10 = {127/136/145/235}, no 8,9, clean-up: no 3 in R6C2 (step 21)
25a. 1 of {127/136} must be in R5C4 -> no 6,7 in R5C4

26. 45 rule on C789 3 outies R158C6 = 22 = 9{58/67}, no 4, 9 locked for C6, clean-up no 5 in R4C4 (step 13), no 7 in R4C8 (step 20), no 7 in R6C9 (step 22), no 2 in R9C5
26a. 6 of {679} must be in R1C6 -> no 6 in R58C6, clean-up: no 5 in R45C7 (step 23a), no 5 in R6C9 (step 22)

27. CPE 5 in D\ must be in R1C1 or R6C6 -> no 5 in R1C6, clean-up: no 6 in R4C8 (step 20)

28. R158C6 (step 26) = 9{58/67}
28a. 8 of {589} must be in R1C6 -> no 8 in R58C6, clean-up: no 3 in R45C7 (step 23a), no 3 in R6C9 (step 22)

29. R4C158 (step 24) = {139/157/238/256}
29a. 5 of {256} must be in R4C8 -> no 5 in R4C1, clean-up: no 6 in R2C4 (step 19)

30. 45 rule on N2 4 innies R1C6 + R2C4 + R3C45 = 18 = {1368/2358/3456} (cannot be {1458} because R3C45 cannot be {14/15}), 3 locked for N2
30a. 3,5 of {2358} must be in R3C45 -> no 2 in R3C45

31. 45 rule on N6 4 innies R4C78 + R5C7 + R6C9 = 12 = 12{36/45}
31a. Killer pair 5,6 in R4C78 + R5C7 + R6C9 and R45C9, locked for N6

32. 19(4) cage at R1C6 = {1369/1378/1468/1567/2368/2458/2467} (cannot be {1279/1459/2359/3457} because R1C6 only contains 6,8) [1/2]
33. 17(4) cage at R2C9 must contain 1/2 = {1349/1358/1367/1457/2348/2357/2456} (cannot be {1259/1268} which contain both 1 and 2)
[Steps 32 and 33 are included for completeness even though they didn’t lead to anything. I spent some time examining them because the 4-cell cage at R1C6 had proved to be a breakthrough area for Vortex Lite.]

34. 16(3) cage in N1 (step 17) = {259/268/367}
34a. If R3C3 = 2 => R2C4 = 2 => R4C1 = 9 (step 19) => R2C2 = 9 -> 16(3) cage in N1 cannot be {268} = {259/367}, no 8
34b. 8 in N1 locked in R2C1 + R3C12 -> no 8 in R4C1, clean-up: no 3 in R2C4 (step 19)
[Mike commented "BTW, another (alternative, not better) way of getting the same result is:
34. from step 26, R158C6 = 22 = 9{58/67}
34a. if {679}, R1C6 = 6
34b. if {589}, R6C6 = 6
34c. -> 6 locked in R16C6
34d. -> CPE: no 6 in R1C1
34e. -> {268} combo blocked for 16(3) cage in N1 (because none of these digits now available in R1C1)"
Then need to do my step 34b.
In one way Mike's version of this step is better because his step 34c can be extended as 34c. -> 6 locked in R16C6, not elsewhere in C6. Para had the extended version of Mike's step in his walkthrough.]


35. R259C4 (step 25) = {127/145/235} (cannot be {136} because R2C4 only contains 2,4,5), no 6, clean-up: no 5 in R6C2 (step 21)
[You may want to save this position to look at Mike’s comment after step 44.]

36. R1C6 + R2C4 + R3C45 (step 30) = {2358/3456} (cannot be {1368} because R2C4 only contains 2,4,5) = 35{28/46}, no 1, 5 locked for N2
36a. 11(3) cage in N2 = 1{28/46}
36b. Killer pair 6,8 in R1C6 and 11(3) cage, locked for N2
36c. 3 in N2 locked in R3C45, locked for R3

37. 45 rule on N8 2 innies R8C6 + R9C4 – 5 = 1 outie R6C5, min R8C6 + R9C4 = 7 -> min R6C5 = 2

38. 45 rule on N5 4 innies R4C5 + R5C46 + R6C5 = 18 = {1278/1359/2367} (cannot be {1269/2358} which clash with R4C4 + R6C6, cannot be {1368} because R5C6 only contains 5,7,9)
38a. 1,2 of {1278} must be in R4C5 + R5C4
38b. 2 of {2367} must be in R4C5
38c. Taking steps 38a and 38b together -> no 2 in R6C5
38d. 7 of {1278/2367} must be in R5C6 -> no 7 in R6C5

39. 45 rule on N124 2 innies R1C6 + R6C2 – 10 = 2 outies R4C5 + R5C4
39a. Min R4C5 +R5C4 = 3 -> min R1C6 + R6C2 = 13, no 4 in R6C2, clean-up: no 7 in R9C4 (step 21)

40. R259C4 (step 35) = {145/235} = 5{14/23}, 5 locked for C4
[Mike commented "This leaves R3C4 = {34} -> R3C4 and R259C4 form killer pair on {34} in C4. This would have been very useful (see comments after step 44), because it knocks out the 3 in R6C4."
I should have spotted that killer pair. I must have done that step fairly automatically having already eliminated the 6 from R259C4 in step 35.
BTW The 3 would have been eliminated from R6C4 by using the extended form of Mike’s step 34c.]


41. 45 rule on N4 2 innies R4C1 + R6C2 – 12 = 1 outie R5C4
41a. R4C1 + R6C2 + R5C4 = {67}1/[68]2/{69}3/[78]3/[98]5

42. 15(3) cage at R5C3 = {159/249/348/357/456} (cannot be {168/267} which clash with R4C1 + R6C2 + R5C4 (step 41a), cannot be {258} which clashes with 15(3) cage at R4C2 – step 7)
42a. 4 of {348/456} must be in R6C3 -> no 6,8 in R6C3

43. 45 rule on R789 3 outies R6C259 = 17 = {269/278/368/458/467} (cannot be {359} because R6C9 only contains 2,4,6)
43a. 3 of {368} must be in R6C5 => R6C6 = 5 when R6C56 = [35] clash with split-cage R4C5 + R5C46 + R6C5 = {1359}
43b. 3 of split-cage R4C5 + R5C46 + R6C5 = {2367} must be in R5C4
43c. Taking steps 43a and 43b together -> no 3 in R6C5
43d. R6C259 = {269/278/458/467}
43e. 2,4 only in R6C9 -> no 6 in R6C9, clean-up: no 5 in R8C6 (step 22)

44. Killer quint {56789} in R4C4 + R5C6 + R6C56 + (R4C6 + R6C4 which need one of 6,7,8) -> no 5 in R5C4
[Alternatively hidden killer triple {123} in R4C5 + R5C4 + (R4C6 + R6C4 which need one of 1,2,3) but I saw the quint first.]
[Mike commented "This candidate could have been eliminated much earlier (after step 35), as follows:
35a. CPE: R5C4 sees all 5's in R6 -> no 5 in R5C4
This could be followed up by:
35b. -> max. R5C4 = 3 -> no 3 in R56C3 (otherwise 15(3) cage sum unreachable)

Note also that the N5 innies (step 38) can only contain 2 of {123}, which (after step 35b above is applied), must go in R4C5 and R5C4 -> no 1,2,3 in R6C5.

Note that, after doing all this (including Mike’s addition to step 40), the 3 in R6 has been locked into R6C78 -> not elsewhere in N6 -> R4C8 = 5, R1C6 = 6."

Mike and Cathy are much better at spotting CPEs than I am.

I saw Mike's elimination of 3 from R56C3, after removing 5 from R5C4, but didn't include it in my walkthrough as it didn't immediately lead to anything; it's so easy to just move on to other moves that make more progress.]


45. R4C5 + R5C46 + R6C5 (step 38) = {1278/1359/2367}
45a. If {1278} => R5C4 = {12}, R5C6 = 7, R45C7 = [41] (step 23a) -> R5C4 + R5C7 clash with R5C1 -> cannot be {1278}
[This was there at step 38 but I didn’t see it then.]
45b. If {1359} => R4C5 = 1, R5C4 = 3, R5C6 + R6C5 = {59}, R45C7 = [61/21]
45c. If {2367} => R4C5 = 2, R5C4 = 3, R5C6 = 7, R6C5 = 6, R45C7 = [41]
45d. -> R5C4 = 3, R5C7 = 1, R6C5 = {569}, R56C1 = [21], clean-up: no 6,8 in R4C6, no 6 in R6C4 (both step 14), no 8 in R6C2 (step 21)
45e. R4C5 + R5C46 + R6C5 = {1359/2367} = 3{159/267}

46. R3C4 = 4 (naked single) -> R3C5 = 3 (hidden single in R3), R4C5 = 2, R5C6 = 7, R6C5 = 6 (both step 45e), R4C6 = 1, R6C4 = 8 (both locked for D/), R4C4 = 9, R6C6 = 5 (both locked for D\), R1C45 = [79], R8C6 = 9, R1C6 = 6 (step 26), [I forgot about R4C8 here, it’s in step 47], clean-up: no 5 in R5C9, no 5 in R9C5, no 2,8 in R9C6, no 2 in R2C4, no 7 in R4C1 (both step 19) -> R2C4 = 5, R4C1 = 6, R4C7 = 4, R6C9 = 2, R9C4 = 2, clean-up: no 3,6 in R1C23 + R2C3 (step 10), no 4,7 in R2C1 + R3C12 (step 11), R6C2 = 9 (step 21)
46a. R1C1 = 3 (naked single, locked for D\), R2C2 + R3C3 = {67} (step 34a, locked for D\)
46b. R1C9 = 5 (naked single, locked for D/), R2C8 + R3C7 = {69} (only remaining combination, locked for N3 and D/), clean-up: no 9 in R5C9
46c. R9C1 = 7 (naked single), R8C2 + R7C3 = {23}, locked for N7, R9C56 = [83], R2C5 = 1, R9C9 = 1, R7C7 + R8C8 = {28}, locked for N9

47. Naked pair {58} in R5C23, locked for R5 and N4 -> R5C8 = 9, R45C9 = [86], R2C8 = 6, R3C7 = 9, R2C2 = 7, R3C3 = 6, R3C9 = 7, R4C23 = [37], R4C8 = 5, R6C3 = 4, R5C3 = 8, R5C2 = 5

and the rest is naked singles


Vortex Lite by Mike (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver; 1-9 cannot repeat on the diagonals:
3x3:d:k:3840:4865:4865:2051:2051:4613:4613:4613:3080:5385:3840:4865:4865:5133:5133:4613:3080:7185:5385:5385:3840:3605:3605:5133:3080:7185:7185:5385:3612:3612:5918:3605:5918:2849:7185:2851:1828:3612:5414:5414:5918:2849:2849:3883:2851:1828:4398:5414:5918:4913:5918:3883:3883:3893:4398:4398:4408:3897:4913:4913:5692:3893:3893:4398:4408:3905:3897:3897:3396:3396:5692:3893:4408:3905:3905:3905:3916:3916:3396:3396:5692:
Solution:
+-------+-------+-------+
| 4 9 7 | 6 2 5 | 8 1 3 |
| 6 8 2 | 1 9 3 | 4 7 5 |
| 1 5 3 | 4 7 8 | 2 9 6 |
+-------+-------+-------+
| 9 7 4 | 5 3 6 | 1 8 2 |
| 2 3 8 | 7 1 4 | 6 5 9 |
| 5 1 6 | 9 8 2 | 7 3 4 |
+-------+-------+-------+
| 3 6 5 | 8 4 7 | 9 2 1 |
| 7 4 9 | 2 5 1 | 3 6 8 |
| 8 2 1 | 3 6 9 | 5 4 7 |
+-------+-------+-------+
Quote:
Mike, lead-in:(Est. rating: 1.25)
Andrew: at one time I was struggling a bit..a fun puzzle. No "advanced techniques"
Walkthrough by Cathy:
Vortex Lite - with thanks to Para for helping me see step 36:

Prelims:

a) 8(2) N2 = {17/26/35}
b) 20(3) N2: no 1,2
c) 28(4) @ r2c9 = {4789/5689} -> r12c8 <> 8,9
d) 7(2) N4 = {16/25/34}
e) 21(3) @ r5c3 = {489/579/678}
f) 11(3) @ r4c7: no 9
g) 11(2) @ r4c9 = {29/38/47/56}
h) 19(3) @ r6c5: no 1
i) 15(2) r9c56 = {69/78}
j) 22(3) N9 = {589/679} -> 9 n/e N9 and D\
k) 13(4) @ r8c6 = {1237/1246/1345} must have 1 -> r8c9 <> 1


1. Outies D\: r4c6 + r6c4 = 15 = {69/78}

2. Outies D/: r4c4 + r6c6 = 7 = {16/25/34}
-> r5c5 = 23 - 15 - 7 = 1
-> r4c4, r6c6 <> 6, r6c1 <> 6, r1c4 <> 7
-> 12(3) = {237/246/345} (no 8,9)

3. Outies N1: r2c4 + r4c1 = 10 = {19/28/37/46/55}

4. Outies N3: r1c6 + r4c8 = 13 = {49/58/67}

5. Outies N7: r6c2 + r9c4 = 4 = {13/22}

6. Outies N9: r8c6 + r6c9 = 5 = {14/23}

7. Outies r123: r4c158 = 20 - > r4c1 <> 1,2; r4c5 <> 2 -> r2c4 <> 8,9

8. Outies r789: r6c259 = 13 Max from r6c29 = 3+4 = 7 -> r6c5 min 6

9. Outies c123: r259c4 = 11 -> r5c4 <> 9 Min from r59c4 = 4+1 = 5 -> r2c4 max 6 -> r4c1 min 4

10. Outies c789: r158c6 = 10 = {127/136/145/235} (r1c6 <> 8,9) -> r4c8 <> 45
Min from r18c6 = 4+1 = 5 -> r5c6 max 5
Combos: [451/523/532/541/631/721] -> r8c6 <> 4 -> r6c9 <> 1

11. Innies N5: r46c5 + r5c46 = 22 = {2569/2578/3469/3478}

12. Killer combo 13(4) @ r8c6 and 22(3) N9:
22(3) = {589/679}
13(4) = {1237/1246/1345} -> one of 5,6,7 within N9
-> 15(4) @ r6c9 <> 5 -> 15(4) = {1248/1347/2346}

13. O-I N2: r1c6 + r2c4 - r4c5 = 3

14. O-I N4: r4c1 + r6c2 - r5c4 = 3

15. O-I N6: r4c8 + r6c9 - r5c6 = 8

16. O-I N8: r8c6 + r9c4 - r6c5 = -4 -> r6c5 - (r8c6+r9c4) = 4 -> r6c5 <> 6

17. HS r8c6 = 1 -> r6c9 = 4 -> r5c1 <> 3, r45c9 <> 7, r4c4 <> 3
-> If 28(4) = {4789}, r3c8 = 4 -> r3c8 <> 7

18. HS r7c9 = 1

19. r6c5 = (78) -> r4c6 + r6c4 = {69} n/e D/, N5

20. 12(3) N3 = {237/345} -> 3 n/e D/, N3 -> 17(3) N7 = {278/458} -> 8 n/e N7

21. Split 11(3) r259c4 = [182/173/452/542] -> r2c4 <> 2,3,6 -> r4c1 <> 4,7,8

22. 11(3) @ r4c7 = {128/137/146/236/245}
Combo analysis: r5c6 <> 5 -> r1c6 <> 4, r4c7 <> 7,8
-> r4c8 <> 9

23. 9 locked to 28(4) within N3 -> r12c7 <> 9
-> 9 locked to r1c23 n/e N1 -> 19(4) = 9{127/136/145/235} -> r1c23, r2c3 <> 8

24. 8 locked to r1c17 -> r7c7 <> 8

25. Killer combo: r7c8+r8c9 = {28/37}, 22(3) N9 = {589/679} -> r8c7, r9c78 <> 7

26. Grouped Turbot (1): r1c8 = r6c8 - r6c1 = r23c1 -> r1c23 <> 1

27. Grouped Turbot (8): r1c1 = r1c7 - r23c9 = r34c8 -> r8c8 <> 8 -> r9c9 <> 5

28. Combo analysis split 20(3) r4c158 -> r4c1 <> 6 -> r2c4 <> 4
-> split 11(3) r259c4 = [182/173/542] -> r5c4 <> 5

29. Grouped x-wing: 8 locked to r7c8, r89c9 in N9; 8 locked to 28(4) within c89
-> r45c9, r56c8 <> 8 -> r45c9 <> 3

30. If 28(4) = {4789} r4c8 = 7 -> r23c9 <> 7 (Can't have both 47 within N3 since 12(3) must have one of 4,7)

31. Killer combo: 11(2) r45c9 = {29/56}, r23c9 (of 28(4)) = {56/58/89} ({68} not possible in r45c9 as neither 5 or 9 in r4c8) Must have one of 5,9
-> r9c9 <> 9, r1c9 <> 5

32. From step 13, r4c5 <> 7 as can’t make 10 from r1c6 + r2c4

33. 1 locked to 18(4) within N3 -> 18(4) = 1{278/458/467}
If {1458}, r1c6 = 5 -> r1c78, r2c7 <> 5

34. Grouped Turbot (9): r4c6 = r6c4 - r6c7 = r7c7 -> r7c6 <> 9 -> r7c5 <> 2,3

35. Pointing pair: 7 in either r5c4 or r6c5 -> r6c3 <> 7

So near and yet so far. Don't know why I couldn't see the next step before. Confused

36. 12(3) N3 = {237} ({345} would block both options for 28(4) which must have one of 4,5 within N3.) -> 2,7 n/e N3 and D/
-> 17(3) N7 = {458} -> 4,5 n/e N7

37. Split 20(3) r4c158 = {389/479/578} ({569} not possible because r4c6 = {69} -> r4c8 <> 6
Andrew noted: r1c6 <> 7 as well

38. 17(4) @ r6c2 = {1367} (only remaining combo)
-> r6c2 = 1 -> r9c4 = 3, r9c3 = 1, r8c3+r9c2 = {29}

39. r6c5 = 8 (from O-I N8) -> r5c4 = 7 -> r2c4 = 1 -> r4c1 = 9
...

Relatively straightforward from here with cage combos and singles.
Phew! Smile
I'll leave Mike's original Vortex for the Masters!
Chain notation by Mike:
mhparker wrote:
Hi Cathy,

To answer your questions:

CathyW wrote:
...aren't the eliminations in common peer cells of the start and finish of the Turbot Chain?

True, but the emphasis is on the type of chain in this case.

CathyW wrote:
Did I get the notation correct?

Yes.

CathyW wrote:
In previous WTs these type of steps have been described in terms of If (a) then (c), if (b) then (c), either case (c).

Actually, it's a bit different to the way you stated it. Traditionally, people unfamiliar with AICs and Nice Loops will write something like:

  • if (a) then ... (b)
  • if (a) then ... not (b) => Contradiction!
  • Conclusion: (a) cannot be true
What's happening is this: One can consider the cell(s) being acted on as an additional node, joined to each end of the AIC by two weak links, thus transforming the chain into a Nice Loop. Because these two weak links are contiguous, the alternating nature of the AIC is broken, so the loop is discontinuous.

Many people intuitively prefer this loop approach (even though they may be unaware that it's a Nice Loop), because they find it more natural to start on a true premise than on a false premise, as is the case with AICs.

Because the loop is discontinuous, if one starts at the discontinuity (the cell(s) being acted on), and works one's way around the loop back to the start, one ends up with a contradiction. The chain proves that if X is true, then X must be false. The result being that X is false.

However, instead of working their way around the loop, back to the start, it's unfortunately common practice in walkthroughs for people to start at the discontinuity and branch out in both directions, meeting up in the middle of the loop somewhere. Whilst this will also prove a contradiction, it's considered less elegant than doing a complete cycle of the loop in one direction.

If you find yourself in this "either/or" situation, it's worth spending a couple of minutes to "normalize" the loop to standard form to just go in one direction. This is always possible.

As an example, consider your contradiction move in your Assassin 59 WT:

Quote:
25. Split 6(2) r3c46:
If [24] -> r1c7 = 2 -> CONFLICT: No place for 2 in N1


Marks pic:

Code:
.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 12346789    12346789  | 1234        56789     | 5678      | 1234        123       | 345789      345789    |
:-----------------------'-----------.           |           |           .-----------'-----------------------:
| 3456789     3456789     345678    | 123567    | 5678      | 1234      | 3456789     123456789   123456789 |
:-----------------------.-----------:           :-----------:           :-----------.-----------------------:
| 12345678    12345678  | 456789    | 25        | 789       | 14        | 12346789  | 123456789   123456789 |
:-----------.           |           :-----------'           '-----------:           |           .-----------:
| 2345678   | 3456789   | 456789    | 12356789    1234789     1235679   | 12346789  | 12345689  | 23456789  |
|           '-----------'-----------+-----------------------------------+-----------'-----------'           |
| 12345678    12345678    12345678  | 12356789    1234789     1235679   | 3456789     123456789   123456789 |
|           .-----------.-----------+-----------------------------------+-----------.-----------.           |
| 12345678  | 345679    | 4789      | 2356789     1234789     235679    | 2467      | 123567    | 12345678  |
:-----------'           |           :-----------.           .-----------:           |           '-----------:
| 1234568     1234568   | 3458      | 79        | 123       | 79        | 3468      | 1234568     1234568   |
:-----------------------'-----------:           :-----------:           :-----------'-----------------------:
| 123456789   123456789   12345678  | 124       | 2356      | 568       | 3456789     123456789   123456789 |
:-----------------------.-----------'           |           |           '-----------.-----------------------:
| 456789      456789    | 5678        1234      | 2356      | 568         234       | 12346789    12346789  |
'-----------------------'-----------------------'-----------'-----------------------'-----------------------'


This is a case of what I mean by going in two directions:

  1. r3c4 = 2 -> r3c12 <> 2
  2. r3c4 = 2 -> r12c6 <> 2 => r1c7 = 2 -> r1c123 <> 2
(Note: "->" = weak link; "=>" = strong link)

However, this can be reworked. The fact that knocking 2 out of two groups of aligned cells in N1 leaves no candidate positions for 2 in N1 should alert one to the fact that there is a strong link between these groups of cells in N1 on the digit 2. We can use this strong link to make up a complete circuit, and move round the loop in a constant direction, starting from (and finishing at) r3c4 as follows:

r3c4 = 2 -> r12c6 <> 2 => r1c7 = 2 -> r1c123 <> 2 => r3c12 = 2 -> r3c4 <> 2 (contradiction)

This is a Nice Loop with two weak links at the discontinuity. To "convert" this to an AIC, we can simply strip off the two weak links at the discontinuity, so that the resulting chain starts and ends with a strong link, as follows:

r12c6 <> 2 => r1c7 = 2 -> r1c123 <> 2 => r3c12 = 2

Converting this to Eureka notation, we have the AIC:

(2)r12c6=r1c7-r1c123=r3c12

In other words, if r12c6 does not contain a 2, then r3c12 must contain a 2, so we can remove a 2 from all common peers of r12c6 and r3c12, as at least one of the ends of the chain must be true.

Note that the AIC consists of two strong links (at either end), with a weak link in the middle, and some of the nodes consist of groups of multiple cells. Therefore, your move was actually a grouped Turbot fish!

It's well worth getting into the habit of trying to "normalize" contradiction moves in this way, if for no other reason than the fact that the WT sounds more impressive and less like T&E when using the correct lingo!

Hope this post has helped and not been totally confusing!
Walkthrough by Andrew:
mhparker wrote:
Vortex Lite Est. rating 1.25

and

Come on guys, the Vortex Lite is really no more difficult than a typical Assassin!

I'll agree with that although at one time I was struggling a bit. Then I had several restarts when I saw things that I ought to have spotted earlier and it became easier.

As Ruud said, a fun puzzle. Thanks Mike!

Here is my walkthrough for Vortex Lite. No "advanced techniques". Step 34 was combination crunching but not heavy stuff, the rest was normal killer moves.

Thanks Mike for the feedback on step 34. I've also done a bit of minor editing.

1. R1C45 = {17/26/35}, no 4,8,9

2. R45C9 = {29/38/47/56}, no 1

3. R56C1 = {16/25/34}, no 7,8,9

4. R9C56 = {69/78}

5. 20(3) cage in N2 = {389/479/569/578}, no 1,2

6. 11(3) cage at R4C7 = {128/137/146/236/245}, no 9

7. 21(3) cage at R5C3 = {489/579/678}, no 1,2,3

8. 19(3) cage at R6C5 = {289/379/469/478/568}, no 1

9. 22(3) cage at R7C7 = 9{58/67}, 9 locked for N9 and D\

10. 28(4) cage at R2C9 = {4789/5689} = 89{47/56}
10a. CPE no 8,9 in R12C8

11. 13(4) cage at R8C6 = {1237/1246/1345} = 1{237/246/345}, no 8,9
11a. CPE no 1 in R8C9

After the preliminary steps the first placement

12. 45 rule on D/ + D\, R5C5 counts toward both diagonals, total 89 -> R5C5 = 1, locked for D/ and D\, clean-up: no 7 in R1C4, no 6 in R6C1

13. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
13a.1 of {128/137} must be in R4C7 -> no 7,8 in R4C7

14. 45 rule on D/ 2 outies R4C4 + R6C6 = 7 = {25/34}

15. 45 rule on D\ 2 outies R4C6 + R6C4 = 15 = {69/78}

16. 12(3) cage in N3 = {237/246/345}, no 8,9

17. 15(3) cage in N1 = {258/267/348} (cannot be {357/456} which clash with R4C4 + R6C6)

18. 45 rule on N1 2 outies R2C4 + R4C1 = 10 = {19/28/37/46/55}

19. 45 rule on N3 2 outies R1C6 + R4C8 = 13 = {49/58/67}, no 1,2,3 in R1C6

20. R8C6 = 1 (hidden single in C6)
20a. 1 in N9 locked in R7C89, locked for R7 and 15(4) cage -> no 1 in R6C9
20b. R7C9 = 1 (hidden single in C9)
[It’s neat the way that fixing R8C6 leads directly to fixing R7C9!]

21. 45 rule on N7 2 outies R6C2 + R9C4 = 4 = [13/22]

22. 45 rule on N9 1 remaining outie R6C9 = 4, clean-up: no 9 in R1C6 (step 19), no 3 in R4C4 (step 14), no 7 in R45C9, no 3 in R5C1
22a. 15(4) cage at R6C9 = 14{28/37}, no 5,6
22b. 4 in N9 locked in 13(4) cage at R8C6 = 14{26/35}, no 7
[Alternatively killer pair 7,8 in R7C8 + R8C9 and 22(3) cage, locked for N9]

23. 28(4) cage at R2C9 (step 10) = {4789/5689}
23a. 4 of {4789} must be in R3C8 -> no 7 in R3C8

24. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
24a. 4 of {245} must be in R5C6 -> no 5 in R5C6

25. 17(3) cage in N7 = {269/278/359/458} (cannot be {368/467} which clash with R4C6 + R6C4)

26. 45 rule on R123 3 outies R4C158 = 20 = {389/479/569/578}, no 1,2, clean-up: no 8,9 in R2C4 (step 18)

27. 45 rule on R789 2 remaining outies R6C25 = 9, R6C2 = {12} -> R6C5 = {78}

28. 45 rule on C123 3 outies R259C4 = 11 = {128/137/236/245} (cannot be {146} because R9C4 only contains 2,3), no 9
28a. 7 of {137} must be in R5C4 -> no 7 in R2C4
28b. 6 of {236} must be in R5C4 -> no 6 in R2C4
28c. Clean-up: no 3,4 in R4C1 (step 18)

29. 45 rule on C789 2 remaining outies R15C6 = 9 = [54/63/72], clean-up: no 5,9 in R4C8 (step 19)

30. 28(4) cage at R2C9 (step 10) = 89{47/56}, 9 locked in R2C9 + R3C89 for N3
30a. 4/5 now in N3 -> 12(3) cage in N3 (step 16) = {237/246} (cannot be {345} which clashes with 28(4) cage) = 2{37/46}, no 5, 2 locked for N3 and D/

31. Killer pair 6,7 in 12(3) cage in N3 and R4C6 + R6C4, locked for D/

32. 5 on D/ locked in 17(3) cage, locked for N7
32a. 17(3) cage (step 25) = {359/458}
[Step 31 wasn’t actually needed but I saw it before I checked for an earlier step with the 17(3) cage.]

33. 45 rule on C1234 6 remaining outies R1348C5 + R46C6 = 25, min R46C6 = 8 -> max R1348C5 = 17, no 9
33a. R1348C5 = 14, 15, 16 or 17 = {2345/2346/2347/2348/2356/2357/2456}, 2 locked for C5

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. If {1467} with R1C6 = 6 => R1C78 + R2C7 = {147} clashes with 12(3) cage
34c. If {1467} with R1C6 = 7 => R1C78 + R2C7 = {146} => R2C9 + R3C89 => {589} => R4C8 = 6 => no 6 in R1C8
34d. -> 18(4) cage at R1C6 = {1458/1467} = 14{58/67}, no 3, 4 locked for N3, no 6 in R1C6 (steps 34a and 34b) no 6 in R1C8 (step 34c), clean-up: no 6 in 12(3) cage (step 30a), no 7 in R4C8 (step 19) , no 3 in R5C6 (step 29)
See discussion of this after the walkthrough

35. Naked triple {237} in 12(3) cage in N3, locked for D/ and N3, clean-up: no 8 in R4C6 + R6C4 (step 15), no 9 in 17(3) cage in N7 (step 32a)

36. 11(3) cage at R4C7 (step 6) = {128/146/236/245} (cannot be {137} because R5C6 only contains 2,4), no 7
36a. 7 in N6 locked in 15(3) cage = 7{26/35}, no 1,8,9

37. R4C7 = 1 (hidden single in N6)
37a. R5C67 = 10 = [28/46]

38. R1C8 = 1 (hidden single in C8), clean-up: no 7 in R1C5
38a. R9C8 = 4 (hidden single in C8)

39. Naked pair {68} in R4C8 + R5C7, locked for N6, clean-up: no 3,5 in R45C9

40. Naked pair {29} in R45C9, locked for C9 and N6, clean-up: no 8 in R7C8 (step 22a)

41. R3C8 = 9 (hidden single in N3)

42. R7C7 = 9 (hidden single in N7)

43. 28(4) cage at R2C9 = {5689}
43a. 5 locked in R23C9, locked for C9 and N3, clean-up: no 8 in R8C8 (step 9)

44. R4C8 = 8 (hidden single in C8), R5C7 = 6, R5C6 = 4, R1C6 = 5 (step 29), clean-up: no 3 in R1C45, no 2 in R89C7 (step 22b), no 2 in R4C4, no 3 in R6C6 (both step 14) -> R4C4 = 5, R6C6 = 2, 2,5 locked for D\, R6C2 = 1, no 5 in R5C1, no 3 in R6C1 -> R56C1 = [25], R45C9 = [29] , clean-up: no 8 in R9C9 (step 9) [Clean-up edited]
44a. R5C8 = 5 (hidden single in R5)
44b. R9C3 = 1 (hidden single in R9)

45. R3C7 = 2, R6C7 = 7 (hidden singles in C7), R6C5 = 8, R6C8 = 3, clean-up: no 7 in R9C6

46. R2C8 = 7, R7C8 = 2, R8C8 = 6, R1C9 = 3, R89C9 = [87], 6,7 locked for D\, clean-up: no 8 in R9C6

47. R5C4 = 7, R5C23 = [38], R4C5 = 3 (hidden single in N5)

48. R4C5 = 3 -> R3C45 = 11 = [47]
48a. R7C6 = 7 (hidden single in C6), R7C5 = 4, R7C3 = 5, R8C2 = 4, R9C1 = 8
48b. R8C5 = 5 (hidden single in C5), R89C7 = [35]

49. R7C4 = 8 (hidden single in C4), R8C4 = 2

and the rest is naked singles and a cage sum




Step 34. Mike commented that the logic was too complicated and should be broken down into smaller steps, for example

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1458}, 5 must be in R1C6 -> no 5 in R1C78+R2C7
34b. -> 5 in N3 locked in 28(4) at R2C9 = {5689} (no 4,7) -> no 6 in R12C8.
34c. Hidden killer pair on {47} in N3, as follows. 12(3) at R1C9 contains 1 of {47}. Only other place for {47} in N3 = 18(4) at R1C6 = {(4/7)..} = {1458/1467} (no 3) -> 4 locked in R1C78+R2C7 for N3
34d. Clean-up: no 6 in R1C6 (step 19), no 6 in 12(3) cage (step 30a), no 3 in R5C6 (step 29)


There is a reason why I missed his step 34a, which I ought to have seen. After typing step 34 with "must have 1" I used Ruud's combination calculator and forgot to enter that so I got {1368/1458/1467/3456}. When I was part way through my sub-steps I realised why {3456} wasn't allowed but forgot to go back and look at {1458} again, having already accepted that it was valid.

Mike's step 34c is a difficult one to find. You either spot it immediately or have to work hard to find it.

One can combine thoughts from both sets of sub-steps and come up with

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. 18(4) cage at R1C6 = {1458/1467}, no 3, 4 locked for N3, clean-up: no 6 in 12(3) cage (step 30a)
34c. 28(4) at R2C9 (step 10) = {5689} (only remaining combination), no 7 -> CPE no 6 in R1C8
34d. Clean-up: no 6 in R1C6 (step 19), no 3 in R5C6 (step 29)

I'll leave it to you to decide whether Mike's step 34c or my step 34a is the easier one to find.


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PostPosted: Sun Jul 06, 2008 10:54 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Transformers X by Para (Sept 07)
Puzzle pic; 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver; 1-9 cannot repeat on the diagonals:
3x3:d:k:6656:6656:6656:4611:1028:4613:5894:5894:5894:4105:6656:4611:4611:1028:4613:4613:5894:6673:4105:6656:5908:5909:5909:5909:4376:5894:6673:4105:4105:5908:5908:5909:4376:4376:6673:6673:4644:4644:2854:5908:1576:4376:4650:3883:3883:3885:4644:2854:2854:1576:4650:4650:3883:4149:3885:3885:8760:8760:8760:8760:8760:4149:4149:6463:6463:8760:3906:3906:3906:8760:5446:5446:6463:6463:6463:4427:4427:4427:5446:5446:5446:
Solution:
+-------+-------+-------+
| 8 2 3 | 4 1 9 | 5 6 7 |
| 4 6 9 | 5 3 7 | 2 1 8 |
| 1 7 5 | 2 8 6 | 3 4 9 |
+-------+-------+-------+
| 2 9 6 | 3 7 8 | 1 5 4 |
| 7 3 1 | 9 4 5 | 8 2 6 |
| 5 8 4 | 6 2 1 | 9 7 3 |
+-------+-------+-------+
| 6 4 2 | 1 9 3 | 7 8 5 |
| 3 5 8 | 7 6 2 | 4 9 1 |
| 9 1 7 | 8 5 4 | 6 3 2 |
+-------+-------+-------+
Quote:
Para, lead-in: It contains a little trick, that could help you in solving..Rating-wise i think .. a 1.75.
mhparker: Standard moves exhausted and a huge forest of candidates still remaining! Looks like I'll need to find that trick Para was talking about..
mhparker: ..some advanced moves ...so I think a 1.75 rating is not a bad estimate
Andrew (in 2012): For the second time this week (the first was for Para’s Killer-X), belated thanks to Para for a fun puzzle! :) In my walkthrough for Transformers-X Lite I wrote "Not sure whether I'll try the harder version. I may just work through Mike's walkthrough." I only tried the harder version in 2012 and felt that I might not have been able to solve it in 2007; then I looked at Mike’s walkthrough for the first time.
Our solving paths started in similar ways; then Mike’s step 14 made his solving path very different from mine.
Rating Hard 1.5.
2021 Revisit to this puzzle here
Walkthrough by mhparker:
CathyW wrote:
Since A68 is out tomorrow, I'll let Mike do the WT for the original.
OK, Cathy, here it is below. Nice to see that we managed to squeeze in WT's for both versions just before the "closing bell" for this week.

Many thanks to Para for creating a very enjoyable Killer-X!


Transformer Killer Walkthrough

1. 4(2)n2 = {13}, locked for c5 and n2
1a. -> 6(2)n5 = {24}, locked for c5 and n5

2. 26(4)n36: no 1

3. 11(3)n45: no 9

4. Outies c12 = r19c3 = 10(2): no 5

5. Outies c89 = r19c7 = 11(2): no 1

6. Innies n8 = r7c456 = 13(3) = h13(3)n8
6a. min. r7c5 = 5 -> max. r7c46 = 8
6b. -> no 8,9 in r7c46

7. Outies r89 = r7c34567 = 22(5)
7a. -> (from step 6) r7c37 = 9(2)
7b. -> no 9 in r7c37

8. Innies r89 = r8c37 = 12(2)
8a. -> no 1,2,6 in r8c37

9. Innies r89 (step 8) "see" all of r78c456
9a. -> these two cells must map to 2 cells of 17(3)n8
9b. -> remaining cell of 17(3)n8 must be 17 - 12 = 5
9c. -> 5 locked in 17(3)n8 for r9 and n8
9d. cleanup: no 7 in r7c46 (step 6), no 6 in r1c7 (step 5)
9e. 17(3)n8 = {5(39/48)} (no 1,2,6,7) = {(4/9}..}
9f. -> r8c37 = {39/48} (no 5,7) = {(3/4)..}

10. Hidden killer pair on {17} in n8
10a. {17} in n8 only within h13(3)n8 and 15(3)n8
10b. neither of these cages can contain both of {17}
10c. -> each of these 2 cages must contain exactly one of {17}
10d. -> 15(3)n8 = {168/267} (no 3,4,9)
10e. -> 6 locked in 15(3)n8 for r8 and n8
10f. h13(3)n8 = {139/148/247} (step 10c) = {(1/4)..}, {(3/4)..}

11. 34(7)n789 contains 2 of {1234} within r7c46
11a. split 9(2) (step 7a) at r7c37 must also contain 1 of {1234}
11b. also, split 12(2) at r8c37 (step 9f) must contain 1 of {34}
11c. -> 34(7)n789 must contain all of {1234}
11d. -> 34(7)n789 = {1234789} (no 5,6)
11e. {127} in 34(7)n789 locked in r7 -> not elsewhere in r7 (r7c1289)
11f. cleanup from step 11d: no 3,4 in r7c37 (step 7a)

12. Outies r789 = r6c19 = 8(2): no 4,8,9

13. Hidden killer pair on {56} in r7 as follows:
13a. only places for {56} in r7 are r7c1289
13b. r7c12 cannot contain both of {56} due to 4 unavailable in r6c1 for 15(3)n47 cage sum
13c. r7c89 cannot contain both of {56} due to 16(3)n69 cage sum
13d. -> r7c12 and r7c89 must each contain exactly one of {56} (no eliminations yet)

14. Innies n7 = r7c12+r78c3 = 20(4), {56} unavailable in r78c3
14a. -> must contain exactly one of {56} (step 13d)
14b. also, cannot contain both of either {14} or {34} due to h13(3)n8 (step 10f)
14c. -> possible combos are: {2(369/459/468)}
14d. -> r7c3 = 2
14e. -> r7c7 = 7 (step 7a), r56c5 = [42]
14f. cleanup: no 6 in r6c19 (step 12), no 8 in r19c3 (step 4), no 4 in r19c7 (step 5)

15. I/O diff. n9: r8c7 = r6c9 + 1
15a. -> r6c9 = {37}, r8c7 = {48}
15b. -> r6c19 (step 12) = [17/53] = {(1/3)..}
15c. r8c37 = {48}, locked for r8 and 34(7)n789

16. Naked single (NS) at r7c5 = 9

17. Naked pair (NP) on {13} at r7c46 -> no 1,3 elsewhere in r7 and n8

18. Naked triple (NT) on {458} at r9c456 -> no 4,8 elsewhere in r9
18a. -> no 6 in r1c3 (step 4), no 3 in r1c6 (step 5)

19. NT on {267} at r8c456 -> no 2,7 elsewhere in r8

20. {48} locked in n7 innies
20a. -> r7c12+r8c3 = {468} (no 5)
20b. 6 locked in r7c12 for r7 and n7
20c. cleanup: no 4 in r1c3 (step 4)
20d. r19c3 = {19/37} = {(1/3)..}

21. 2 unavailable to 11(3)n45 = {1(37/46)} (no 5,8)
21a. 4 only in r6c3
21b. -> no 6 in r6c3
21c. r6c34 cannot be {13}, blocked by r6c19 (step 15b)
21d. r56c3 cannot be {13}, blocked by r19c3 (step 20d)
21e. -> no 7 in r5c3, no 7 in r6c4
21f. CPE: no 1 in r6c12

22. NS at r6c1 = 5
22a. -> r6c9 = 3 (step 15b)
22b. cleanup: no 8 in r7c12, no 4 in r7c89

23. Hidden single (HS) in n7 at r8c2 = 5

24. HS in n7 at r8c3 = 8
24a. -> r8c7 = 4

25. 18(3)n56 = {189} (only possible combo, since {347} all unavailable)
25a. CPE: no 1,8,9 in r6c8

26. 11(3)n45 and 18(3)n56 form grouped X-Wing on 1 in r56
26a. -> no 1 elsewhere in r56

27. Outies r1234 = r5c46 = 14(2) = {59/68} (no 3,7)

28. 7 in n5 locked in r4 -> not elsewhere in r4

29. 7 in n6 locked in 15(3)n6 = {267} (only possible combo)
29a. 2 locked in r5c89 for r5 and n6
29b. 6 locked for n6

30. 5 in d\ locked in r3c3+r4c4 -> no 5 in r35c4 (CPE)
30a. cleanup: no 9 in r5c6 (step 27)

31. HS in r5 at r5c6 = 5
31a. -> r5c4 = 9 (step 27)

32. HS in d\ at r3c3 = 5
32a. split 9(2) at r4c34 = [18]/{36}

33. HS in c5 at r9c5 = 5

34. HS in 18(3)n56 at r6c7 = 9

35. NP on {18} at r45c7 -> no 1,8 elsewhere in c7 and n6
35a. cleanup: no 3 in r9c7 (step 5)

36. Hidden pair (HP) on {29} in n4/r4 at r4c12
36a. -> r4c12 = {29} = 11 total
36b. -> r23c1 = 5(2) = {14}, locked for c1 and n1
36c. cleanup: no 9 in r9c3 (step 4)

Last couple of moves just to quickly get down to all singles:

37. NP on {45} at r4c89 = 9 total
37a. -> r23c9 = 17 = {89}, locked for c9 and n3

38. I/O diff. n3: r12c6 = r3c7 + 13
38a. -> r3c7 = 3
38b. and r12c6 = {79}, locked for c6 and n2
38c. -> r2c7 = 2 (cage-split of 18(3)n23)

Now all naked and hidden singles to end
Andrew’s 2012 walkthrough:
I solved Transformers-Lite when it first appeared but didn’t try the harder version then, so I’m trying it now for the first time.

Prelims

a) R12C5 = {13}
b) R56C5 = {15/24}
c) 11(3) cage at R5C3 = {128/137/146/236/245}, no 9
d) 26(4) cage at R2C9 = {2789/3689/4589/4679/5678}, no 1

1. Naked pair {13} in R12C5, locked for C5 and N2, clean-up: no 5 in R56C5

2. Naked pair {24} in R56C5, locked for C5 and N5

[Para said "It contains a little trick, that could help you in solving. There's always ways around these tricks but if you notice it all the better. The trick is something new to me but actually not that hard. But it has all to do with the cage pattern."]
3. 45 rule on R89 2 innies R8C37 = 12 = {39/48/57}, no 1,2,6
3a. Using "the trick", R8C37 "see" all the cells in N8 except for R9C456 -> R9C456 must contain the same two numbers as R8C37
3b. R9C456 = 17(3) cage, R8C37 = 12 -> 17(3) cage at R9C4 must also contain 5, locked for R9 and N8
3c. 17(3) cage = {359/458}, no 1,2,6,7
3d. R8C37 = {39/48}, no 5,7

4. 45 rule on N8 3 innies R7C456 = 13, R8C37 = 12 (step 3) -> R7C37 = 9, no 9 in R7C37

5. 15(3) cage at R8C4 = {168/267} (cannot be {249/348} which clash with R8C37), no 3,4,9, 6 locked for R8 and N8

6. R7C456 = 13 (step 4) = {139/148/247} (cannot be {238} which clashes with 15(3) cage at R8C4)
6a. R7C5 = {789} -> no 7,8,9 in R7C46

7. R7C456 (step 6) = {139/148/247} contains one of 3,4, R8C37 = {39/48} contains 3,4 -> 3,4 locked in R7C456 + R8C37 for 34(7) cage at R7C3, no 3,4 in R7C37
7a. R7C37 = 9 (step 4) = {18/27}
7b. R7C34567 = {139}{27}/{148}{27}/{247}{18}, 1,2,7 locked for R7

8. 45 rule on R1234 2 outies R5C46 = 14 = {59/68}

9. 45 rule on R789 2 outies R6C19 = 8 = {17/26/35}, no 4,8,9

10. 45 rule on C12 2 outies R19C3 = 10 = {19/28/37/46}, no 5

11. 45 rule on C89 2 outies R19C7 = 11 = {29/38/47}/[56], no 1, no 6 in R1C7

12. Hidden killer pair 5,6 in R7C12 and R7C89 for R7, R7C12 cannot contain both of 5,6 because no 4 in R6C1, R7C89 cannot contain both of 5,6 because 16(3) cage at R6C9 cannot be 5{56} -> R7C12 and R7C89 must each contain one of 5,6
12a. 16(3) cage at R6C9 must contain one of 5,6 in R7C89 -> no 5,6 in R6C9 (because 16(3) cage cannot contain both of 5,6), clean-up: no 2,3 in R6C1 (step 9)
12b. 15(3) cage at R6C1 = {159/168/357/456} (cannot be {348} which doesn’t contain 5 or 6)

13. 18(3) cage at R5C1 = {189/279/369/378/459/468/567}
13a. 1 of {189} must be in R5C12 (R5C12 cannot be {89} which clashes with R5C46), no 1 in R6C2

14. Consider placements for R56C5
14a. R56C5 = [24], 2 placed for both diagonals => R7C37 = {18} (step 7a), R7C456 (step 7) = {247} => 15(3) cage at R6C1 (step 12b) cannot be {456} and 1 of {168} must be in R6C1 => no 6 in R6C1
or R56C5 = [42], no 2 in R6C9 => no 6 in R6C1 (step 9)
-> no 6 in R6C1, clean-up: no 2 in R6C9

15. 45 rule on N7 2 innies R78C3 = 1 outie R6C1 + 5
15a. R78C3 cannot total 12, which clashes with R8C37 (CCC within 34(7) cage at R7C3) -> no 7 in R6C1, clean-up: no 1 in R6C9 (step 9)
15b. R6C1 = {15} -> R78C3 = 6,10 = [24/19/28/73], no 8 in R7C3, clean-up: no 1 in R7C7 (step 7a)
15c. 15(3) cage at R6C1 (step 12b) = {159/168/456}, no 3

16. 3 in R7 only in R7C689, CPE no 3 in R8C7, clean-up: no 9 in R8C3 (step 3d)
16a. R78C3 (step 15b) = [24/28/73], no 1, clean-up: no 8 in R7C7 (step 7a)

17. Naked pair {27} in R7C37, locked for R7, CPE no 2,7 in R3C37 + R5C5 using diagonals -> R5C5 = 4, placed for both diagonals, R6C5 = 2
17a. 1 in R7 only in R7C46, locked for N8

18. 15(3) cage at R8C4 (step 5) = {267} (only remaining combination), locked for R8

19. 11(3) cage at R5C3 = {128/137/146/236/245}
19a. 2 of {128} must be in R5C3 -> no 8 in R5C3
19b. 7 of {137} must be in R6C34 (R6C34 cannot be {13} which clashes with R6C19), no 7 in R5C3
19c. 2,4 of {245} must be in R56C3 -> no 5 in R56C3

20. 1 in N9 only in 21(5) at R8C8 = {12369/12459/12468/13458/13467} (cannot be {12378/12567} which clash with R7C7)
20a. Using "the trick" again, R8C37 and 17(3) cage at R9C4 must both contain either 3 or 4 -> one of 3,4 locked for R89 -> 21(5) cage at R8C8 cannot contain both of 3,4
20b. 21(5) cage = {12369/12459/12468} (cannot be {13458/13467} which contain both of 3,4), 2 locked for N9 -> R7C7 = 7, placed for D\, R7C3 = 2, placed for D/

21. 11(3) cage at R5C3 = {137/146}, no 5,8, CPE no 1 in R6C1
21a. 4 of {146} must be in R6C3 -> no 6 in R6C3

22. R6C1 = 5, R6C9 = 3 (step 9)

23. 15(3) cage at R6C1 (step 15c) = {456} (only remaining combination) -> R7C12 = {46}, locked for R7 and N7

24. R78C3 (step 16a) = [28] (only remaining permutation), R8C7 = 4 (step 3d), R7C5 = 9
24a. Naked pair {13} in R7C46, locked for N8
24b. Naked pair {58} in R7C89, locked for N9

25. R8C2 = 5 (hidden single in N7), placed for D/

26. 18(3) cage at R5C7 = {189} (only remaining combination), CPE no 1,8,9 in R6C8

27. Caged X-Wing for 1 in 11(3) cage at R5C3 and 18(3) cage at R5C7, no other 1 in R56
27a. 1 in N6 only in R456C7, locked for C7

28. 15(3) cage at R5C8 = {249/267} (cannot be {258} because R6C8 only contains 4,6,7, cannot be {456} = {56}4 which clashes with R5C46), no 5,8, 2 locked for R5 and N6
28a. 2 in R4 only in R4C12, locked for 16(4) cage at R2C1, no 2 in R23C1

29. 17(4) cage at R3C7 = {1358/1367}, no 9, 1 locked for R4, clean-up: no 5 in R5C4 (step 8)

30. 1 in N4 only in R56C3, locked for C3 and 11(3) cage at R5C3, no 1 in R6C4
30a. 11(3) cage at R5C3 (step 21) = {137/146}
30b. R6C4 = {67} -> no 6,7 in R56C3
30c. 18(3) cage at R5C1 = {369/378/468}
30d. Killer pair 3,4 in 18(3) cage and 11(3) cage, locked for N4

31. 1 in N5 only in R46C6, locked for C6 -> R7C46 = [13]

31. R4C4 = 3 (hidden single in C4), placed for D\
31a. 23(4) cage at R3C3 = {3569/3578} -> R3C3 = 5

33. Naked pair {19} in R8C89, locked for R8 and N9 -> R8C1 = 3

34. 17(4) cage at R3C7 (step 29) = {1358/1367} -> R3C7 = 3
34a. 6 of {1367} must be in R4C7 (R45C6 cannot be [76] which clashes with R6C6), no 6 in R45C6, clean-up: no 8 in R5C4 (step 8)
34b. R9C8 = 3 (hidden single in R9)

35. 23(4) cage at R3C3 (step 31a) = {3569} (only remaining combination), no 7
35a. Naked pair {69} in R4C3 + R5C4, CPE no 6,9 in R4C5 + R5C12

36. 18(3) cage at R5C1 (step 30c) = {378} (only remaining combination, cannot be {369/468} because 4,6,9 only in R6C2) -> R5C2 = 3, R5C1 + R6C2 = {78}, locked for N4

37. R56C3 = [14], R6C4 = 6 (cage sum), R5C4 = 9, R4C3 = 6, R5C7 = 8, R5C6 = 5, R4C7 = 1, R4C6 = 8 (cage sum), placed for D/, R4C5 = 7, R8C5 = 6, R3C5 = 8, 17(3) cage at R9C4 = [854], R5C1 = 7, R6C2 = 8, R6C67 = [19]

38. R34C5 = [87] = 15 -> R3C46 = 8 = [26]

39. Naked pair {79} in R12C6, locked for N2, R3C7 = 2 (cage sum)
39a. R12C4 = {45} -> R3C3 = 9 (cage sum), R12C6 = [97]

40. R4C12 = {29} = 11 -> R23C1 = 5 = {14}, locked for C1 and N1

and the rest is naked singles without using the diagonals.

I’ll rate my walkthrough for Transformers Killer-X at Hard 1.5. My hardest steps are the original "trick", which is effectively a "clone pair", a short forcing chain and another use of the "trick".
Transformers X Lite by Para (Sept 07)
Puzzle pic; 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:6656:6656:6656:5379:2564:5125:6406:6406:6406:4361:6656:5379:5379:2564:5125:5125:6406:4369:4361:6656:4372:3861:3861:3861:4888:6406:4369:4361:4361:4372:4372:3861:4888:4888:4369:4369:3876:3876:4902:4372:3368:4888:2346:4395:4395:2605:3876:4902:4902:3368:2346:2346:4395:5429:2605:2605:8248:8248:8248:8248:8248:5429:5429:7487:7487:8248:4162:4162:4162:8248:5958:5958:7487:7487:7487:3659:3659:3659:5958:5958:5958:
Solution:
+-------+-------+-------+
| 7 2 5 | 8 6 3 | 9 4 1 |
| 1 3 6 | 7 4 9 | 8 5 2 |
| 8 9 4 | 1 2 5 | 7 6 3 |
+-------+-------+-------+
| 3 5 1 | 9 7 6 | 2 8 4 |
| 2 6 9 | 3 8 4 | 5 1 7 |
| 4 7 8 | 2 5 1 | 3 9 6 |
+-------+-------+-------+
| 5 1 3 | 4 9 2 | 6 7 8 |
| 6 4 7 | 5 3 8 | 1 2 9 |
| 9 8 2 | 6 1 7 | 4 3 5 |
+-------+-------+-------+
Quote:
Para, lead-in: Rating-wise i think .. about a 1.00-1.25
CathyW: Had to have a big hint from Para about "The Trick" but have now solved the Transformer Lite. "The Trick" is clever - seems obvious once you know!! I would agree with 1 - 1.25 for the Lite.
Andrew: An interesting puzzle with a fairly difficult cage pattern and, of course, a neat trick by Para..Even with "The Trick" I rate it a solid 1.25. No way it's a 1.0!
Walkthrough by CathyW:
Had to have a big hint from Para about "The Trick" but have now solved the Transformer Lite. "The Trick" is clever - seems obvious once you know!! The puzzle would indeed have been very difficult to solve without it. Many thanks Para for this puzzle. :)

Since A68 is out tomorrow, I'll let Mike do the WT for the original.

Prelims:

a) 21(3) @ r1c4 and r6c9 = {489/579/678}
b) 10(2) r12c5: no 5
c) 20(3) @r1c6 = {389/479/569/578}
d) 13(2) r56c5 = {49/58/67}
e) 9(3) @r5c7 = {126/135/234}
f) 19(3) @r5c3: no 1
g) 10(3) @r6c1 = {127/136/145/235}
h) 32(7) @r7c3 = 123{4589/4679/5678}

1. Outies c12: r19c3 = 7 = {16/25/34}

2. Outies c89: r19c7 = 13 = {49/58/67}

3. Outies r1234: r5c46 = 7 = {16/25/34}

4. Outies r789: r6c19 = 10 = [19/28/37/46/64]
-> r6c1 <> 5,7; r6c9 <> 5

5. Outies N2: r2c37 + r4c5 = 21 = {399/489/579/588/669/678}
-> r4c5 = (4…9) -> r3c456 is max 11 -> r3c456 <> 9

6. Innies N8: r7c456 = 15

7. Innies r89: r8c37 = 8 = {17/26/35}
-> r7c37 = 9 = {18/27/36/45}
-> 29(5) N7 = {9…}

8. O-I N7: r78c3 – r6c1 = 6
-> r78c3 = 7, 8, 9, 10, 12
-> r78c7 = 5, 7, 8, 9, 10

9. O-I N9: r78c7 – r6c9 = 1

10. Split 9(2) r7c37 <> 4,5:
a) 32(7) = 1235678 – no 4
b) 32(7) = 1234679 – no 5
c) 32(7) = 1234589 -> 9 locked to split 15(3) r7c456 -> split 9(2) = {18}

11. Split 15(3) r7c456 = {168/249/258/267/357}
{159/348/456} blocked by split 8(2) and split 9(2) as follows:
a) {159) forces split 8(2) = {26} no option left for 9(2)
b) {348} forces split 9(2) = {27} no option left for 8(2)
c) {456} forces split 8(2) = {17} no option left for 9(2)

12. O-I r12: r3c28 – r2c19 = 12 = (15-3), (16-4), (17-5)
-> r3c28 = (6789); r2c19 = (1234)

13. “The Trick”: r8c37 = 8
-> one of {17/26/35} must be within 14(3)
-> 14(3) = 6{17/35} -> 6 n/e N8/r9
-> r8c37 = {17/35}
Clean up: r1c3 <> 1, r1c7 <> 7

14. 32(7) must have 2 which is now locked to r7c3-7 -> r7c12 <> 2

15. 32(7) must have 3 which must be in one of r78c37 -> r7c456 <> 3

16. 16(3) r8c456 = {178/259/349/358}; {457} blocked by 14(3)/split 8(2)

17. split 15(3) r7c456 = {249/258}
-> Must have 2 thus not elsewhere in N8/r7
-> r7c37 <> 7
-> 29(5) N7 must have 2 -> 29(5) = 29{378/468/567} -> 29(5) <> 1 -> r1c3 <> 6
-> 16(3) = {178/349/358}

18. Killer pair {13}: r7c12 can’t be both {13} -> r6c1 <> 6 -> r6c9 <> 4

19. From step 8:
r78c3 = 7, 8, 9, 10 = [61], [17] ([35] blocked by split 7(2) r19c3), [63/81], [37]
r78c7 = 10, 9, 8, 7 = [37], [81] ([63] blocked since r78c3 can’t be [35], [17/35], [61]
-> r8c3 <> 5, r8c7 <> 3

20. 3 locked to r78c3, r7c7 within 32(7) -> r7c12, r3c3 <> 3
-> 3 locked to r7c37 -> r7c37 = {36} n/e r7 -> r8c37 = {17} n/e r8
-> r3c7 <> 3
-> 14(3) r9c456 = {167} n/e r9
Andrew noted r3c37 <>6 from the same logic.

21. 10(3) @r6c1 can’t have both {17} within r7c12 -> r6c1 <> 2 -> r6c9 <> 8

22. r78c3 = 7, 9, 10 = [61], [not possible], [37]
-> r6c1 <> 3 -> r6c9 <> 7
-> 10(3) @r6c1 = {145} -> 5 n/e r7/N7
-> r8c3 = 7 -> r8c7 = 1
-> split 15(3) = {249} n/e N8/r7
-> 16(3) = {358} n/e r8
-> r6c1 = 4 -> r6c9 = 6
-> r7c3 = 3, r7c7 = 6
-> split 7(2) r19c3 = [52]
-> split 13(2) r19c7 = {49}/[85]
Clean up: 13(2) r56c5 = [49]/{58}

23. 19(3) @ r5c3 = [892/982/685] -> 8 n/e c3/N4
-> 1 locked to r34c3 -> r45c4 <> 1 -> r5c6 <> 6
-> 4 locked to r23c3 -> r12c2 <> 4
-> 1 locked to r456c6 -> r39c6 <> 1
-> split 7(2) r5c46 <> {25}

24. 9(3) @r5c7 = {135/234} -> r6c8 <> 3
Analysis: r6c6 <> 5, r5c7 <> 2

25. 7 locked to r4c456 n/e r4

26. 7 locked to r23c7 n/e N3

27. Innies N3: r23c79 = 20 = {7…}

28. Innies N1: r23c13 = 19 = {4…}
Split 19(4) = {1468/2467} -> 26(5) = 359{18/27}
-> r3c1 = (78)
-> r2c3 = 6, r3c3 = 4 -> r4c3 = 1 -> r45c4 = [93] (only option)

All singles from here
Walkthrough by Andrew:
I only got round to starting Transformers Lite yesterday and finished it today.

An interesting puzzle with a fairly difficult cage pattern and, of course, a neat trick by Para.
CathyW wrote:
"The Trick" is clever - seems obvious once you know!!
Very true!
Para wrote:
... and a "Lite" version for during the lunch break
That would be a very long lunch break! It took me a lot longer than that.

There is a lot of similarity between Cathy's solving path and mine. Must be a narrow solving path needing to make good progress in the bottom third before expanding to the rest of the grid.

Even with "The Trick" I rate it a solid 1.25. No way it's a 1.0!


Here is my walkthrough for Transformers Lite

1. R12C5 = {19/28/37/46}, no 5

2. R56C5 = {49/58/67}, no 1,2,3

3. 21(3) cage at R1C4 = {489/579/678}, no 1,2,3

4. 20(3) cage at R1C6 = {389/479/569/578}, no 1,2

5. 19(3) cage at R5C3 = {289/379/469/478/568}, no 1

6. 9(3) cage at R5C7 = {126/135/234}, no 7,8,9

7. 10(3) cage at R6C1 = {127/136/145/235}, no 8,9

8. 21(3) cage at R6C9 = {489/579/678}, no 1,2,3

9. 32(7) cage at R7C3 = 123{4589/4679/5678}

[Extremely unusual! Nothing has been locked by the preliminary steps.]

10. Para said "It contains a little trick, that could help you in solving. There's always ways around these tricks but if you notice it all the better. The trick is something new to me but actually not that hard. But it has all to do with the cage pattern."

I suppose it could be called Extended CPE.

11. 45 rule on R8 2 innies R8C37 = 8 = {17/26/35}

12. Using "The Trick" the pair in R8C37 must be in R9C456 = 6{17/35} -> R8C37 = {17/35}, no 2,6
12a. 6 locked in R9C456 for R9 and N8

13. 45 rule on N8 3 innies R7C456 = 15
13a. R7C37 = 32 – 8 – 15 = 9 = {18/27/36/45}, no 9

14. R7C456 = {249/258/348} (cannot be {159/357} which clash with R8C37 within 32(7) cage), no 1,7

15. Hidden killer triple 1,2,3 in 32(7) cage which must contain all of 1,2,3 (step 9) -> R7C37, R7C456 and R8C37 must each contain one of 1,2,3 -> no 4,5 in R7C37 = {18/27/36}

16. R7C456 = {249/258} (cannot be {348} => R7C37 = {27} so clash with R8C37 within 32(7) cage) = 2{49/58}, no 3, 2 locked for R7 and N8, clean-up: no 7 in R7C37 (step 13a)

17. R8C456 = {178/349/358} (cannot be {457} which clashes with R8C37)

18. Killer pair 1,3 in R8C37 and R8C456, locked for R8

19. Killer pair 3,7 in R8C37 and R8C456, locked for R8

20. 45 rule on R789 2 outies R6C19 = 10 = [19/28/37/46/64], no 5, no 7 in R6C1

21. 45 rule on R1234 2 outies R5C46 = 7 = {16/25/34}, no 7,8,9

22. 45 rule on C12 2 outies R19C3 = 7 = {25/34}/[61], no 7,8,9, no 1 in R1C3

23. 45 rule on C89 2 outies R19C7 = 13 = {49/58}/[67], no 1,2,3, no 7 in R1C7

24. 45 rule on N2 3 outies R2C37 + R4C5 = 21, max R2C37 = 17 -> min R4C5 = 4
24a. Min R4C5 -> max R3C456 = 11, no 9

25. 45 rule on N7 2 innies R78C3 – 6 = 1 outie R6C1
25a. R78C3 cannot total 12 -> no 6 in R6C1, clean-up: no 4 in R6C9 (step 20)
25b. No 3 in R6C1 because R7C12 = {16} clashes with R78C3 = [63/81], clean-up: no 7 in R6C9 (step 20)
25c. If R6C1 = 1 => R78C3 = [61] => R7C12 = {45} => R6C9 = 9 (step 20) => R7C89 = {48/57} clashes with R7C12 -> no 1 in R6C1, no 9 in R6C9

26. 10(3) cage at R6C1 (step 7) = {127/145/235} (cannot be {136} because R6C1 only contains 2,4), no 6
26a. R6C1 = {24} -> no 4 in R7C12
26b. R7C12 = {15/17/35}

27. R6C1 = {24} -> R78C3 = 8,10 (step 25) = [17/35/37], no 6,8 in R7C3, no 1,3 in R8C3, clean-up: no 1,3 in R7C7 (step 13a), no 5,7 in R8C7 (step 12)
27a. 6 in N7 locked in R8C12, locked for R8

28. Naked quad {1357} in R7C123 + R8C3, locked for N7, clean-up: no 2,4,6 in R1C3 (step 22)

29. 21(3) cage at R6C9 (step 8) = {489/678} (cannot be {579} because R6C9 only contains 6,8), no 5 = 6{78}/8{49}/8{67}
29a. 6 in N9 locked in R7C789
29b. R7C789 = {469/678}

30. 9(3) cage at R5C7 (step 6) = {126/135/234}
30a. R56C7 cannot be {13} which would clash with R8C7 -> no 5 in R6C6

31. 2,5 in N9 locked in 23(5) cage = {12578/23459}
31a. R9C456 (step 12) = 6{17/35} -> R9C789 must contain {17/35}
31b. {12578} contains {17} in R9C789 => 5 in R8C89, R8C7 = 3 => R8C3 = 5 (step 12) clashes with 5 in R8C89 -> 23(5) cage cannot be {12578}
31c. 23(5) cage = {23459}, locked for N9 -> R8C7 = 1, R8C3 = 7 (step 12), R7C3 = 3 (locked for D/), R7C7 = 6 (step 13a, locked for D\), R1C3 = 5, R9C3 = 2 (step 22), clean-up: no 7 in R6C5
31d. R9C789 (step 31a) must contain {35}, locked for R9 and N9

32. Naked pair {78} in R7C89, locked for R7, R6C9 = 6, R6C1 = 4 (step 20), clean-up: no 7,9 in R5C5, no 5 in R7C456 (step 16)
32a. 4 in C3 locked in R23C3, locked for N1

33. Naked triple {249} in R7C456, locked for N8

34. Naked triple {358} in R8C456, locked for R8

35. 9(3) cage at R5C7 (step 6) = {135/234} = 3{15/24}
35a. CPE no 3 in R6C8
35b. 4 of {234} in R5C7 -> no 2 in R5C7

36. 19(3) cage at R5C3 (step 5) = {289/568}, no 7
36a. 2 of {289} and 5 of {568} in R6C4 -> no 8,9 in R6C4
36b. 8 locked in R56C3, locked for C3 and N4
36c. 7 in N5 locked in R4C456, locked for R4

37. 1 in C3 locked in R34C3, locked for 17(4) cage -> no 1 in R45C4, clean-up: no 6 in R5C6 (step 21)

38. 1 in N5 locked in R456C6, locked for C6

39. 7 in N6 locked in 17(3) cage = 7{19/28}, no 3,4,5

40. 7 in N4 locked in 15(3) cage = 7{26/35}, no 1,9

41. 17(3) cage in N6 contains 8/9 -> R4C789 must contain 8/9
41a. 13(2) cage in N5 contains 8/9 -> R4C456 must contain 8/9
41b. Killer pair 8,9 in R4C456 and R4C789, locked for R4

42. Hidden pair {89} in R56C3, locked for C3 -> R6C4 = 2 (locked for D/)

43. 9(3) cage at R5C7 (step 35) = {135} (only remaining combination), no 4 -> R6C6 = 1 (locked for D\)
43a. Naked pair {35} in R56C7, locked for C7 and N6, clean-up: no 8 in R1C7 (step 23)

44. R4C7 = 2 (hidden single in C7), clean-up: no 8 in 17(3) cage in N6 (step 39)
44a. Naked triple {179} in 17(3) cage in N6, locked for N6
44b. Naked pair {48} in R4C89, locked for R4 and 17(4) cage at R2C9
[Guess I should have done more on the 17(4) cage at this stage! I was just thinking ahead to the next two steps.]

45. Naked pair {49} in R19C7, locked for C7
45a. Naked pair {78} in R23C7, locked for N3

46. R3C3 = 4 (naked single, locked for D\), R2C3 = 6, R4C3 = 1, clean-up: no 4 in R1C5, no 4,5,9 in R12C4 (step 3), no 9 in R6C5

47. Naked pair {78} in R12C4, locked for C4 and N2, clean-up: no 2,3 in R12C5

48. R34C3 = [41] -> R45C4 = 12 = [93] (9 locked for D\) -> R8C8 = 2 (locked for D\), R56C7 = [53], R5C6 = 4

49. R5C5 = 8 (naked single, locked for D/ and D\) -> R6C5 = 5, R8C456 = [538], R56C3 = [98], R6C2 = 7, R6C8 = 9

50. R2C2 = 3 (locked for locked for D\) -> R1C1 = 7, R9C9 = 5, R12C4 = [87], R23C7 = [87]
50a. R9C8 = 3 (hidden single in N9)
50b. R3C9 = 3 (hidden single in C9)
50c. R1C6 = 3 (hidden single in C6), R2C6 = 9, clean-up: no 1 in R12C5 = [64], R4C5 = 7, R4C6 = 6 (locked for D/), R7C456 = [492]

51. R9C1 = 9, R8C2 = 4 (both locked for D/), R1C9 = 1, R2C8 = 5

and the rest is naked singles.

Not sure whether I'll try the harder version. I may just work through Mike's walkthrough.


Last edited by Ed on Sun Nov 08, 2009 2:11 am, edited 1 time in total.

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PostPosted: Sun Jul 06, 2008 10:59 pm 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Assassin 68 by Ruud (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:5120:5120:5120:3331:3332:2565:7430:7430:7430:6153:5120:3331:3331:3332:2565:2565:7430:3857:6153:5120:6676:6677:6677:6677:3608:7430:3857:6153:6153:6676:6676:6677:3608:3608:3857:3857:3620:3620:3622:6676:6677:3608:5674:4907:4907:2861:3620:3622:3622:1585:5674:5674:4907:4917:2861:2861:2872:2872:1585:3643:3643:4917:4917:2623:4416:2872:5186:5186:5186:3643:4166:1351:2623:4416:4416:4427:4427:4427:4166:4166:1351:
Solution:
+-------+-------+-------+
| 5 3 1 | 2 7 4 | 6 8 9 |
| 4 9 8 | 3 6 5 | 1 2 7 |
| 7 2 6 | 9 1 8 | 3 4 5 |
+-------+-------+-------+
| 8 5 9 | 7 3 6 | 4 1 2 |
| 6 7 2 | 4 5 1 | 8 9 3 |
| 3 1 4 | 8 2 9 | 5 7 6 |
+-------+-------+-------+
| 2 6 7 | 1 4 3 | 9 5 8 |
| 9 4 3 | 5 8 7 | 2 6 1 |
| 1 8 5 | 6 9 2 | 7 3 4 |
+-------+-------+-------+
Quote:
CathyW: took a couple of hours including doing the WT..a reasonable challenge...Estimated rating 1-1.25
Para: I agree with Cathy about the rating.
Andrew: I'll rate it as 1.25. I found it a difficult cage pattern because I didn't find any innies/outies that helped
Walkthrough by CathyW:
Good to see you here Frank! I've only just printed out the puzzle to look at in lunch break later.
ttfn
Cathy x

Edit:
Here's my WT (took a couple of hours including doing the WT - Yes it was an extended lunch break!!) - a reasonable challenge if you don't use Frank's method! Estimated rating 1-1.25 (Para's trick doesn't apply either!)

Prelims

a) 13(2) N2 = {49/58/67}
b) 10(3) @r1c6 = {127/136/145/235}
c) 26(4) @r3c3: no 1
d) 14(4) @r3c7: no 9
e) 22(3) @r5c7 = {589/679} -> r6c89 <> 9
f) 19(3) @r5c8, r6c9: no 1
g) 11(3) @r6c1, r7c3: no 9
h) 6(2) r67c5 = {15/24}
i) 10(2) N7: no 5
j) 20(3) N8: no 1,2
k) 5(2) N9 = {14/23}

1. Outies c12: r19c3 = 6 = {15/24}

2. Outies c89: r19c7 = 13 = {49/58/67}

3. Outies r1234: r5c456 = 10 = {127/136/145/235}

4. Outies r789: r6c159 = 11 = {128/137/146/236/245}

5. Innies r89: r8c37 = 5 = {14/23}

6. Innies N8: r7c456 = 8 = {125/134} -> 1 n/e r7/N8

7. Outies N7: r6c1 + r7c4 = 4 = {13/22} -> max from r6c15 = 3+5=8 -> r6c9 <> 2,3

8. Outies N9: r6c9 + r7c6 = 9 = [45/54/63/72/81]

9. 9 locked to r7c789
-> 16(3) N9 does not contain 9 -> r1c7 <> 4

10. 17(3) @r8c2: r9c3 max 5 -> r89c2 min 12 -> r89c2 <> 1,2
Options: {179/269/278/458/467/539}

11. 11(3) @r7c3: r78c3 = 8; 9; 10 = [71/62]; [81/63]; [82/73/64] -> r7c3 = (678)
(combo [542] blocked by split 6(2) r19c3)

12. 14(3) @r7c6: max from r7c6+r8c7 = 5+4=9 -> r7c7 = (6789)

13. 1 locked to r4c789 n/e r4

14. Innies c1234: r389c4 = 20 = {389/479/569/578} (r3c4 <> 1,2; r9c4 <> 2)

15. Innies c6789: r389c6 = 17

16. 11(3) @r6c1 = [128/182/137/173/146/164/236/263/326/362]
[245/254] blocked by split 8(3) in r7c456 -> r7c12 <> 5
17(3) @r8c2 = 5{39/48}
-> r9c3 = (45) -> r1c3 = (12).

17. 17(3) r9c456 = {269/278/368/467}; {359/458} blocked by split 8(3)
-> r9c456 <> 5

18. Innies N7: r7c123 + r8c3 = 18 = {1278/1368/1467/2367} -> r8c3 <> 4 -> r8c7 <> 1
Mike noted that {1368} blocked by 10(2)n9 (or 17(3)n9) which I missed at the time I did the puzzle. Would have been helpful if I'd seen it!


19. Innies N9: r7c789 + r8c7 = 24 = {2589/2679/3579/4569} -> r7c89 <> 2,3,4
({3489} blocked by 5(2) r89c9)
Thanks to Andrew: for completeness {3678} is blocked by r7c3. {4578} is also impossible because r8c3 = 1, r7c3 = 6 cannot give a valid combination in the 11(3) cage.


20. 19(3) @ r6c9 = {469/478/568} -> r6c9 <> 7 -> r7c6 <> 2
Must have at least one of 6,8 within r7c89 -> 16(3) @r8c8 <> {268} -> r89c8 <> 2

21. KP {68}: split 18 (4) r7c123 and r7c89 must have at least one of 6,8
-> r7c7 <> 6,8
14(3) = [194/392/572/374/473]
-> 22(3) @r5c7 can’t have both {79} within r56c7 -> r6c6 <> 6

22. Outies N8: r7c37 + r6c5 = 18 = {189/279/567} -> r6c5 <> 4 -> r7c5 <> 2

23. Split 11(3) r6c159 = [128/218/326/254} -> 2 n/e r6; r6c9 <> 5 -> r7c6 <> 4
-> r8c7 <> 3 (see options for 14(3) in step 21) -> r8c3 <> 2

24. KP {24} in N9 between 5(2) and r8c7 -> 16(3) <> 4 -> r1c7 <> 9

25. 2 locked to r789c1+r7c2 in N9 -> r6c1 <> 2 -> r7c4 <> 2 -> r7c3 = 7
-> r89c1 <> 3; -> r7c7 = 9 -> r6c6 = 9; r7c6 <> 5, r6c9 <> 4

26. NP {13} r7c46 n/e N8/r7 -> r7c5 = 4 -> r6c5 = 2

27. HS: r9c6 = 2 -> r8c9 <> 3, r8c1 <> 8

28. r7c12 = {26/28} -> r8c1 <> 2; r9c1 <> 8

29. r7c89 = {56/58} -> 16(3) N9 <> 5 -> 16(3) = 7{18/36} -> r1c7 <> 8

30. Naked Quad {5678} r1569c7 -> r234c7 = (1234)

31. KP {68} in N6 between r6c9 and r56c7 -> 6,8 n/e N6
-> 19(3) @ r5c8 = {379} n/e N6
-> r56c7 = {58} n/e c7/N6; r19c7 = {67}
-> 9 locked to r5c89 n/e r5
-> r4c789 = {124} n/e r4
-> r6c9 = 6 -> r7c6 = 3 -> r7c4 = 1 …

Fairly straightforward from here with singles and cage combos.
Walkthrough by Para:
Hi all

It was a really fun puzzle. I enjoyed solving it. I agree with Cathy about the rating. About a 1.00-1.25.
Cathy wrote:
Para's trick doesn't apply either!
Yeah that would be getting a bit one-sided. Also the center nonets lose their transformers look a bit, and start looking more like Dr. Eggman from the Sonic game. (Childhood memories, ahhhh(yeah i played Sega))

Here's my Walk-Through

Walk-through Assassin 68

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

2. 10(3) at R1C6 = {127/136/145/235}: no 8,9

3. 26(4) at R3C3 = {2789/3689/4589/4679/5678}: no 1

4. 14(4) at R3C7 = {1238/1247/1256/1346/2345}: no 9

5. 22(3) at R5C7 = {589/679}: no 1,2,3,4; 9 locked in cage -->> R6C89: no 9

6. 19(3) at R5C8 and R6C9 = {289/379/469/478/568}: no 1

7. 11(3) at R6C1 and R7C3 = {128/137/146/236/245}: no 9

8. R67C5 = {15/24}: no 3,6,7,8,9

9. R89C1 = {19/28/37/46}: no 5

10. 20(3) at R8C4 = {389/479/569/578}: no 1,2

11. R89C9 = {14/23}: no 5,6,7,8,9

12. 45 on R89: 2 innies: R8C37 = 5 = {14/23}: no 5,6,7,8,9

13. 16(3) at R8C8 = {169/178/259/268/358/367/457} = {1|2|3|4..}: {349} blocked by R89C9
13a. Killer Quad {1234} in R8C7 + R89C9 + 16(3) at R8C8 -->> locked for N9

14. 45 on N7: 2 outies: R6C1 + R7C4 = 4 = {13/22}: no 4,5,6,7,8,9

15. 11(3) at R6C1 = {128/137/146/236/245}: needs one of {5678} in R7C12
15a. 11(3) at R7C3 = {128/137/146/236/245}: needs one of {5678} in R7C3 -->> R7C3 = {5678}
15b. Killer Quint {56789} in R7C12 + R7C3 + R7C789 -->> locked for R7
15c. Clean up: R6C5: no 1

16. 45 on N8: 3 innies: R7C456 = 8 = {134}: no 2 -->> locked for R7 and N8
16a. 2 in R7 locked for N7 and 11(3) cage at R6C1
16b. 11(3) at R6C1 = [1]{28}/[3]{26}: no 5,7
16c. Clean up: R8C7: no 3(step 12); R89C1: no 8

17. 45 on N9: 2 outies: R6C9 + R7C6 = 9 = [54/63/81] -->> R6C9 = {568}
17a. 19(3) at R6C9 = {568}(last combo): no 7,9
17b. R7C7 = 9(hidden); R6C6 = 9(last place in 22(3) cage at R5C6)
17c. R7C3 = 7(hidden)
17d. 5 in R7 locked for N9 and 19(3) at R6C9
17e. Clean up: R89C1: no 3

18. 11(3) at R7C3 = 7{13}: no 4
18a. Clean up: R8C7: no 1

19. 14(3) at R7C6 = 9[14/32]: R7C6: no 4
19a. R7C5 = 4(hidden); R6C5 = 2

20. 22(3) at R5C7 = 9{58/67} -->> R56C7 = {58/67} = {6|8..}
20a. Killer Pair {68} at R56C7 + R6C9 -->> locked for N6

21. 19(3) at R5C8 = {379}(last combo): no 2,4,5; locked for N6; 9 locked in R5C89 for R5
21a. R56C7 = {58}(last combo): no 6 -->> locked for C7 and N6
21b. R6C9 = 6

22. R7C89 = {58} -->> locked for R7 and N9
22a. 11(3) at R6C1 = {236} -->> R6C1 = 3; R7C12 = {26} -->> locked for N7
22b. R6C8 = 7
22c. R89C1 = {19}(last combo) -->> locked for C1 and N7
22d. R8C3 = 3; R7C46 = [13]; R8C7 = 2; R9C7 = 7(hidden)

23. 16(3) at R8C8 = 7[63](last combo): R89C8 = [63]
23a. R5C89 = [93]

24. 45 on C89: 1 outie: R1C7 = 6

25. 20(3) at R8C4 = {578}(last combo) -->> locked for R8 and N8
25a. R8C2 = 4; R89C9 = [14]; R89C1 = [91]; R4C9 = 2

26. 45 on C12: 2outies: R19C3 = 6 = [15](only combo)
26a. R9C2 = 8

27. 14(3) at R5C3 = {248}(last combo) -->> R5C3 = 2; R6C34 = {48} -->> locked for R6
27a. R56C7 = [85]; R6C2 = 1

28. Naked Pair {14} in R4C78 -->> locked for R4

29. 14(3) at R5C1 = 1{67}(last combo): R5C12 = {67} -->> locked for R5 and N4
29a. R6C3 = 4(hidden); R6C4 = 8

30. 6 in C3 locked for N1

31. 14(4) at R3C7 = {1346}(last possible combo): no 5,7; R4C6 = 6; R3C7 = 3
31a. R4C5 = 3(hidden); R4C4 = 7(hidden); R9C6 = 2; R8C4 = 5
31b. R5C456 = [451]; R4C78 = [41]; R2C7 = 1; R3C5 = 1(hidden)

32. 13(3) at R1C5 = [76](last combo)

33. 15(4) at R2C9 = 12{57}(last combo): R23C9 = {57} -->> locked for C9 and N3

And the rest is all naked and hidden singles

greetings

Para
Walkthrough by Andrew:
So many things to do! I only started Assassin 68 yesterday evening and wrapped it up today.

Gary's post about this puzzle can be put more clearly as
r6c9+r7c6=9, r6c9, r7c6 <> 9
Min r4c789 = 6
Min r56c7 = 13
-> Max r6c9 = 7 -> r7c6 <> 1

I've only had a glance at Cathy's and Para's walkthoughs; I'll look at them properly later.

Edit. I've now worked through Cathy's and Para's walkthoughs. Although we did some things in a different order, Para's and my walkthroughs are very similar. If I'd gone through his one first I probably wouldn't have posted mine. Still, having posted it, there's no point in deleting it. Also I see that Ed has already quoted all three walkthroughs in his ratings thread.

One interesting thought. Para composed his Transformer-Xs. Cathy had solved Transformer Lite before she solved A68, which may have made it easier for her to solve A68. I haven't yet tried either Transformer-X but my solution for A68 was remarkably similar to Para's one.


I'll rate it as 1.25. I found it a difficult cage pattern because I didn't find any innies/outies that helped; maybe they will be more help for the variants with their different cage totals. Fortunately there was enough help from other steps.


Here is my walkthrough for Assassin 68

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

2. R67C5 = {15/24}

3. R89C1 = {19/28/37/46}, no 5

4. R89C9 = {14/23}

5. 10(3) cage at R1C6 = {127/136/145/235}, no 8,9

6. 22(3) cage at R5C7 = 9{58/67}, CPE no 9 in R6C89

7. 19(3) cage at R5C8 = {289/379/469/478/568}, no 1

8. 11(3) cage at R6C1 = {128/137/146/236/245}, no 9

9. 19(3) cage at R6C9 = {289/379/469/478/568}, no 1

10. 11(3) cage at R7C3 = {128/137/146/236/245}, no 9

11. R8C456 = {389/479/569/578}, no 1,2

12. 26(4) cage at R3C3 = {2789/3689/4589/4679/5678}, no 1

13. 14(4) cage at R3C7 = {1238/1247/1256/1346/2345}, no 9

14. 1 in N6 locked in R4C789, locked for R4

15. 3 in C5 locked in R34589C5
15a. 45 rule on C5 5 innies R34589C5 = 26 = {13589/13679/23489/23678} (cannot be {23579/34568} which clash with R67C5)

16. 45 rule on N8 3 innies R7C456 = 8 = 1{25/34}, 1 locked for R7 and N8

17. R9C456 = {269/278/359/368/467} (cannot be {458} which clashes with R7C456)

18. 45 rule on R89 2 innies R8C37 = 5 = {14/23}

19. 11(3) cage at R7C3 (step 10) = {128/137/146/236/245}
19a. 7 of {137} and 6 of {236} must be in R7C3 -> no 3 in R7C3

20. 45 rule on R1234 3 outies R5C456 = 10 = {127/136/145/235}, no 8,9

21. 45 rule on C12 2 outies R19C3 = 6 = {15/24}

22. 17(3) cage in N7, max R9C3 = 5 -> min R89C2 = 12, no 1,2

23. 45 rule on C89 2 outies R19C7 = 13 = {49/58/67}, no 1,2,3

24. 45 rule on C1234 3 innies R389C4 = 20 = {389/479/569/578}, no 1,2

25. 45 rule on N7 2 outies R6C1 + R7C4 = 4 = {13/22}

26. 11(3) cage at R7C3 (step 10) = {128/137/146/236/245}
26a. Max R7C4 + R8C3 = 7 -> no 2 in R7C3
26b. 6 of {146} and 5 of {245} must be in R7C3 -> no 4 in R7C3
[Alternatively R7C3 cannot be 4 which would make the 11(3) cage [434])

27. 45 rule on N9 2 outies R6C9 + R7C6 = 9, no 2,3 in R6C9

28. 9 in R7 locked in R7C789, locked for N9, clean-up: no 4 in R1C7 (step 23)

29. 14(3) cage at R7C6 = {149/158/239/248/257/347/356} (cannot be {167} because 6,7 only in R7C7)
29a. Max R7C6 + R8C7 = 9 -> no 2,3,4 in R7C7
29b. 6,7,8 of {356/257/158} must be in R7C7 -> no 5 in R7C7
[Alternatively R7C7 cannot be 5 which would make the 14(3) cage [554])

30. 16(3) cage in N9 = {178/268/358/367/457}
30a. Killer quad {1234) in R8C7, R89C9 and 16(3) cage -> no 2,3,4 in R7C89
[This could have been done after step 18 but I didn’t see it then. It probably didn’t make much difference that I didn’t see it until now.]

31. 19(3) cage at R6C9 = {469/478/568}
31a. 4 of {478} must be in R6C9 -> no 7 in R6C9, clean-up: no 2 in R7C6 (step 27)

32. 45 rule on N7 4 innies R7C123 + R8C3 = 18 = {1278/1458/1467/2358/2367/2457/3456} (cannot be {1368} which clashes with R89C1)
32a. R8C3 = {1234}, R7C3 = {5678} -> R7C12 must contain one of 5,6,7,8
32b. Killer quint {56789} in R7C12, R7C3, R7C789 -> no 5 in R7C56, clean-up: no 2 in R7C45 (step 16), no 2 in R6C1 (step 25), no 1,4 in R6C5, no 4 in R6C9 (step 27)

33. 19(3) cage at R6C9 (step 31) = {568} (only remaining combination)

34. R7C7 = 9 (hidden single in R7), clean-up: no 4 in R9C7 (step 23)
34a. R7C6 + R8C7 = 5 = [14/32/41], no 3 in R8C7

35. R6C6 = 9 (only remaining place for 9 in 22(3) cage)

36. Naked quad {5678} in R1569C7, locked for C7

37. Naked triple {134} in R7C456, locked for R7 and N8

38. 2 in N8 locked in R9C56, locked for R9, clean-up: no 4 in R1C3 (step 21), no 8 in R8C1, no 3 in R8C9
38a. R9C456 = 2{69/78}, no 5

39. 5 in N8 locked in R8C456, locked for R8
39a. R8C456 = 5{69/78}

40. 2 in R7 locked in R7C12, locked for N7, clean-up: no 8 in R9C1
40a. 11(3) cage at R6C1 (step 8) = {128/236}, no 5,7

41. R7C3 = 7 (hidden single in R7), clean-up: no 3 in R89C1
41a. R7C4 + R8C3 = 4 = {13}, no 4 [Forgot to clean-up for R8C7. If I’d remembered, R67C5 would have been fixed two moves earlier.]

42. 5 in R7 locked in R7C89, locked for N9 and 19(3) cage at R6C9 -> no 5 in R6C9, clean-up: no 8 in R1C7 (step 23)

43. 16(3) cage in N9 = {178/268/367}, no 4

44. 45 rule on R789 3 outies R6C159 = 11 = {128/236} = 2{18/36} -> R6C5 = 2, R7C5 = 4, clean-up: no 9 in R12C5, no 1 in R8C6 (step 34a)

45. R9C6 = 2 (hidden single in R9)

46. 2 in N2 locked in R12C4, locked for 13(3) cage at R1C4 -> no 2 in R2C3
46a. 13(3) cage = 2{38/47/56}, no 1,9
46b. 7 only in R12C4 -> no 4 in R12C4

47. 5 in N7 locked in R9C23
47a. 17(3) cage = {359/458}, no 1,6, clean-up: no 5 in R1C3 (step 21)

48. R5C456 (step 20) = {136/145}, no 7, 1 locked for R5 and N5

49. R56C7 = {58/67}, R6C9 = {68}
49a. Killer pair {68} in R56C7 and R6C9, locked for N6

50. 19(3) cage in N6 = {379} (only remaining combination), locked for N6, clean-up: no 6 in R56C7

51. Naked pair {58} in R56C7, locked for C7 and N6 -> R6C9 = 6, R6C1 = 3 (step 44), R6C8 = 7

52. R9C7 = 7 (hidden single in N9), R1C7 = 6, clean-up: no 7 in R2C5, no 8 in R9C45 (step 38a)
52a. R89C8 = 9 = {18/36}, no 2

53. Naked pair {58} in R7C89, locked for R7 and N9, clean-up: no 1 in R89C8 (step 52a)

54. Naked pair {36} in R89C8, locked for C8 and N9, clean-up: no 2 in R8C9 -> R5C89 = [93], clean-up: no 6 in R5C456 (step 48), no 2 in R8C9

55. Naked pair {14} in R89C9, locked for C9 and N9 -> R4C9 = 2, R8C7 = 2, R7C6 = 3 (step 34a), R7C4 = 1, R8C3 = 3, clean-up: no 9 in 17(3) cage in N7 (step 47a)

56. 17(3) cage in N7 = {458}, locked for N7, clean-up: no 6 in R89C1

57. Naked pair {19} in R89C1, locked for C1

58. Naked pair {69} in R9C45, locked for R9 and N8 -> R89C1 = [91], R89C8 = [63], R89C9 = [14], R9C23 = [85], R8C2 = 4, R1C3 = 1 (step 21)

59. Naked triple {145} in R5C456, locked for R5 and N5 -> R56C7 = [85], R6C4 = 8, R56C3 = 6 = [24], R6C2 = 1, clean-up: no 6,7 in R12C4 (step 46a)

60. Naked pair {67} in R5C12, locked for N4

61. 6 in C3 locked in R23C3, locked for N1

62. 14(4) cage at R3C7 = {1346} (only remaining combination), no 5,7 -> R4C6 = 6, R3C7 = 3

60. 26(4) cage at R3C3 = {4589/4679/5678} (cannot be {3689} because R5C4 only contains 4,5), no 3 -> R4C4 = 7, R4C5 = 3, R8C4 = 5
60a. 26(4) cage = {4679/5678} = 67{49/58} -> R3C3 = 6, R2C3 = 8, R4C3 = 9, R5C4 = 4, R5C56 = [51], R4C12 = [85], R3C4 = 9, R9C45 = [69], R2C5 = 6, R1C5 = 7, R8C56 = [87], R3C5 = 1

61. Naked pair {45} in R23C6, locked for C6 -> R3C6 = 8, R2C7 = 1, R4C78 = [41]

62. R4C12 = [85] -> R23C1 = 11 = {47}
62a. Naked pair {47} in R23C1, locked for C1 and N1 -> R3C2 = 2, R1C1 = 5, R5C12 = [67], R7C12 = [26]

63. R4C89 = [12] -> R23C9 = 12 = {57}
63a. Naked pair {57}, locked for C9 and N3

and the rest is naked singles

I think I missed some hidden singles in the later steps but it probably didn't make much difference.


I'll try to find time to have a go at A68V1.5 (I'll probably "pass" on V2 and V3) and at least one of Para's Transformers


Assassin 68 thread T&E, hypothetical-AIC discussion
T&E Walkthrough by Frank:
Nice one. Here is my jiggery-pokery (T & E) walkthrough:

1. Outies of C12 => R19C3 = 6

2. Innies of R89 => R8C37 = 5

3. Innies of N8 => R7C456 = 8

4. Outies of N7 => R6C1+R7C4 = 4

=> R6C1+R7C4 = [13], [22], or [31]

Now comes the jiggery-pokery part Smile

[13] quickly leads to a contradiction

[22] quickly leads to a contradiction

So R6C1+R7C4 = [31]

And the rest is plain sailing
gary w question:
I wonder if the moves outlined below constitute an AIC or a hypothetical?
In any event,having seen this route I was disinclined to look for others!

prelims...

r6c9+r7c6=9
r6c1+r7c4=4 > r6c1,r7c4=13/22/31
r8c3=r8c7=5 > 1/4 2/3
r6c159=11
r7c456=8 > (1/3/4) (1/2/5)

r6c9 <> 8 otherwise in N6 R4C789 would be 5 maximum..impossible

thus r7c6<> 1

If R7C5=1 >R6C5=5 >R6C9=4 >R6C1=2 >R7C456=215
thus r7c12=(3/6)..X

Now,both r78c3 and r78c7 =9 and the 5 at r7c6 > r8c3+r8c7<> (1/4)>r8c3=3
Contradiction wth X.

Thus r7c4=1.This breaks puzzle.
mhparker answering questions raised:
Hi all,

This post is aimed at answering one or two of the questions above.
frank wrote:
Here is my jiggery-pokery (T & E) walkthrough
Howard S wrote:
Why is Frank's approach deemed to be T&E?
Unfortunately, I'm not able to answer that question in this specific case, because Frank did not provide any detail as to why or how each of the two combinations he mentioned "quickly leads to a contradiction". This is a pity, because this is the most interesting bit! Furthermore, the logic leading to the contradiction is not obvious enough to render an explanation superfluous. Although Frank could no doubt provide more information here, it's my experience that, on having to explain the logic to others, the logic used suddenly appears to be nowhere near as simple as one originally thought!
gary w wrote:
I wonder if the moves outlined below constitute an AIC or a hypothetical?

Thanks very much for listing the steps you took in some detail, Gary. Much appreciated. To answer your question: hypothetical - yes, AIC - no.

In order to qualify as an AIC, three key criteria need to be met:

  1. The implications must be expressible as a linear chain or loop. That is, there should be no branching.
  2. Each link must not depend on any previous links. Note that, since each implication is in itself bidirectional, this requirement makes the entire chain bidirectional.
  3. The chain should start on a false premise and end on a true premise. The idea is to show that at least one of the premises at either end of the chain must be true.
Note that, although your logic is great as a hypothetical, it fails to meet any of the above requirements for being an AIC. In particular:

The rejection of [72] for R78C3 depends on remembering that we placed a 2 in R7C4 in an earlier link in the chain. Similarly, the rejection of [54] for the same two cells relies on remembering that the starting premise placed a 5 in R7C6 (as part of the [215] permutation for the N8 innies). Last but not least, the rejection of [81] here requires creating a branch to visit the 14(3) cage at R7C6.

For comparison, to see an example of an AIC in practice, consider the following position reached on my solving path for the [b]A68:

Code:
.-----------------------------------.-----------.-----------.-----------.-----------------------------------.
| 123456789   123456789   12        | 123456789 | 456789    | 1234567   | 56789       123456789   123456789 |
:-----------.           .-----------'           |           |           '-----------.           .-----------:
| 123456789 | 123456789 | 12345689    123456789 | 456789    | 1234567     1234567   | 123456789 | 123456789 |
|           |           :-----------.-----------'-----------'-----------.-----------:           |           |
| 123456789 | 123456789 | 2345689   | 3456789     123456789   123456789 | 12345678  | 123456789 | 123456789 |
|           '-----------:           '-----------.           .-----------'           :-----------'           |
| 23456789    23456789  | 2345689     23456789  | 23456789  | 2345678     12345678  | 123456789   123456789 |
:-----------------------+-----------.           |           |           .-----------+-----------------------:
| 123456789   123456789 | 12345689  | 234567    | 1234567   | 1234567   | 56789     | 23456789    23456789  |
:-----------.           |           '-----------+-----------+-----------'           |           .-----------:
| 13        | 123456789 | 12345689    2456789   | 1245      | 56789       56789     | 2345678   | 45678     |
|           '-----------+-----------------------:           :-----------------------+-----------'           |
| 2468        2468      | 7           13        | 1245      | 12345       2345689   | 2345689     2345689   |
:-----------.-----------:           .-----------'-----------'-----------.           :-----------.-----------:
| 124689    | 34589     | 13        | 3456789     3456789     3456789   | 24        | 12345678  | 1234      |
|           |           '-----------+-----------------------------------+-----------'           |           |
| 124689    | 34589       45        | 3456789     23456789    23456789  | 45678       12345678  | 1234      |
'-----------'-----------------------'-----------------------------------'-----------------------'-----------'


From this position, I found the following grouped AIC (Eureka notation):

(3)r8c3=(3)r89c2,(5)r9c3,(1)r1c3-(1=3)r8c3

This AIC forms a discontinuous loop, with two strong links on the same candidate at the discontinuity (r8c3).

In plain English, this AIC can be expressed as follows:

"If R8C3 is not 3, then R89C2 must contain a 3 (only other place for 3 in N7), implying that r9c3 must be a 5 (because only combination for 17(3) cage containing a 3 is {359}, which would force a 5 into R9C3). This in turn forces a 1 into R1C3 (outies C12 = R19C3 = 6(2)), which prevents a 1 from existing in R8C3, implying that R8C3 must be a 3 (only other candidate in bivalue cell)."

Thus, this logic shows that if R8C3 is not a 3 (starting premise), then it must be a 3, which is a contradiction. The result is that we can immediately place a 3 in this cell!

The above AIC consists of two strong links, 1 weak link and 2 direct links, as follows:

1. R8C3 <> 3 => R89C2 contain a 3 (strong link, N7)
2. R89C2 = {3..} => R9C3 = 5 (direct link based on 17(3) cage combinations)
3. R9C3 = 5 => R1C3 = 1 (direct link, C12 outies)
4. R1C3 = 1 => R8C3 <> 1 (weak link, C3)
5. R8C3 <> 1 => R8C3 = 3 (strong link, bivalue cell R8C3)

The important thing to note is that all of the above implications work independently of each other, regardless of what has gone before. Furthermore, the logic is representable as a linear implication chain (forming a closed loop in this case). In addition, the chain begins on a false premise (R8C3 is not 3) and ends on a true premise (R8C3 is 3). Thus all key requirements for an AIC listed above are fulfilled.[/color][/quote]

Note that AICs represent an advanced solving technique. The difficulties come from getting used to the cryptic Eureka notation and in distinguishing bona fide AICs from other intuitive contradiction moves that look deceptively similar. Like many formalisms, they require a deep understanding of the basic principles.


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PostPosted: Sun Jul 06, 2008 11:03 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Assassin 68 v2 by Ruud (Sept 07) Edit: image corrected. Thanks for noticing Børge.
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:5376:5376:5376:6147:2308:5637:4870:4870:4870:3593:5376:6147:6147:2308:5637:5637:4870:5137:3593:5376:5396:5397:5397:5397:6424:4870:5137:3593:3593:5396:5396:5397:6424:6424:5137:5137:4644:4644:2598:5396:5397:6424:3882:3883:3883:4653:4644:2598:2598:3121:3882:3882:3883:3893:4653:4653:3384:3384:3121:3899:3899:3893:3893:3647:3136:3384:3650:3650:3650:3899:2886:3911:3647:3136:3136:3147:3147:3147:2886:2886:3911:
Solution:
+-------+-------+-------+
| 5 7 2 | 8 6 9 | 3 4 1 |
| 4 1 9 | 7 3 5 | 8 6 2 |
| 3 6 8 | 2 1 4 | 9 5 7 |
+-------+-------+-------+
| 2 5 4 | 6 9 7 | 1 3 8 |
| 1 9 6 | 3 5 8 | 7 2 4 |
| 7 8 3 | 1 4 2 | 6 9 5 |
+-------+-------+-------+
| 9 2 1 | 5 8 6 | 4 7 3 |
| 8 4 7 | 9 2 3 | 5 1 6 |
| 6 3 5 | 4 7 1 | 2 8 9 |
+-------+-------+-------+
Quote:
Ruud, lead-in: This Transformer pattern was very fruitful. Here are some variations..(rated 2.0-3.0?)
Para: It is probably a high 1.75 because it uses no hypotheticals, mostly creative combination work.
mhparker: I would rate it around 1.75 (i.e., roughly A60RP level)
sudokuEd: lots of combination pecking....Still no-where near A60 RP Lite in difficulty. But a very nice 1.75! Para and especially Mike found lots of things I missed
Andrew (in 2013): Another variant I came back to this year. My solving path was more like Ed's than the other two. However I missed 45 rule on R89 4 outies R7C3467=16 so didn't see the interactions between this and R7C456=19, which Ed, Mike and Para all used. Instead my solving path reduced hidden 8(3) cage R389C6 to one combination.
Para wrote:
I checked it against Mike's walk-through. Exactly the same opening digit. I think you are kinda forced to get to that one, because this pattern forces you to work in one part of the puzzle, just because the rest doesn't give away anything.
An interesting thought, but Ed's first placement was different and mine was different from all three of you.
Moderator edit:SSv3.6.3 gives this puzzle a 1.35 and stepping through it, nothing too complicated which suggests all 4 solutions below missed something big.
2021 Revisit to this puzzle here
Walkthrough by Para:
Good morning all.

Here's last nights effort on Assassin 68V2. It is probably a high 1.75 because it uses no hypotheticals, mostly creative combination work.
Solved this one in my second attempt. First try i went to about step 25. Then couldn't see the next "obvious" step. The second attempt broke it open. Luckily spotted a logical flaw somewhere along and could correct it. The flaw did lead to the correct solution.
Now only got the V3 left. Hope i can work it in a similar way, because i can't think of any other way to tackle this pattern.

Enjoy! :D

Ok this is the final draft. I couldn't get that stupid site to work, kept making mistakes. Just too difficutl so split it up in smaller steps. Now i am almost sure it is without mistakes. Other wise i give up.

Walk-Through Assassin 68V2

1. R12C5 = {18/27/36/45}: no 9

2. 19(5) at R1C7 = {12349/12358/12367/12457/13456}: 1 locked for N3

3. 14(4) at R2C1 = {1238/1247/1256/1346/2345}: no 9

4. 10(3) at R5C3 = {127/136/145/235}: no 8,9

5. R67C5 = {39/48/57}: no 1,2,6

6. R89C1 = {59/68}: no 1,2,3,4,7

7. 11(3) at R8C8 = {128/137/146/236/245}: no 9

8. R89C9 = {69/78}: no 1,2,3,4,5

9. 24(3) at R1C4 = {789} -->> locked within cage: R2C56: no 7,8,9
9a. Clean up: R1C5 no 1,2

10. 22(3) at R1C6 = {589/679}: no 1,2,3,4; 9 locked within cage -->> R2C4: no 9
10a. R2C6 needs one of {56} -->> R1C6 + R2C7: no 5,6

11. Naked Triple {789} in R2C347 -->> locked for R2

12. Naked Triple {789} in R1C46 + R2C4 -->> locked for N2
12a. 9 in N2 locked for R1
12b. Clean up: R2C5: no 1,2
12c. Killer Pair {56} in R12C5 + R2C6 -->> locked for N2
12d. 1 and 2 in N2 locked for R3 and 21(5) at R3C4

13. 21(5) at R3C4 needs {12} and one of {34} in R3C456 -->> 21(5) = {12369/12378/12459}:{12468} blocked by R12C5(removes all combinations for this cage) -->> R45C5 = {59/69/78}: no 3,4

14. R12C5 = {5|6..}; R45C5 = {5|6|7..} -->> R67C5 = {39/48}: {57} blocked by R12C5: R45C5: no 5,7
14a. Killer Pair {89} in R45C5 + R67C5 -->> locked for C5
14b. Killer Pair {34} in R12C5 + R67C5 -->> locked for C5

15. 45 on R89: 2 innies: R8C47 = 12 = {39/48/57}: no 1,2,6

16. 45 on N7: 2 outies: R6C1 + R7C4 = 12 = {39/48/57/66}: no 1,2

17. 45 on N9: 2 outies: R6C9 + R7C6 = 11 = {29/38/47/56}: no 1

18. 45 on C12: 2 outies: R19C3 = 7 = {16/25/34}: no 7,8,9

19. 45 on C89: 2 outies: R19C7 = 5 = {14/23}: no 5,6,7,8

20. 45 on C6789: 3 innies: R389C6 = 8 = {125/134}: no 6,7,8,9; 1 locked for C6
20a. 1 in N5 locked for C4

21. 45 on R9: 2 innies and 2 outies: R8C28 + 10 = R9C19: Max R9C19 = 17 -->> Max R8C28 = 7: no 7,8,9

22. 45 in N1: 2 innies and 2 outies: R4C12 + 10 = R23C3: Max R23C3 = 17 -->> Max R4C12 = 7: no 7,8; Min R4C12 = 3 -->> Min R23C3 = 13: no 3

23. 13(3) at R7C3 = {139/148/157/238/247/256/346}: Min R7C4 + R8C3 = 7 -->> R7C3: no 7,8,9
23a. When {157/256}, {12} in R7C3 -->> R7C3: no 5

24. 45 on C123: 2 outies and 3 innies: R67C4 + 15 = R234C3: Max R234C3 = 24 -->> Max R67C4 = 9: R6C4: no 7; R7C4: no 9; Min R67C4 = 4 -->> Min R234C3 = 19: no 1
24a. Clean up: R6C1: no 3

25. R12C5 = {4|6..}; R67C5 = {4|9..} -->> R45C5 = {59/78}: {69} blocked by R12C5 + R67C5: no 6

26. 45 on N8: 3 innies: R7C456 = 19 -->> R7C46 = 10/11/15/16

27. 45 on R89: 4 outies R7C3478 = 16: R7C46 = 10/11: (15/16) blocked-->> R7C37 = 5/6: no 6,7,8,9
27a. R7C46 = 10/11 -->> R7C6: no 9; R7C5: no 3,4(step 26)
27b. Clean up: R6C5: no 8,9; R6C9: no 2

28. R7C456 = 19: [892]/{3[9]7}/{4[9]6}/{4[8]7}/{5[8]6}: R7C6: no 8
28a. Clean up: R6C9: no 3

29. 18(3) at R6C1 = {279/378/459/468}: {189/369/567} blocked by R89C1: no 1

30. 45 on N9: 1 outie + 2 innies: R7C6 + 4 = R7C89: R7C89: no 4: Min R7C89 = 6

31. 45 on N7: 1 outie + 2 innies: R7C4 + 6 = R7C12: R7C12: no 6: Min R7C12 = 9

32. 45 on N89: 4 innies: R7C4589 = 22 [mod edit: actually = 23 -> following step not valid which may make the rest of the solution invalid also; let Ed know if you've worked out what to do about this step. Thanks!]: Min R7C89 = 6 -->> Max R7C45 = 16: R7C4: no 8
32a. Clean up: R6C1: no 4; R7C6: no 2(step 28); R6C9: no 9

33. 18(3) at R6C1 = [5]{49}/[6]{48}/[7]{29}/[7]{38}/[8]{37}/[9]{27}/[9]{45}

34. 13(3) at R7C3 = [139/148]/[1]{57}/[238]/[265]: [2]{47} and {3[6]4} blocked by combination for 18(3) at R6C1: R7C3 = {12}; R8C3 = {5789}
34a. Clean up: R8C7: no 8,9

35. 15(3) at R7C6 = [627/645/654/357/537/735/753]: R7C6: no 4; R7C7: no 1
35a. Clean up: R6C9: no 7

36. 15(3) at R6C9 = [4]{29}/[5]{19/28/37}: [4]{38/56},[6]{27} and [8]{25} blocked by combinations for 15(3) at R7C3,{168} blocked by 15(2) at R8C9: R6C9 = {45}, R7C89 = {19/28/29/37}: no 5,6
36a. Clean up: R7C6: no 3,5

37. R7C456 = [397/496/487/586]: R7C6: no 6,7
37a. R7C6 = 6(hidden); R2C6 = 5; R6C9 = 5(step 17)
37b. Clean up: R7C4: no 3(step 37); R6C1: no 6,9
37c. R1C6 + R2C7 = {89}: R1C89 + R2C4: no 8
37d. R2C4 = 7

38. R12C5 = {36}(last combo); locked for C5 and N2
38a. R67C5 = [48]; R7C4 = 5(step 37); R6C1 = 7(step 16)

39. R7C12 = {29}(last combo) -->> locked for R7 and N7
39a. R89C1 = {68}(last combo) -->> locked for C1 and N7
39b. R8C3 = 7; R7C3 = 1; R8C7 = 5(step 15); R7C7 = 4
39c. R9C5 = 7(hidden)

40. 12(3) at R9C4 = 7{14/23}: no 9
40a. R8C4 = 9(hidden); R1C46 = [89]; R2C37 = [98]
40b. R9C9 = 9(hidden); R8C9 = 6; R89C1 = [86]; R9C8 = 8(hidden)

41. R7C89 = {37} -->> locked for N9

42. R19C7 = [32](step 19; last combo)
42a.R12C5 = [63]; R8C8 = 1; R8C56 = [23]; R8C2 = 4; R9C46 = [41]
42b. R6C456 = [214]; R3C9 = 7; R7C89 = [73]; R1C2 = 7(hidden)

43. 7 in C6 locked within 25(4) cage at R3C6
43a. R4C6 = 7(hidden); R5C7 = 7(hidden)

44. R6C67 = [26](last combo)
44a. R45C7 = [91]; R6c34 = [31]; R5C3 = 6

45. 14(4) at R2C1 = {2345}: no 1

And the rest is all singles.

I checked it against Mike's walk-through. Exactly the same opening digit. I think you are kinda forced to get to that one, because this pattern forces you to work in one part of the puzzle, just because the rest doesn't give away anything.

greetings

Para
Walkthrough by mhparker:
sudokuEd wrote:
Hmm - must be rusty: :wink: only got to step 25 on the second attempt.

Who's this "Ed" guy? Reminds me of someone I used to know around here... :wink:

Here's my V2 attempt. Will be interesting to compare notes with Ed + Para:

[Edit: I would rate it around 1.75 (i.e., roughly A60RP level).]

Assassin 68 V2 Walkthrough

1. 9(2)n2 = {18/27/36/45} (no 9)

2. 19(5)n3 = {12349/12358/12367/12457/13456}
2a. -> 1 locked for n3

3. 14(3)n14 = {1238/1247/1256/1346/2345} (no 9)

4. 10(3)n45 = {127/136/145/235} (no 8,9)

5. 12(2)n58 = {39/48/57} (no 1,2,6)

6. 14(2)n7 = {59/68} (no 1,2,3,4,7)

7. 11(3)n9 = {128/137/146/236/245} (no 9)

8. 15(2)n9 = {69/78} (no 1,2,3,4,5)

9. 24(3)n12 = {789}
9a. CPE: no 7,8,9 in r2c56
9b. Cleanup: no 1,2 in r1c5

10. 22(3)n23 = {(58/67)9} (no 1,2,3,4)
10a. -> must have exactly 1 of {56}, which must go in r2c6
10b. -> no 5,6 in r1c6+r2c7
10c. 22(3)n23 and 24(3)n12 form grouped X-Wing on 9 in r12
10d. -> no 9 elsewhere in r12

11. 9 in r1 locked in n2 -> not elsewhere in n2

12. Naked Triple (NT) on {789} at r1c46+r2c4
12a. -> no 7,8,9 elsewhere in n2
12b. Cleanup: no 1,2 in r2c5

13. 9(2)n2 (step 1) now {36/45} = {(3/4)..},{(5/6)..}
13a. -> 9(2)n2 and r2c6 form killer pair on {56}
13b. -> no 5,6 elsewhere in n2

14. Naked Triple (NT) on {789} at r2c347
14a. -> no 7,8 elsewhere in r2 (Note: 9 already eliminated in step 10d)

15. Outies n2: r2c37+r45c5 = 31(5)
15a. 9 of r2 locked in r2c37 = {79/89} = 16 or 17
15b. -> r45c5 = 14 or 15 = {59/68/69/78} (no 1,2,3,4) = {(8/9)..}
15c. -> 9(2)n2 (step 12) and r45c5 together block {57} combo for 12(2)n58
15d. -> 12(2)n58 = {39/48} (no 5,7) = {(3/4)..},{(8/9)..}
15e. -> 9(2)n2 and 12(2)n58 form killer pair on {34} -> no 3,4 elsewhere in c5
15f. 12(2)n58 and r45c5 (step 15b) form killer pair on {89} -> no 8,9 elsewhere in c5

16. {12} in n2 locked in r3 -> not elsewhere in r3

17. Innies r89: r8c37 = 12(2) = {39/48/57} (no 1,2,6)

18. Outies c12: r19c3 = 7(2) = {16/25/34} (no 7,8,9)

19. Outies c89: r19c7 = 5(2) = {14/23} (no 5,6,7,8)

20. Innies c6789: r389c6 = 8(3) = {125/134} (no 6,7,8,9)
20a. -> 1 locked for c6

21. 1 in n5 locked in c3 -> not elsewhere in c3

22. Outies n7: r6c1+r7c4 = 12(2) = {39/48/57/66} (no 1,2)
22a. min. r7c4+r8c3 = {34} = 7
22b. -> max. r7c3 = 6 (no 7,8,9)

23. Outies n9: r6c9+r7c6 = 11(2) = {29/38/47/56} (no 1)

24. I/O diff. n7: r7c12 = r7c4 + 6
24a. -> no 6 in r7c12 (IOU)

25. I/O diff. n9: r7c89 = r7c6 + 4
25a. -> no 4 in r7c89 (IOU)

26. 18(3)n47 = {279/378/459/468} (no 1)
26a. Note: {189/369/567} all blocked by 14(2)n7 (step 6)

27. 12(3)n7 = {129/138/147/237/246/345}
27a. Note: {156} blocked by 14(2)n7 (step 6)

28. 13(3)n78 can only contain at most 1 of {56789} in n7 (due to no {12} in r7c4)
28a. 14(2)n7 contains exactly 2 of {56789}
28b. 12(3)n7 contains exactly 1 of {56789} (step 27)
28c. -> 18(3)n47 cannot contain two of {56789} in n7
28d. -> r6c1 = {56789} (step 26)
28e. furthermore, 13(3)n78 must now contain exactly 1 of {56789} in n7
28f. -> no 8,9 in r7c4

29. Outies r89: r7c3467 = 16(4)

30. Innies n8: r7c456 = 19(3)

31. Combining steps 29 and 30: r7c5 = r7c37 + 3
31a. -> r7c5 = min. 6
31b. -> r7c5 = {89}, r6c5 = {34}
31c. -> r7c37 sum to 5 or 6
31d. -> no 6,7,8,9 in r7c37

32. Revisit step 30: r7c456 = {379/469/478/568} (no 2)
32a. only 1 of {89}, which must go in r7c5
32b. -> no 8,9 in r7c6
32c. cleanup: no 2,3,9 in r6c9 (step 23)

33. Hidden killer pair on {89} in c6 at r1456c6
33a. r456c6 cannot contain both of {89} due to r45c5 (step 15b)
33b. -> r1c6 = {89} (no 7) r456c6 = {(8/9)..}
33c. -> r45c5 and r456c6 form killer pair on {89} -> no 8,9 elsewhere in n5

34. 7 in n2 locked in r12c4 -> no 7 elsewhere in c4
34a. no 7 in r2c3
34b. cleanup: no 5 in r6c1 (step 22)

35. Outies r789: r6c159 = 16(3) = [637/736/745/835/934]
35a. -> no 8 in r6c9
35b. cleanup: no 3 in r7c6 (step 23)

36. {12} unavailable in r7c4+r8c3
36a. -> no 5 in r7c3 (permutations, 13(3) cage)

37. Innies n78: r7c1256 = 25(4) = {2689/3589/3679/4579/4678}
37a. (Note: cannot be {1789}, because 1 unavailable)
37b. -> must contain 2 of {789}
37c. only other place for {789} in r7 is r7c89
37d. -> r7c1256 and r7c89 form hidden killer triple on {789} in r7
37e. -> r7c89 = {(7/8/9)...}
37f. -> no 5,6 in r7c89; no 7 in r6c9
37g. cleanup: no 4 in r7c6 (step 23)

38. 15(3)n69 = {159/249/258/348/357} (no 6)
38a. Note: {168/267} both blocked by 15(2)n9 (step 8)
38b. cleanup: no 5 in r7c6 (step 23)
38c. from step 35, r6c159 now [745/835/934]
38d. -> no 6 in r6c1
38e. -> no 6 in r7c4 (step 22)

39. Hidden single (HS) in r7 at r7c6 = 6
39a. -> r6c9 = 5 (step 23)
39b. cleanup: no 9 in r6c1 (step 38c) -> no 3 in r7c4 (step 22); no 9 in r8c7 -> no 3 in r8c3 (step 17)

40. Naked single (NS) at r2c6 = 5
40a. cleanup: no 4 in 9(2)n2 (step 1); no 7 in r2c7 (step 10)

41. HS in r2 at r2c4 = 7

42. HS in c5 at r6c5 = 4
42a. -> r7c5 = 8, r6c1 = 7 (step 38c), r7c4 = 5 (step 22)

43. HS in c4 at r1c4 = 8
43a. -> r1c6 = 9; r2c37 = [98]

The rest solves easily now.
Walkthrough by sudokuEd:
Here's a very different way of tackling Assassin 68V2. Para and especially Mike found lots of things I missed: various '45's, implied subsets, combo conflicts and IOU's. Well done.

So mine is more combination pecking. Hopefully easy enough to follow. Still no-where near A60 RP Lite in difficulty. But a very nice 1.75! Sadly, my way doesn't help much with the V3.

Assassin 68 V2
Preliminaries
i. 24(3)n2 = {789}
ii. 9(2)n2: no 9
iii. 22(3)n2 = {589/679}
iv. 19(5)n3 must have 1: 1 locked for n3
v. 14(4)n1: no 9
vi. 10(3)n4: no 8 or 9
vii. 12(2)n5: no 1,2 or 6
viii. 14(2)n7 = {59/68}
ix. 11(3)n9: no 9
x. 15(2)n9 = {69/78}


1. 24(3)n2 = {789} -> no 7,8,9 in r2c56
1a. no 1 or 2 in r1c5

2. 22(3)n2 = 9{58/67} = [5/6..]
2a. r2c6 = {56} -> no 5/6 elsewhere in 22(3)n2 (step 2)
2b. 22(3) must have 9 -> no 9 r2c4
2c. Generalized X-Wing 9 in 24(3) & 22(3)-> 9 locked for r12

3. Naked triple {789} in r12c4 + r1c6: all locked for n2
3a. no 1 or 2 in r2c5

4. 9(2)n2 = {36/45} = [4/6,5/6..]
4a. Killer pair 5/6 with r2c6: both locked for n2

5. 1 & 2 in n2 only in r3: both locked for r3 and no 1 or 2 in r45c5
5a. 21(5) must have 3 of {1234} in r3c456 = 12{369/378/459} ({12468} blocked: clashes with [4/6] needed in 9(2)n2 which 'sees' each cell in 21(4))
5b. r45c5 = {69/78/59}(no 3,4) = [8/9..]

6. Naked Triple {789} r2c347: all locked for r2

7. deleted

8. {57} combo. in 12(2)n5 is blocked: forces r45c5 to {69} (step 5b) but {56} clashes with 5/6 needed in 9(2)n2
8a. 12(2)n5 = {39/48} = [8/9..]
8b. Killer pair {89} in r4567c5: locked for c5
8c. Killer pair {34} in 9(2) & 12(2): both locked for c5

9. "45" c12: r19c3 = h7(2): no 789

10. "45" c1234: r389c4 = h15(3)
10. max. r3c4 = 4 -> min. r89c4 = 11 (no 1)

11. "45" c89: r19c7 = h5(2) = {14/23}

12. "45" c6789: r389c6 = h8(3) = 1{25/34}
12a. 1 locked for c6

13. 1 in n5 only in c4: 1 locked for c4
13a. h15(3)r389c4 = {249/258/267/348/357/456}

14. "45" r9: r8c1289 = h19(4)
14a. min r8c19 = 11 -> max r8c28 = 8 (no 8,9)

15. "45" r89: r8c37 = h12(2) = {39/48/57} (no 1,2 or 6)

16. "45" r89: r7c3467 = h16(4)

17. "45" n8: r7c456 = h19(3) -> r7c5 = r7c37 + 3 (step 16)
17a. min r7c37 = {12} = 3 -> min r7c5 = 6
17b. r67c5 = [39/48]
17c. r7c5 = 8/9 -> r7c37 = 5/6 = {14/23/15/24}(no 6789}
17d. r7c37+r7c5 = {14/23}[8]/{15/24}[9]

18. 14(3)n8 = {149/158/167/239/248/257/356} ({347} clashes with h12(2)r8c37)

19. "45" r789: r6c159 = h16(3)
19a. max r6c5 = 4 -> min r6c19 = 12 (no 1,2)

20. "45" n78: 2 outies r6c15 - 5 = r7c6
20a. max r6c15 = [94] = 13 -> max r7c6 = 8. However this means 2 8's in r7c56
20b. -> r6c15 max = [93] = 12 -> max r7c6 = 7
20c. min. r6c9 = 4 (step 19)
20d. max r7c56 = [97] = 16 -> min r7c4 = 3 (step 17)

21. "45" n9: 2 outies = 11

Should have done this next one after 17b.
23. "45" r1234: r5c456 = h16(3) & remembering r45c5 = {59/68/78} (step 5b)
23a. r5c456 = {169/178/268/358/367/457} ({259} blocked by clash with r4c5;{349} blocked by r6c5)
23b. 1 only in r5c4 -> no 9 r5c4

24. "45" n1:r4c12 + 10 = r23c3
24a. min r4c12 = 3 -> min r23c3 = 13
24b. no 3 r3c3
24c. max r23c3 = 17 -> max r4c12 = 7
24d. r4c12, no 7,8

25. 12(3)n7 = {129/138/147/237/246/345}({156} blocked by 14(2))

26. A nice straight chain shows a complex hidden single 5 in r1245c5. Here's how.
26a. r45c5 = {59/69/78} (step 5b) & 9(2) = [5/6..]
26b. -> r1245c5 = Killer pair 5/6 or r45c5 = {78} -> 12(2) = [39] -> 9(2) = {45}
26c. ->5 locked for c5

27. 12(3)n8 = {129/138/147/156/237/246} = [1/6/7..]({345} blocked by r9c5)

28. 14(3)n8 = {149/158/239/248/257/356} ({167} blocked by 12(3) step 27)
28a. 1 must be in r8c5 for {149/158} -> no 1 r8c6
28b. 4 must be in r8c6 for {149/248} -> no 4 r8c4
28c. 6 must be in r8c5 for {356} -> no 6 r8c4

29. deleted

30. deleted

31. deleted

32. OK - much more creative. Maybe this is a Killer AIC.
32a. when r3c4 = 4 -> 9(2)n2 = {36} -> r2c6 = 5 -> h8(3)r389c6 = [1]{34}
32b. this is useful for next step

33. h15(3)r389c4 = {249/258/267/348/357/456}
33a. = [294/4{29}/2{58}/276/384/357] ([4]{38} blocked by 3 in r89c6 step 32a; [456] blocked by {34} in r89c6 (step 32a- forces r8c5 = 6: 2 6's n8;[375] blocked by 14(3) must be [725] but 2 5s n8)
33b. no 3 r89c6;

34. h15(3)r389c4 = {249/258/267/348/357) = [7/8/9..]
34a. hidden killer triple 7/8/9 with r12c4: all locked for c4

35. h19(3)r7c456 = {379/469/478/568}(no 2)
35a. 3 in {379} only in r7c4 -> no 3 r7c6

36. 13(3)n7 = {139/148/157/238/247/256/346}
36a. 1 and 2 only in r7c3 -> no 5 r7c3
36b. r7c34 = [13/14/15/23/24/26/36/46]

37. "45" n9: 2 outies = 11
37a. min r7c6 = 4 -> max r6c9 = 7

38. "45" n7: 2 outies = 12
38a. max r7c4 = 6 -> min r6c1 = 6

39. h16(3)r6c159 = [934/835/637/745] = [3/5..] ([736] blocked by [55] forced into r7c46 steps 37, 38)
39a. no 6 r6c9 -> no 5 r7c6 (step 37)
39b. & following up on step 37, 38: h19(3)r7c456 = [397/496/694/586] = {379/469/568} = [3/6..]
39c. -> from step 36b.r7c34 = [13/14/15/23/24/26/36]([46] blocked by [694] in h19(3))
39d. no 4 r7c3
39e. max r7c34 = 9 -> min r8c3 = 4
39f. no 9 r8c7 (h12(2)r8c47)

40. 14(3)n8 = {149/158/239/248/257}(no 6) = [1/2..]({356} blocked by h19(3) step 39b)

41. 12(3)n8 = {138/147/156/237/246}(no 9) ({129} blocked by 14(3) step 40)

42. r89c5 must have 1/2 for c5 (but cannot have both)
42a. When 14(3)n8 = {257} -> 12(3)n8 can only be {138} = [813] -> r8c5 !=2 (step 42) -> 14(3) = [572]
42b. -> 7 must be in r8c5 when 14(3) = {257} -> no 7 r8c4
42c. no other combo with 7, & {257} cannot combine with any other combo's in 12(3)
42d. -> no 7 r8c4

43. from step 33. h15(3)r389c4 = [294/492/258/285/384/357](no 6 r9c4) ([276] blocked by r8c4)

44. 12(3)n8 = {138/147/156/237/246}
44a. 5 in {156} must be in r6c4 -> no 5 r9c6

45. h8(3)r389c6 = 1{25/34}.
45a. 5 in {125} only in r8c6 -> no 2 r8c6

46. 14(3)n8 = {149/158/239/248/257}
46a. 5 in {158/257} must be in r8c6
46b. no 5 r8c4

47. from step 43. h15(3)r389c4 = [294/492/285/384]
47a. no 2 r8c4; no 7,8 r9c4

48. 7 in c4 only in r12c4: 7 locked for n2 and not in r2c3

48. 14(3)n7 = {149/158/239/248}(no 7)

49. Naked pair {12} r38c5: locked for c5

50. from step 39c. r7c34 = [13/14/15/23/24/26/36]
50a. 13(3) = [139/148/157/247/265/364] ([238] combined with h12(2)r8c37 clashes with [3/4/8] needed in 14(3)n8;
50b. r7c34 = [13/14/15/24/26/36]

51. from step 39b h19(3)r7c456 = [397/496/694/586]
51a. combining with step 50b -> h16(4)r7c3467+r7c5 = [1375][9]/[1465][9]/[1564][8] ([2464][9]/[2644][9]/[3643][9] clash)
51b. -> r7c3 = 1 finally!
51c. r7c4 = {345}, r7c6 = {67}, r7c7 = {45}
51d. 13(3)n7 = [1][39/48/57]
51e. r8c3 = {789}
51f. r8c7 = {345}
51g. 15(3)n8 = {357/456} = [753/654/645]
51h. r6c1: no 6 ("45"n7)
51i. r6c9: no 7 ("45"n9)
51j. r7c456 = [397/496/586] -> h16(3)r6c159 = [934/835/745]

52. 6 in c4 only in n5: 6 locked for n5
52a. r45c5 = {78/59}

53. from step 51g. r78c7 = [53/54/45]: 5 locked for n9 & c7
53a. = 8/9

54. "45"n9: 4 innies = 19 = 5{239/248/347}(no 6)
54a. from step 53, r7c89 = {29/28/37} (no 4)
54b. 15(3)n6 = {249/258/357}

55. 11(3)n9 must have 1 for n9 = 1{28/37/46}

56. 12(3)n7 = {237/246/345}

56. from 51d. r7c4 + r8c3 = [39/48/57]

57. deleted

58. c34: generalized swordfish (or is triple X-wing?) {789} in 24(3)n2(X3), r8c34(X2), 21(4)n1(X1)
58a. -> no 7 in 10(3)n4 = {136/145/235}
58b. -> 21(4)n1 can only 1 of {789} = {1569/2469/2468/3459/3468/3567}

59. h7(2)r19c3 = {25/34} = [2/3,3/5,4/5..]
59a. -> 10(3)n4 = {136/235}(no 4) = {36}[1]/{25}[3] ({145} blocked by h7(2))
59b. r6c4 = {13}

60. 6 in c4 only in r45c4 in 21(4)n1
60a. no 6 r34c3

61. 6 in c3 only in r56c3 -> 10(3) = {36}[1]
61a. {36} locked for n4 & c3

62. h7(2)r19c3 = {25} locked for c3

63. 18(3)r5c1 = {189/279/459} = 9{..}
63a. 9 locked for n4
63b. no 3 r7c4
63c. no 9 r8c3

64. from 51j. r7c456 = [496/586] -> h16(3)r6c159 = [835/745]
64a. r6c9 = 5, r7c6 = 6, r9c5 = 7
64b. r45c5 = {59}: both locked for c5 & n5
64c. 9(2)n2 = {36}: 3 locked for n2, c5


things feel easy from here!
Andrew's walkthrough in 2013:
Prelims

a) R12C5 = {18/27/36/45}, no 9
b) R67C5 = {39/48/57}, no 1,2,6
c) R89C1 = {59/68}
d) R89C9 = {69/78}
e) 24(3) cage at R1C4 = {789}
f) 22(3) cage at R1C6 = {589/679}
g) 10(3) cage at R5C3 = {127/136/145/235}, no 8,9
h) 11(3) cage at R8C8 = {128/137/146/236/245}, no 9
i) 14(4) cage at R2C1 = {1238/1247/1256/1346/2345}, no 9

1. 24(3) cage at R1C4, CPE no 7,8,9 in R2C56, clean-up: no 1,2 in R1C5
1a. 22(3) cage = {589/679}
1b. R2C6 = {56} -> no 5,6 in R1C6 + R2C7
1c. Naked triple {789} in R1C46 + R2C4, locked for N2, clean-up: no 1,2 in R2C5
1d. Killer pair 5,6 in R12C5 and R2C6, locked for N2
1e. Naked triple {789} in R2C347, locked for R2
1f. Caged X-Wing for 9 in 24(3) cage at R1C4 and 22(3) cage at R1C6, no other 9 in R12
1g. 9 in R1 only in R1C46, locked for N2

2. 45 rule on R89 2 innies R8C37 = 12 = {39/48/57}, no 1,2,6

3. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 12 = {39/48/57}/[66], no 1,2

4. 45 rule on N9 2(1+1) outies R6C9 + R7C6 = 11 = {29/38/47/56}, no 1

5. 45 rule on C12 2 outies R12C3 = 7 = {16/25/34}, no 7,8,9

6. 45 rule on C89 2 outies R12C7 = 5 = {14/23}

7. 19(5) cage at R1C7 = {12349/12358/12367/12457/13456}, 1 locked for N3

8. 9 in R2 only in R2C37 = {79/89} = 16,17
8a. 45 rule on N2 4(2+2) outies R2C37 + R45C5 = 31 -> R45C5 = 14,15 = {59/68/69/78}, no 1,2,3,4
8b. 1,2 in N2 only in 21(5) cage at R3C4, locked for R3
8c. 21(5) cage = {12369/12378/12459} (cannot be {12468} which clashes with R12C5)

9. 18(3) cage at R6C1 = {279/378/459/468} (cannot be {189/369/567} which clash with R89C1), no 1

10. 45 rule on C6789 3 innies R389C6 = 8 = {125/134}, 1 locked for C6
10a. 1 in N5 only in R456C4, locked for C4

11. R12C5 = {36/45}, 21(5) cage at R3C4 = {12369/12378/12459} -> combined cage R12C5 + 21(5) cage = {36}(12459}/{45}{12369}/{45}{12378}, 5 locked for C5 (because no 5 in R3C46), clean-up: no 7 in R67C5
11a. Killer pair 3,4 in R12C5 and R67C5, locked for C5
11b. Killer pair 8,9 in R45C6 and R67C5, locked for C5

12. Min R7C4 + R8C3 = 7 -> max R7C3 = 6

13. R6C1 + R7C4 = 12 (step 3)
-> R67C1 cannot total 12 (CCC) -> no 6 in R7C2
and R6C1 + R7C2 cannot total 12 (CCC) -> no 6 in R7C1

14. R6C9 + R7C6 = 11 (step 4)
-> R67C9 cannot total 11 (CCC) -> no 4 in R7C8
and R6C9 + R7C8 cannot total 11 (CCC) -> no 4 in R7C9

[With hindsight the eliminations in steps 13 and 14 are more easily obtained by
45 rule on N7 2 innies R7C12 = 1 outie R7C4 + 6, IOU no 6 in R7C12, and
45 rule on N9 2 innies R7C89 = 1 outie R7C6 + 4, IOU no 4 in R7C89
as in Para’s and Mike’s walkthroughs.]


15. 12(3) cage at R9C4 = {129/138/147/156/237/246} (cannot be {345} because no 3,4,5 in R9C5)
15a. 14(3) cage at R8C4 = {149/158/239/248/257/347/356} (cannot be {167} which clashes with 12(3) cage at R9C4)
15b. 1 of {149/158} must be in R8C5 -> no 1 in R8C6
15c. 6 of {356} must be in R8C5 -> no 6 in R8C4

16. 45 rule on C1234 3 innies R389C4 = 15 = {249/258/267/348/357/456}
16a. 12(3) cage at R9C4 (step 15) = {129/138/147/156/237/246}, 14(3) cage at R8C4 (step 15a) = {149/158/239/248/257/347/356}
16b. 12(3) cage and 14(3) cage cannot simultaneously contain 5 and 6 in R89C4 (because 12(3) cage = {156} blocks 5 from R8C4 and 12(3) cage = {246} blocks 14(3) cage = {257/356} while 14(3) cage = {158} must have 8 in R8C4)
-> R389C4 = {249/258/267/348/357}
16c. Killer triple 7,8,9 in R12C4 and R389C4, locked for C4, clean-up: no 3,4,5 in R6C1 (step 3)

17. 45 rule on N8 3 innies R7C456 = 19 = {379/469/478/568} (cannot be {289} because R7C4 only contains 3,4,5,6), no 2, clean-up: no 9 in R6C9 (step 4)
17a. 3 of {379} must be in R7C4 -> no 3 in R7C56, clean-up: no 9 in R6C5, no 8 in R6C9 (step 4)
17b. 7 of {478} must be in R7C6, 8 of {568} must be in R7C5 -> no 8 in R7C6, clean-up: no 3 in R6C9 (step 4)

18. Hidden killer triple 5,6,7 in R12C5, R45C5 and R89C5 for C5, R12C5 contains one of 5,6, R45C5 contains one of 5,6,7 -> R89C5 must contain one of 6,7
18a. Killer pair 6,7 in R7C456 and R89C5, locked for N8

19. 7 in C4 only in R12C4, locked for N2 and 24(3) cage at R1C4, no 7 in R2C3

20. 12(3) cage at R9C4 (step 15) = {129/138/147/156/237/246}
20a. 5 of {156} must be in R9C4 -> no 5 in R9C6

21. R389C6 (step 10) = {125/134}
21a. 5 of {125} must be in R8C6 -> no 2 in R8C6

22. R389C6 (step 10) = {125/134}
22a. Consider placements for 3 in N8
3 in R789C4, no 3 in R3C4, R12C5 = {36} => R2C6 = 5 => R389C6 = {134}
or 3 in R789C4, no 3 in R3C4, R3C6 = 3 => R389C6 = {134}
or 3 in R89C6 => R389C6 = {134}
-> R389C6 = {134}, locked for C6, clean-up: no 7 in R6C9 (step 4)
[With hindsight, the following alternative forcing chain is simpler
22a. Consider combinations for R12C5 = {36/45}
R12C5 = {36}, locked for N2 => R2C6 = 5 => R389C6 = {134}
or R12C5 = {45}, locked for C5 => R67C5 = [39] => 3 in C6 only in R389C6 = {134}
-> R389C6 = {134}, locked for C6, clean-up: no 7 in R6C9 (step 4)]

[Not yet cracked, but after that it’s a bit easier.]

23. 14(3) cage at R8C4 (step 15a) = {149/239/248/347/356} (cannot be {158/257} because R8C6 only contains 3,4)
23a. 8,9 of {239/248} must be in R8C4 -> no 2 in R8C4
23b. 12(3) cage at R9C4 (step 15) = {129/147/156/237} (cannot be {138/246} which clash with 14(3) cage), no 8
23c. 4 of {147} must be in R9C4 -> no 4 in R9C6
23d. R9C6 = {13} -> no 3 in R9C4, no 1 in R9C5

24. 2 in C6 only in R456C6, locked for N5
24a. 2 in C4 only in R39C4 -> R389C4 (step 16) = {249/258}, no 3
24b. 8 of {258} must be in R8C4 -> no 5 in R8C4

25. 14(3) cage at R8C4 (step 23) = {149/239/248/347}, no 6
25a. R7C456 (step 17) = {379/478/568} (cannot be {469} which clashes with 14(3) cage)
25b. 7 of {379} must be in R7C6 -> no 9 in R7C6, clean-up: no 2 in R6C9 (step 4)
25c. 4 of {478} must be in R7C4 -> no 4 in R7C5, clean-up: no 8 in R6C5

26. R389C4 (step 24a) = {249/258}
26a. R7C456 (step 25a) = {379/568} (cannot be {478} = [487] which clashes with R389C4 = [285]), no 4, clean-up: no 8 in R6C1 (step 3)

27. 18(3) cage at R6C1 (step 9) = {279/459/468} (cannot be {378} which clashes with R7C456), no 3
27a. 18(3) cage = {279/459} (cannot be {468} = [648] which clashes with R7C456 because R6C1 + R7C4 = [66], step 3), no 6,8, clean-up: no 6 in R7C4 (step 3)
27b. 18(3) cage = {279/459}, CPE no 9 in R89C1, clean-up: no 5 in R89C1

28. Naked pair {68} in R89C1, locked for C1 and N7

29. R7C456 (step 26a) = {379/568}
29a. R7C4 = {35} -> no 5 in R7C6, clean-up: no 6 in R6C9 (step 4)

30. 6 in C4 only in R456C4, locked for N5

31. 2,7 in C6 only in R4567C6
31a. 45 rule on C789 4 outies R4567C6 = 1 innie R2C7 + 15
31b. R2C7 = {789} -> R4567C6 = 22,23,24 containing both of 2,7 = {2578/2579/2678} (cannot be {2679} = {279}6 which clashes with R45C5)
31c. R4567C6 = {2578/2579/2678} = 22,23 -> R2C7 = {78}
[Cracked.]

32. R2C3 = 9 (hidden single in R2)
32a. Naked pair {78} in R12C4, locked for C4 and N2 -> R1C6 = 9

33. R389C4 (step 24a) = {249} (only remaining combination), locked for C4, 9 locked for N8 -> R7C5 = 8, R6C4 = 4, R6C9 = 5, R7C6 = 6 (step 4), R2C6 = 5, R2C7 = 8 (cage sum), R12C4 = [87]

34. Naked pair {36} in R12C5, locked for N2
34a. Naked triple {124} in R3C456, locked for R3
34b. Naked triple {278} in R456C6, locked for N5
34c. Naked pair {59} in R45C5, locked for N5

35. R7C4 = 5 (hidden single in N8), R6C1 = 7 (step 3)

36. R6C1 = 7 -> R7C12 = 11 = {29}, locked for R7 and N7

37. R7C4 = 5 -> R78C3 = 8 = [17], clean-up: no 8 in R9C9

38. Naked pair {37} in R7C89, locked for N9 -> R7C7 = 4, R8C7 = 5 (cage sum), clean-up: no 8 in R8C9
38a. Naked pair {69} in R89C9, locked for C9 and N9

39. R12C7 (step 6) = {23} (only remaining combination) -> R1C7 = 3, R9C7 = 2, R12C5 = [63], R3C9 = 7, R7C89 = [73]

40. R9C5 = 7 -> R9C46 = 5 = [41], R89C8 = [18], R89C1 = [86], R8C6 = 3, R8C2 = 4

41. 7 in C6 only in R45C6, locked for 25(4) cage at R3C7, no 7 in R4C7
41a. R5C7 = 7 (hidden single in C7), R6C67 = 8 = [26], R6C34 = [31], R5C3 = 6 (cage sum)

42. R45C4 = [63] = 9 -> R34C3 = 12 = [84]

43. R6C8 = 9 -> R5C89 = 6 = {24}, locked for R5 and N6, R4C7 = 1, R4C89 = [38], R2C9 = 2 (cage sum)

44. Naked pair {25} in R4C12, locked for N4 and 14(4) cage at R2C1 -> R3C1 = 3, R2C1 = 4 (cage sum)

and the rest is naked singles.

[After 45 rule on 45 rule on N8 3 innies R7C456 = 19, Para, Mike and Ed all found 45 rule of R89 4 outies R7C3467 = 16 and then used interactions with R7C456, giving very different solving paths.]

My hardest steps are probably the combination analysis in step 16b, also steps 26a and 27a.


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PostPosted: Sun Jul 06, 2008 11:06 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Assassin 68 v3 by Ruud (Sept 07) Edit: image corrected. Thanks for noticing Børge.
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
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:
Solution:
+-------+-------+-------+
| 6 1 4 | 7 5 8 | 9 3 2 |
| 7 2 3 | 6 9 4 | 5 1 8 |
| 9 5 8 | 3 2 1 | 6 7 4 |
+-------+-------+-------+
| 4 6 1 | 9 8 2 | 3 5 7 |
| 2 3 9 | 5 4 7 | 8 6 1 |
| 5 8 7 | 1 3 6 | 4 2 9 |
+-------+-------+-------+
| 1 9 2 | 8 6 5 | 7 4 3 |
| 3 4 5 | 2 7 9 | 1 8 6 |
| 8 7 6 | 4 1 3 | 2 9 5 |
+-------+-------+-------+
Quote:
Ruud: estimated rating 3-4
mhparker: it looks it's going to be a strong candidate for the Unsolvables list!
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.
Walkthrough by Andrew:
Prelims

a) R12C5 = {59/68}
b) R67C5 = {18/27/36/45}, no 9
c) R89C1 = {29/38/47/56}, no 1
d) R89C9 = {29/38/47/56}, no 1
e) 9(3) cage at R5C8 = {126/135/234}, no 7,8,9
f) 8(3) cage at R9C4 = {125/134}
g) 19(3) cage at R8C8 = {289/379/469/478/568}, no 1
h) 26(4) cage at R2C1 = {2789/3689/4589/4679/5678}, no 1
i) 18(5) cage at R1C1 = {12348/12357/12456}, no 9
j) 18(5) cage at R3C4 = {12348/12357/12456}, no 9

Steps resulting from Prelims
1a. 8(3) cage at R9C4 = {125/134}, 1 locked for R9 and N8, clean-up: no 8 in R6C5
1b. 18(5) cage at R1C1 = {12348/12357/12456}, 1,2 locked for N1

2. 45 rule on R89 2 innies R8C37 = 6 = {15/24}

3. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 13 = {49/58/67}, no 1,2,3

4. 45 rule on N9 2(1+1) outies R6C9 + R7C6 = 14 = [59/68/77/86/95], no 1,2,3,4
4a. Min R7C6 + R8C7 = 6 -> max R7C7 = 7

5. 45 rule on C12 2 outies R19C3 = 10 = [19]/{28/37/46}, no 5

6. 45 rule on C89 2 outies R19C7 = 11 = {29/38/47/56}, no 1

7. 45 rule on C1234 3 innies R389C4 = 9 = {126/135/234}, no 7,8,9
[Note. R389C4 cannot be 2{34} (which clashes with 8(3) cage = {134}, CCC)]

8. 18(3) cage at R8C4 = {279/369/378/468/567} (cannot be {459} which clashes with 8(3) cage at R9C4)
8a. 2 of {279} must be in R8C4 -> no 2 in R8C56

[When I originally looked at this puzzle, I’d overlooked that there was the 18(5) cage at R3C8. It looks like anything to help will be welcome.]
9. 18(5) cage at R3C4 = {12348/12357} (cannot be {12456} which clashes with R12C5), no 6

[I was a bit slow in spotting these IOUs. Since they both eliminate the same number in the same row, I’ve moved them here to simplify some later steps.]
10. 45 rule on N7 2 innies R7C12 = 1 outie R7C4, IOU no 2 in R7C12

11. 45 rule on N9 2 innies R7C89 = 1 outie R7C6 + 2, IOU no 2 in R7C89

12. 1 in N9 only in R7C789 + R8C7
12a. 45 rule on N9 4 innies R7C789 + R8C7 = 15 = {1239/1248/1257/1347/1356}
[Note that R7C789 + R8C7 = {1257} can only be R78C7 = {25}, R7C89 = {17} from cage values]
12b. 6 of {1356} must be in R7C89 (R78C7 cannot be {16} because 13(3) cage at R7C6 cannot be 6{16}, cannot be [65] from cage totals and no 3,6 in R8C7) -> no 6 in R7C7
12c. R7C789 + R8C7 = {1239/1248/1347/1356} (cannot be {1257} = {25}{17} because 16(3) cage at R6C9 + R78C7 = 8{17}{25} clashes with R89C9)
12d. 5 of {1356} must be in R78C7 (R7C89 cannot be {56} because 16(3) cage at R6C9 cannot be 5{56}) -> no 5 in R7C89
12e. R89C9 = {29/47/56} (cannot be {38} which clashes with R7C789 + R8C7), no 3,8 in R89C9

13. R7C789 + R8C7 (step 12d) = {1239/1248/1347/1356}, R8C37 (step 2) = {15/24}
[Note that R7C789 + R8C7 = {1239} can only be R78C7 = [32], R7C89 = {19} from cage values]
13a. 45 rule on N8 3 innies R7C456 = 19
13b. R7C456 = 19 -> R7C6 cannot be 4 more than R8C3 (because R7C34 cannot equal R7C45, CCC)
13c. 13(3) cage at R7C6 cannot be [931] because R8C37 = [51] would make R7C6 4 more than R8C3 -> no 9 in R7C6, clean-up: no 5 in R6C9 (step 4)
13d. 13(3) cage cannot be [832] because R8C37 = [42] would make R7C6 4 more than R8C3 -> R78C7 cannot be [32]
13e. R7C789 + R8C7 = {1248/1347/1356} (cannot be {1239} = [32]{19} because R78C7 cannot be [32], step 13d) -> no 9 in R7C89
[Steps 13c and 13d could be done using forcing chains, but they are clearer this way.]

14. R7C789 + R8C7 (step 13e) = {1248/1347/1356}, R8C37 (step 2) = {15/24}
[Note that R7C789 + R8C7 = {1248} can only be R78C7 = {24}, R7C89 = {18} from cage values]
14a. Consider combinations for 18(3) cage at R8C4 (step 8) = {279/369/378/468/567}
18(3) cage = {279/468} => R8C37 = {15} => R78C7 cannot be {24}
or 18(3) cage = {369} => 8 in N8 only R7C456, locked for R7 => R7C89 cannot be {18}
or 18(3) cage = {378/567}, 7 locked for N8 => 13(3) cage at R7C6 cannot be 7{24}
-> R7C789 + R8C7 = {1347/1356} (cannot be {1248} because either R78C7 not {24} or R7C89 not {18}), no 2 in R78C7, no 8 in R7C89, 3 locked for R7 and N9, clean-up: no 6 in R6C5, no 4 in R8C3 (step 2)

15. R8C37 (step 2) = {15/24}
15a. R7C6 cannot be 4 more than R8C3 (step 13b)
15b. 13(3) cage at R7C6 cannot be [634] because R8C37 = [24] would make R7C6 4 more than R8C3) -> no 6 in R7C6, clean-up: no 8 in R6C9 (step 4)
15c. 13(3) cage = [571]/7{15}/8{14}, no 3 in R7C7
15d. 13(3) cage = [571]/7{15}/8{14}, 1 locked for C7 and N9

16. R7C456 = 19 (step 13a) = {289/478/568} (cannot be {469} because R7C6 only contains 5,7,8), 8 locked for R7 and N8
16a. 18(3) cage at R8C4 (step 8) = {279/369/567}, no 4

17. R7C456 = (step 16) = {289/478/568}
17a. 2 in R7 only in 15(3) cage at R7C3 = [285] or R7C456 = {289} = [928] -> R7C4 = {89}, clean-up: R6C1 = {45} (step 3)
17b. Min R7C4 = 8 -> max R78C3 = 7, no 7,9 in R7C3

18. R7C456 = (step 16) = {289/478/568}
18a. 2,4,6 only in R7C5 -> R7C5 = {246}, clean-up: R6C5 = {357}

19. R7C789 + R8C7 (step 12a) = {1347/1356}, R7C456 (step 14) = {289/478/568}
19a. Consider placements for R7C6
R7C6 = 5 => no 5 in R8C7
or R7C6 = 7 => R7C45 = [84], R7C3 = 2 (hidden single in R7) => R8C3 = 5 (cage sum) => no 5 in R8C7
or R7C6 = 8 => R78C7 = 5 = {14}, no 5 in R8C7
-> no 5 in R8C7, clean-up: no 1 in R8C3 (step 2)
19b. Min R7C4 = 8 -> max R78C3 = 7, min R8C3 = 2 -> max R7C3 = 5

20. R7C456 (step 16) = {289/478/568}
20a. Consider permutations for R6C1 + R7C4 (step 3) = [49/58]
R6C1 + R7C4 = [49] => R7C456 = [928]
or R6C1 + R7C4 = [58] => no 5 in R6C5 => no 4 in R7C5
-> R7C456 = {289/568}, no 4 in R7C5, no 7 in R7C6, clean-up: no 5 in R6C5, no 7 in R6C9 (step 4)

21. 13(3) cage at R7C6 = [571]/8{14}, no 5 in R7C7
21a. 16(3) cage at R6C9 = 6{37}/9{34}, no 6 in R7C89
21b. Naked quad {1347} in R7C789 + R8C7, locked for N9, 7 also locked for R7, clean-up: no 4,7,8 in R1C7 (step 6)
21c. Killer pair 6,9 in R6C9 and R89C9, locked for C9

22. 15(3) cage at R6C1 = {159/456}, CPE no 5 in R89C1, clean-up: no 6 in R89C1

23. 4 in N8 only in 8(3) cage at R9C4 = {134}, 3,4 locked for R9 and N8, clean-up: no 6,7 in R1C3 (step 5)
23a. 18(3) cage at R8C4 (step 16a) = {279/567}, 7 locked for R8
23b. Killer pair 2,5 in R8C3 and 18(3) cage, locked for R8
23c. 2,5 in N9 only in R9C789, locked for R9, clean-up: no 8 in R1C3 (step 5)
23d. Min R9C23 = 13 -> max R8C2 = 4
23e. Clean-up: no 8,9 in R8C1, no 9 in R9C1, no 6,9 in R9C9

24. R8C8 = 8 (hidden single in R8), clean-up: no 3 in R1C7 (step 6)

25. 2 in N7 only in 15(3) cage at R7C3 = {249/258}, no 1, 2 locked for C3, clean-up: no 8 in R9C3 (step 5)
25a. 15(3) cage = {258} must be [28]5 (cannot be [58]2 which clashes with R7C6), no 5 in R7C3

26. 15(3) cage at R6C1 = {159/456} = 4{56}/5{19} (cannot be 5{46} which clashes with R7C35, ALS block), no 4 in R7C12

[I think I’ve got as far as I can for now in R789, apart from a few minor candidate eliminations which don’t help at this stage, so I’ll have to move on to other parts of the cage pattern.]

27. R389C4 (step 7) = {126/135/234}
27a. 5 of {135} must be in R8C3 -> no 5 in R3C4

28. Consider permutations for R67C5 = [36/72]
R67C5 = [36] => R12C5 = {59}, locked for C5 => R8C5 = 7
or R67C5 = [72]
-> 7 in R68C5, locked for C5

29. 18(5) cage at R3C4 (step 9) = {12348/12357}
29a. 7 of {12357} must be in R3C6 -> no 5 in R3C6

30. 8 in C5 only in R12345C5, CPE no 8 in R3C6

31. R12C5 and 18(5) cage at R3C4 form combined 32(7) cage = {1234589/1235678} (cannot be {1234679} because R12C5 can only contain one of 6,9), 5 locked for C5

32. 45 rule on C6789 3 innies R389C6 = 13 = {139/157/247/346} (cannot be {256} because R9C6 only contains 1,3,4)

33. R389C6 (step 32) = {139/157/247/346}, R7C456 (step 20a) = {289/568}, 18(3) cage at R8C4 (step 23a) = {279/567}
33a. Consider placements for R8C6 = {5679}
R8C6 = 5 or 6 => 18(3) cage = {567}, no 9
or R8C6 = 7 => R389C6 = {247} = [274] => R7C5 = 2 (hidden single in C5) => R7C456 = {289}, locked for N8
or R8C6 = 9
-> no 9 in R8C5

34. 9 in C5 only in R12C5 = {59}, locked for C5 and N5
34a. 18(5) cage at R3C4 (step 9) = {12348} (only remaining combination), no 7
34b. 6 in C5 only in R78C5, locked for N8
34c. Naked pair {25} in R8C34, locked for R8

35. R389C4 (step 7) = {135/234}, 3 locked for C4
35a. R8C4 = {25} -> no 2 in R3C4

36. R389C6 (step 32) = {139/247}
36a. 2 of {247} must be in R3C6 -> no 4 in R3C6

37. 18(5) cage at R3C4 (step 34a) = {12348}
37a. Consider placements for 2 in N8
37b. R7C5 = 2 => R3C6 = 2 (only remaining place for 2 in 18(5) cage)
or R8C4 = 2
-> 2 in R3C6 or R8C4, CPE no 2 in R12C4

38. 16(3) cage at R1C4 = {169/178/367/457} (cannot be {349/358} because 3,5,9 only in R2C4)
38a. 3,5,9 of {169/367/457} must be in R2C4 -> no 4,6 in R2C4

39. 17(3) cage at R1C6 = {269/278/368/458/467} (cannot be {179} = {17}9 which clashes with 16(3) cage at R1C4, cannot be {359} because 5,9 only in R2C7), no 1 in R12C6

40. 24(4) at R2C9 = {3489/3579/3678/4578} (cannot be {1689/2679/4569} because 6,9 only in R4C8, cannot be {2589} which clashes with R9C9), no 1,2
40a. 6,9 of {3489/3579/3678} must be in R4C8 -> no 3 in R4C8

41. 1 in N6 only in 9(3) cage at R5C8 = {126/135}, no 4

42. Consider combinations for R389C6 (step 36) = {139/247}
R389C6 = {139}, 3 locked for C6
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36}, locked for C6
-> no 3 in R456C6
[Taking these a bit further]
42a. R389C6 = {139} => R7C4 = 8, R78C3 = [25]
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36}, R2C7 = 8 (cage sum), 7 in N2 only in 16(3) cage at R1C6 = {457} (cannot be {17}8) => R2C3 = 5
-> 5 in R2C3 or R8C3, locked for C3
Also R7C4 = 8 or 16(3) cage at R1C6 = {457} -> no 8 in R12C4
42b. 16(3) cage at R1C4 (step 38) = {169/178/367/457}
42c. 3,5,8 of {178/367/457} must be in R2C3 -> no 7 in R2C3
[And a bit further still]
42d. R389C6 = {139}, 1 locked for C6
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36}, R456C6 = {159} => R45C6 = {15/19} (cannot be {59} because min R34C7 = 5), 1 locked for C6
-> no 1 in R6C6
42e. R389C6 = {139} => R8C9 = 9, R7C4 = 8, R7C6 = 5 => R78C7 = 8 = [71]
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36} => R2C7 = 8 (cage sum)
-> no 7 in R2C7

43. 45 rule on R789 3 outies R6C159 = 17 = [476/539]
43a. 18(3) cage at R5C7 = {279/378/459/468/567} (cannot be {369} which clashes with R6C9)
43b. 2 of {279} must be in R6C67 (R6C67 cannot be {79} which clashes with R6C159) -> no 2 in R5C7

44. 16(3) cage at R1C4 (step 38) = {169/178/367/457}
44a. 17(3) cage at R1C6 (step 39) = {269/278/368/458/467}
44b. 5 of {458} must be in R2C7, 4 of {467} must be in R12C6 (R12C6 cannot be {67} which clashes with 16(3) cage) -> no 4 in R2C7

45. R19C7 (step 6) = {29/56}
45a. Consider combinations for R389C6 (step 36) = {139/247}
R389C6 = {139} => R8C6 = 9 => R8C9 = 6 => R9C9 = 5 => R9C78 = {29} => R19C7 = {29}, locked for C7
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36} => R2C7 = 8 (cage sum)
-> no 2,9 in R2C7

46. 26(4) cage at R2C1 = {2789/3689/4679/5678} (cannot be {4589} which clashes with R6C1)
46a. 2,5 of {2789/5678} must be in R234C1 (R234C1 cannot be {678/789} which clash with R9C1) -> no 2,5 in R4C2

47. R19C7 (step 45) = {29/56}
47a. 45 rule on N3 4 innies R23C79 = 23 = {3479/3578/4568} (cannot be {2489/2579} because 2,9 only in R3C7, cannot be {2678/3569} which clash with R19C7), no 2

48. R19C7 (step 45) = {29/56}, R23C79 (step 47a) = {3479/3578/4568}
48a. Consider combinations for R389C6 (step 36) = {139/247}
R389C6 = {139} => R8C6 = 9 => R8C9 = 6 => R9C9 = 5 => R9C78 = {29} => R19C7 = {29}, locked for C7
or R389C6 = {247} = [274] => 18(3) cage at R8C4 = {567} => R7C6 = 8 => R12C6 = {36} => R2C7 = 8 (cage sum) => R23C79 = {3578/4568}
-> no 9 in R3C7
48b. R23C79 = {3578/4568}, 5,8 locked for N3, clean-up: no 6 in R9C7, no 5 in R9C8

[At last some more placements …]
49. R19C7 (step 45) = {29}/[65], R9C78 = {29}/[56]
49a. Consider placements for 6 in N6
6 in R456C7 => R19C7 = {29} => R9C78 = {29}
or 6 in R456C8 => R9C78 = {29} => R19C7 = {29}
-> R19C7 = {29}, locked for C7, R9C78 = {29}, locked for R9 and N9 -> R89C9 = [65], clean-up: no 1 in R1C3 (step 5)
[Cracked. The rest is straightforward.]

50. R8C56 = [79], R7C46 = [85], R8C4 = 2, R67C5 = [36]
50a. R6C9 = 9 -> R7C89 = 7 = {34}, locked for R7 and N9 -> R78C3 = [25], R78C7 = [71]
50b. R7C12 = {19} = 10 -> R6C1 = 5

51. R8C6 = 9 -> R389C6 (step 36) = {139}, 1,3 locked for C6
51a. 3 in N2 only in R3C46, locked for R3

52. 1,2 in N6 only in 9(3) cage at R5C8 = {126}, 6 locked for C8 and N6

53. 6 in N3 only in R23C7 -> R23C79 (step 48b) = {4568} (only remaining combination) -> R23C7 = {56}, locked for C7, R23C9 = {48}, locked for C9 and N3 -> R7C89 = [43], R4C9 = 7, R4C8 = 5 (cage total)

54. 18(3) cage at R5C7 (step 43a) = {378/468} -> R6C6 = {67}, R56C7 = [38]/{48}, 8 locked for C7

55. 17(3) cage at R1C6 (step 39) = {458/467} (cannot be {278} because R2C7 only contains 5,6) -> R12C6 = {47/48}, 4 locked for C6 and N2
55a. Naked pair {13} in R3C46, locked for R3 and N2
55b. Naked pair {67} in R12C4, locked for C4 and N2, R2C3 = 3 (cage sum), R1C3 = 4, R9C3 = 6 (step 5), R12C6 = [84], R2C7 = 5 (cage sum), R23C9 = [84], R3C7 = 6, R45C6 = [27], R4C7 = 3 (cage sum)

56. R6C6 = 6, R5C8 = 6 (hidden single in N6)

57. R45C4 = [95] (hidden pair in N5) = 14 -> R34C3 = 9 = [81], R56C3 = [97], R6C4 = 1 (cage sum)

58. 45 rule on N4 2 remaining innies R4C12 = 10 = {46}, locked for R4, N4 and 26(4) cage at R2C1
58a. Naked pair {79} in R23C1, locked for C1 and N1, R9C1 = 8 -> R8C1 = 3

and the rest is naked singles.

I used a lot of forcing chains so I'll rate my walkthrough at least 2.5. I haven't changed the rating in the tables, which appears to be based on Ruud's estimate.
Assassin 68 v1.5 by mhparker (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:5120:5120:5120:4099:4356:3077:7686:7686:7686:6921:5120:4099:4099:4356:3077:3077:7686:3857:6921:5120:5140:5397:5397:5397:4120:7686:3857:6921:6921:5140:5140:5397:4120:4120:3857:3857:3364:3364:3622:5140:5397:4120:3114:5675:5675:4653:3364:3622:3622:1585:3114:3114:5675:3637:4653:4653:4152:4152:1585:4923:4923:3637:3637:3391:2112:4152:4930:4930:4930:4923:4166:2631:3391:2112:2112:2891:2891:2891:4166:4166:2631:
Solution:
+-------+-------+-------+
| 1 4 7 | 3 9 6 | 8 2 5 |
| 3 2 9 | 4 8 5 | 1 6 7 |
| 8 6 5 | 2 7 1 | 4 9 3 |
+-------+-------+-------+
| 9 7 6 | 8 5 3 | 2 1 4 |
| 2 8 4 | 1 6 7 | 3 5 9 |
| 5 3 1 | 9 4 2 | 7 8 6 |
+-------+-------+-------+
| 4 9 3 | 5 2 8 | 6 7 1 |
| 6 1 8 | 7 3 9 | 5 4 2 |
| 7 5 2 | 6 1 4 | 9 3 8 |
+-------+-------+-------+
Quote:
mhparker, lead-in: (Est. rating: 1.5)
CathyW: nothing majorly advanced required but it was quite a slog...I would rate this puzzle as 1.25 - 1.5. It took me a couple of hours on the 1st attempt..(the solution came mostly from nibbling away at the cage combinations)
Andrew in 2011: ..seems to have a fairly narrow solving path; Cathy and I used many similar steps.....I'll rate my walkthrough ... at 1.25; my solving path was long but there weren't any difficult steps
Walkthrough by CathyW:
V1.5 WT Easier still on the 3rd attempt!!

Prelims

a) 17(2) r12c5 = {89} n/e N2/c5
b) 27(4) @ r2c1 = {3789/4689/5679} Must have 9 -> r56c1 <> 9 (see all cells of 27(4))
c) 22(3) N6 = {589/679} 9 n/e N6
d) 6(2) r67c5 = {15/24}
e) 19(3) @ r7c6 and r8c4: no 1
f) 13(2) r89c1 = {49/58/67}
g) 8(3) N7 = {125/134} 1 n/e N7
h) 11(3) r9c456: no 9
i) 10(2) r89c9: no 5


1. Outies c12: r19c3 = 9 = [45/54/63/72/81]

2. Outies c89: r19c7 = 17 = {89} n/e c7
a) NP {89} r1c57 n/e r1 -> r9c3 <> 1 -> 1 locked r89c2 n/e c2
b) r89c8 of 16(3) = (1…7)
c) r7c6 = (89) (because max from r78c7 = 6+7 = 13 thus r7c6 is min 6 but can’t be 6 or 7.) -> r78c7 <> 2
d) 19(3) r8c456: {289}blocked by r7c6 -> r8c456 <> 2

3. KP {89} N8: r7c6 and 19(3) r8c456 = {379/469/478/568} must have at least one of 8,9 -> 11(3) r9c456 <> 8, r7c4 <> 8,9

4. Innies r89: r8c37 = 13 = [67/76/85/94]

5. Outies r789: r6c159 = 15 = {168/258/267/348/357/456}

6. Outies N7: r6c1 + r7c4 = 10 = [82]/{37/46/55}

7. Outies N9: r6c9 + r7c6 = 14 = [59/68]
a) split 15(3) = [816/825/726]/{456}
b) KP r6c9 and 22(3) N6: r4c789, r56c7 <> 5,6
c) combo analysis 14(3) @ r6c9: r7c89 <> 4, 9

8. Innies N8: r7c456 = 15 = [258/348/618/249/429/519/528]

9. Outies r9: r8c1289 = 13 = 7{123}/{1246/1345}
Clean up: r9c1 <> 4,5; r9c9 <> 1,2,3

10. 16(3) @ r7c3: r7c4+r8c3 is min 2+6 = 8 -> r7c3 <> 9

11. 16(3) @ r1c4: r12c4 is max 6+7 = 13 -> r2c3 is min 3

12. 9 locked r9c79 in N9 -> r9c1 <> 9 -> r8c1 <> 4
a) split 13(4) = 7{123}/6{124}/5{134} (r8c289 = (1234)) -> r9c9 <> 4
b) ca 16(3) N9: r9c8 <> 1,2

13. 2 locked to split 13(4) in r8c1289
a) split 13(4) = 7{123}/6{124}
b) 13(2) r89c1 = {67} n/e N7/c1
c) split 15(3) r6c159 = [816/825/456/546]

14. 18(3) @ r6c1 = {459} only available combo.
a) 9 locked r7c12 -> r8c3 = 8, r7c6 = 8, r78c7 = [65] (because r8c37=13), r6c9 = 6
b) r6c15 = {45} n/e r6 -> r7c5 = (12)
c) r7c89 = {17} n/e r7/N9 -> r7c5 = 2, r6c5 = 4, r6c1 = 5
d) r7c12 = {49} n/e N7 -> 8(3) N7 = {125}, r7c34 = [35]
e) 10(2) N9 = [28] only option -> r8c2 = 1, r9c7 = 9, r89c8 = {34} n/e c8, r1c7 = 8, r1c5 = 9, r2c5 = 8
f) clean up: r1c3 = (47)

15. 19(3) r8c456 = {379/469} -> r8c46 <> 6

16. 11(3) r9c456 = {137/146}

17. 27(4) @ r2c1 = {389}7 ({4689} blocked by r7c1 since 6 would have to go in r4c2)
a) {389} n/e c1 -> r7c1 = 4, r7c2 = 9 -> r15c1 = {12}
b) r56c2 of 13(3) N4 = [48]/{38} 8 n/e N4/c2
c) HS r3c1 = 8

18. Innies N1: r2c1+r23c3 = 17 = [359/395/971/962]
a) 20(5) N1 = {12467/13457/23456}

19. Innies N3: r79c23 = 15 = {1239/1257/1347} must have 1 -> 30(5) <> 1

20. Innies c1234: r389c4 = 15 = 2[76/94] only options.
a) 11(3) r9c456 = {146}
b) 19(3) r8c456 = [739] -> r9c456 = [614]
c) NS: r89c1 = [67], r89c8 = [43]

21. Innies c6789: r3c6 = 1

Straightforward from here
Alt start by gary w:
Using the same prelims as Cathy details;

r7c4<>1 > 9@r6c1 then cannot put 9 in 27/4 cage.
IF (!!??) 1 is @ r7c5 >r6c5=5 >r6c9=6 >>r7c6=8 and r6c1=4 >r7c4=6 >r7c12=5/9 so can't complete the 14/3 cage at N6/9.

Thus 1 goes in the 11/3 cage N8 > 1 @ r8c2.

Although still a bit tough this breaks the puzzle.
Having done all the prelims I saw this quite easily.No pencil marks or "guessing".

What I'ld like your opinion on is;how far does a hypothetical (doesn't sound too bad) have to extend before it becomes T and E (ugh!)?

From step 1? Only if you need pencil marks to reach a contradiction?I rather like solving them this way (when I can !) rather than ..how can I put this?....the use of the apparently tortuous numbers of combos that the experts use.I can see why this method doesn't appeal to the logical or aesthetic sense but at least I can solve them and still leave a LITTLE of the day over to do other things!!
2011 Walkthrough by Andrew:
Thanks Mike for another variant. This one wasn't too hard although I may have thought otherwise if I'd tried it when it was originally posted; that was only a couple of months after we moved to Lethbridge so I was probably still settling in and limiting myself to basic Assassins. I see that Ruud also posted V2 and V3; haven't tried them (yet).

This puzzle seems to have a fairly narrow solving path; Cathy and I used many similar steps.

Here is my walkthrough for A68 V1.5.

Prelims

a) R12C5 = {89}
b) R67C5 = {15/24}
c) R89C1 = {49/58/67}, no 1,2,3
d) R89C9 = {19/28/37/46}, no 5
e) 22(3) cage in N6 = {589/679}
f) 8(3) cage in N7 = {125/134}
g) 19(3) cage at R7C6 = {289/379/469/478/568}, no 1
h) 19(3) cage at R8C4 = {289/379/469/478/568}, no 1
i) 11(3) cage in N8 = {128/137/146/236/245}, no 9
j) 27(4) cage at R2C1 = {3789/4689/5679}, no 1,2

Steps resulting from Prelims
1a. Naked pair {89} in R12C5, locked for C5 and N2
1b. 22(3) cage in N6 = {589/679}, 9 locked for N6
1c. 8(3) cage in N7 = {125/134}, 1 locked for N7
1d. 27(4) cage at R2C1 = {3789/4689/5679}, CPE no 9 in R56C1
1e. Max R12C4 = {67} = 13 -> min R2C3 = 3

2. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 10 = {28/37/46}/[19/55], no 1 in R7C4

3. 45 rule on N9 2(1+1) outies R6C9 + R7C6 = 14 = [59/68/77/86], R6C9 = {5678}, R7C6 = {6789}
3a. Min R6C9 = 5 -> max R7C89 = 9, no 9 in R7C89

4. 45 rule on R89 2 innies R8C37 = 13 = {49/58/67}, no 1,2,3

5. 45 rule on R789 3 outies R6C159 = 15 = {168/258/267/348/357/456}
5a. 1 of {168} must be in R6C5 -> no 1 in R6C1, clean-up: no 9 in R7C4 (step 2)

6. 45 rule on C12 2 outies R19C3 = 9 = [45/54/63/72/81], no 1,2,3,9 in R1C3

7. 45 rule on C89 2 outies R19C7 = 17 = {89}, locked for C7, clean-up: no 4,5 in R8C3 (step 4)
7a. Min R9C7 = 8 -> max R89C8 = 8, no 8,9 in R89C8
7b. Min R7C4 + R8C3 = 8 -> max R7C3 = 8

8. Naked pair {89} in R1C57, locked for R1, clean-up: no 1 in R9C3 (step 6)
8a. 1 in N7 only in R89C2, locked for C2

9. 19(3) cage at R7C6 = {379/469/478/568} (cannot be {289} because 8,9 only in R7C6), no 2
9a. 8,9 only in R7C6 -> R7C6 = {89}, clean-up: no 7,8 in R6C9 (step 3)

10. Killer pair 5,6 in 22(3) cage and R6C9, locked for N6

11. R6C159 (step 5) = {168/258/267/456} (cannot be {348} because R6C9 only contains 5,6, cannot be {357} because 3,7 only in R6C1), no 3, clean-up: no 7 in R7C4 (step 2)
11a. 7,8 of {258/267} must be in R6C1 -> no 2 in R6C1, clean-up: no 8 in R7C4 (step 2)

12. 45 rule on C1 2 outies R47C2 = 2 innies R15C1 + 13
12a. Min R15C1 = 3 -> min R47C2 = 16
12b. Max R47C2 = 17 -> max R15C1 = 4
12c. -> R15C1 = 3,4 = {12/13}, R47C2 = 16, 17 = {79/89}, 9 locked for C2

13. 16(3) cage at R7C3 must contain one of 7,8,9 in R78C3
13a. Killer triple 7,8,9 in R7C2, R78C3 and R89C1, locked for N8
[I originally analysed the permutations for 18(3) cage at R6C1; the killer triple is neater.]

14. 45 rule on N8 3 innies R7C456 = 15 = {159/168/249/258} (cannot be {267/357/456} because R7C6 only contains 8,9, cannot be {348} which clashes with 11(3) cage in N8), no 3, clean-up: no 7 in R6C1 (step 2)
14a. 19(3) cage at R8C4 = {379/469/568} (cannot be {289} which clashes with R7C6, cannot be {478} which clashes with R8C37), no 2
14b. R7C456 = {168/249/258} (cannot be {159} which clashes with 19(3) cage at R8C4)
14c. 11(3) cage in N8 = {137/146/245} (cannot be {128/236} which clash with R7C456), no 8
[I preferred this way for the 11(3) cage, rather than killer pair 8,9 in R7C6 and 19(3) cage at R8C4 locked for N8, because it eliminated an extra combination.]

15. 18(3) cage at R6C1 = {369/378/459/567} (cannot be {279} because 7,9 only in R7C2, cannot be {468} which clashes with R89C1), no 2
15a. 7,9 only in R7C2 -> R7C2 = {79}

16. R15C1 = {12} (hidden pair in C1) -> R47C2 (step 12) = {79}, locked for C2
16a. Max R5C1 = 2 -> min R56C2 = 11, no 2 in R56C2

17. 16(3) cage at R7C3 = {259/268/349/358/367/457}
17a. 3 of {367} must be in R7C3, 7 of {457} must be in R8C3 -> no 7 in R7C3

[I should probably have spotted the next step earlier.]
18. 1,2 in R8 only in R8C289
18a. 45 rule on R9 4 outies R8C1289 = 13 = {1237/1246}, no 5,8,9, clean-up: no 4,5,8 in R9C1, no 1,2 in R9C9
18b. 7 of {1237} must be in R8C1 -> no 7 in R8C89, clean-up: no 3 in R9C9

19. 8 in N7 only in R78C3, locked for C3
19a. 16(3) cage at R7C3 (step 17) = {268/358} (only combinations containing 8), no 4,7,9, clean-up: no 6 in R6C1 (step 2), no 4,6 in R8C7 (step 4)
19b. 3 of {358} must be in R7C3 -> no 5 in R7C3

20. 9 in R8 only in 19(3) cage at R8C4, locked for N8 -> R7C6 = 8, R6C9 = 6 (step 3), clean-up: no 7 in 22(3) cage in N6, no 4 in R89C9
20a. Naked triple {589} in 22(3) cage, locked for N6
20b. 19(3) cage at R8C4 (step 14a) = {379/469}, no 5
20c. R7C456 (step 14b) = {168/258}, no 4, clean-up: no 2 in R6C5
20d. R6C9 = 6 -> R7C89 = 8 = {17/35}, no 2,4

21. R8C3 = 8 (hidden single in N7), R8C7 = 5 (step 4), R7C7 = 6 (step 9)

22. 16(3) cage at R7C3 (step 19a) = {358} (last remaining combination) -> R7C34 = [35], R6C1 = 5 (step 2), R7C1 = 4, R7C2 = 9 (step 15), R4C2 = 7, clean-up: no 5,6 in R1C3 (step 6)
22a. 5 in N6 only in R5C89, locked for R5

23. Naked pair {17} in R7C89, locked for R7 and N9 -> R7C5 = 2, R6C5 = 4, clean-up: no 3 in R8C9, no 9 in R9C9

24. R89C9 = [28], R8C2 = 1, R9C7 = 9, R1C7 = 8, R12C5 = [98]
24a. Naked pair {67} in R89C1, locked for C1
24b. Naked pair {34} in R89C8, locked for C8

25. 13(3) cage at R5C1 = {148/238} (cannot be {346} because 4,6 only in R5C2), no 6, 8 locked for C2 and N4

26. R3C1 = 8 (hidden single in C1)

27. 7 in N6 only in R56C7, locked for C7 and 12(3) cage at R5C7, no 7 in R6C6
27a. 12(3) cage at R5C7 = {147/237}, no 9
27b. 1 of {147} must be in R6C6 -> no 1 in R56C7
27c. 1 in N6 only in R4C789, locked for R4

28. R4C4 = 8 (hidden single in R4)

29. 14(3) cage at R5C3 = {149/167/239} (cannot be {347} because 3,7 only in R6C4)
29a. 4,6 of {149/167} must be in R5C3 -> no 1 in R5C3
29b. 3 of {239} must be in R6C4 -> no 2 in R6C4

30. 45 rule on N4 2 remaining innies R4C13 = 1 outie R6C4 + 6
30a. R6C4 = {1379} -> R4C13 = 7,9,13,15 = [34/36/94/96] -> R4C3 = {46}

31. Min R4C34 = [48] = 12 -> max R3C3 + R5C4 = 8, no 9 in R3C3 + R5C4
31a. 9 in N1 only in R2C13, locked for R2

32. 15(4) cage at R2C9 = {1239/1347} (cannot be {1257} which clashes with R7C9), no 5, 3 locked for C9
32a. 9 of {1239} must be in R3C9, 1 of {1347} must be in R4C8 -> no 1 in R3C9

33. 6,8 in N3 only in 30(5) cage = {25689/45678}, no 1
33a. 5 of {25689} must be in R1C9, 4 of {45678} must be in R1C9 -> R1C9 = {45}
33b. 9 of {25689} must be in R3C8 -> no 2 in R3C8

34. 5 in C5 only in R34C5, locked for 21(5) cage at R3C4, no 5 in R3C6
34a. 21(5) cage = {12567/23457}, 2 locked for R3 and N2

35. 6 in N4 only in R45C3, locked for C3, CPE no 6 in R5C4

36. 45 rule on N1 3 remaining innies R2C13 + R3C3 = 17 = {179/359}, no 4
36a. 1 of {179} must be in R3C3 -> no 7 in R3C3
36b. R3C3 = {15} -> no 5 in R2C3

37. 16(3) cage at R1C4 = {169/349/367}
37a. R2C3 = {79} -> no 7 in R12C4

38. 20(4) cage at R3C3 must contain 8 = {1478/1568/3458} (cannot be {2378/2468} because R3C3 only contains 1,5), no 2

39. R3C4 = 2 (hidden single in C4)

40. 45 rule on C1234 2 remaining innies R89C4 = 13 = [67/76/94], no 1,3, no 4 in R8C4
40a. 19(3) cage at R8C4 (step 20b) = {379/469}
40b. 6 of {469} must be in R8C5 -> no 6 in R8C46, clean-up: no 7 in R9C4

41. 11(3) cage in N8 (step 14c) = {146} (only remaining combination, cannot be {137} because R9C4 only contains 4,6), locked for R9 and N8 -> R89C1 = [67], R89C8 = [43]

42. 21(5) cage at R3C4 (step 34a) = {12567} (cannot be {23457} which clashes with R8C5), no 3,4
42a. R8C5 = 3 (hidden single in C5)
42b. 7 in C5 only in R35C5, locked for 21(5) cage at R3C4, no 7 in R3C6

43. 45 rule on C6789 3 innies R389C6 = 14 = {149/167}, 1 locked for C6
43a. 12(3) cage at R5C7 (step 27a) = {237} (only remaining combination), no 4

44. 4 in N6 only in R4C79, locked for R4 -> R4C3 = 6, R4C5 = 5

45. 20(4) cage at R3C3 (step 38) = {1568} (only remaining combination) -> R3C3 = 5, R5C4 = 1, R5C1 = 2, R1C1 = 1, R9C23 = [52]

46. 13(3) cage at R5C1 (step 25) = {238} (only remaining combination), 3 locked for C2 and N4 -> R4C1 = 9, R2C1 = 3, R56C3 = [41], R6C4 = 9 (step 29), R1C3 = 7, R2C3 = 9

47. 16(3) cage at R1C4 (step 37) = {349} (only remaining combination), R12C4 = [34], R8C46 = [79], R9C456 = [614], R3C6 = 1 (step 43)

48. Naked pair {23} in R46C6, locked for C6
48a. Naked pair {67} in R5C56, locked for R5 -> R5C7 = 3

and the rest is naked singles.


Rating comment. I'll rate my walkthrough for A68 V1.5 at 1.25; my solving path was long but there weren't any difficult steps.


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PostPosted: Sun Jul 06, 2008 11:11 pm 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Assassin 69 by Ruud (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:8704:8704:1538:1538:3076:4357:4357:5639:5639:5129:8704:8704:8704:3076:5639:5639:5639:5137:5129:5129:4884:4884:3076:4631:4631:5137:5137:5129:4884:4884:3614:5151:2848:4631:4631:5137:2084:2084:4902:3614:5151:2848:4650:2859:2859:4397:4902:4902:3614:5151:2848:4650:4650:5173:4397:4397:4902:3385:3386:2875:4650:5173:5173:2367:2367:3393:3385:3386:2875:3397:1606:1606:2120:2120:3393:3393:3386:3397:3397:3407:3407:
Solution:
+-------+-------+-------+
| 4 7 5 | 1 3 8 | 9 6 2 |
| 1 8 6 | 9 2 4 | 3 7 5 |
| 9 2 3 | 5 7 6 | 4 1 8 |
+-------+-------+-------+
| 8 4 7 | 2 9 1 | 5 3 6 |
| 2 6 9 | 8 5 3 | 1 4 7 |
| 3 5 1 | 4 6 7 | 8 2 9 |
+-------+-------+-------+
| 5 9 4 | 6 1 2 | 7 8 3 |
| 6 3 8 | 7 4 9 | 2 5 1 |
| 7 1 2 | 3 8 5 | 6 9 4 |
+-------+-------+-------+
Quote:
CathyW: A69 is proving extremely tricky!..estimated rating 1.5
mhparker: ... Rating probably around 1.25
gary w: Enjoyed going through Cathy's wt too. Solving path seemed very similar to Mike's... the path is quite tightly constrained.....I really appreciate the effort in putting the wts
Andrew: My solving path was quite a lot different than the others. ..I'll agree with Mike's estimated rating of 1.25
Walkthrough by mhparker:
Hi folks,

This was a nice puzzle. Thanks very much, Ruud! :D

Edit: Checked through solution & simplified some steps. Rating probably around 1.25.

Edit: Modified WT to make UR-move optional.
Edit: Incorporated feedback from Andrew (Thanks, Andrew!)

Assassin 69 Walkthrough

1. 34(5)n12 = {46789}
1a. CPE: no 4,6,7,8,9 in r2c1

2. 6(2)n12 and 6(2)n9 = {15/24} = {(4/5)..}

3. 17(2)n23 = {89}, locked for r1
3a. -> {89} of 34(5)n12 (step 1) locked in r2c234
3b. -> no 8,9 elsewhere in r2

4. 22(5)n23 = {(15/24)367} ({89} unavailable)
4a. CPE: no 3,6,7 in r2c9
4b. 34(5)n12 and 22(5)n23 form grouped X-Wing on {67} in r12
4c. -> no 6,7 elsewhere in r12 (r12c5)
4d. max. r12c5 = 9
4e. -> min. r3c5 = 3 (no 1,2)

5. 8(2)n4 and 8(2)n7 = {17/26/35} (no 4,8,9)

6. 11(2)n6 and 11(2)n8 = {29/38/47/56}: no 1

7. 20(3)n5 and 20(3)n69 = {389/479/569/578} (no 1,2)

8. 11(3)n5 = {128/137/146/236/245} (no 9)

9. 13(2)n8 and 13(2)n9 = {49/58/67} (no 1,2,3)

10. 9(2)n7 = {18/27/36/45} (no 9)

11. Innies n8: r9c46 = 8(2) = {17/26/35} (no 4,8,9)
11a. 8(2)n7 and r9c46 require 2 of {567}
11b. -> {67} combo blocked for 13(2)n9
11c. -> 13(2)n9 = {49/58} = {(4/5)..}

12. 13(2)n9 (step 11c) and 6(2)n9 (step 2) form killer pair on {45} -> no 4,5 elsewhere in n9

13. Innie/outie (I/O) diff. r89: r8c46 = r7c5 + 15
13a. -> r7c5 = {12}, r8c46 = 16 or 17 = {79/89}
13b. -> 9 locked in r8c46 for r8 and n8
13c. cleanup: no 7,8 in r7c4; no 5,6,7,8 in r7c6

14. Innies r89: r8c456+r9c5 = 28(4) = {(47/56)89} (no 1,2,3) = {(5/7)..}

15. 3 in n8 locked in r9c46 (step 11) (-> {35}) or 11(2)n8 (-> {38})
15a. -> r9c46+r78c6 require {5/8}
15b. -> {58} combo for 13(2)n8 blocked
15c. -> 13(2)n8 = {49/67} = {(4/6)..}, {(4/7)..}

16. 13(3)n8 = {148/256} (no 7) = {(1/6)..}, {(4/6)..}
(Note: {157} blocked by r89 innies (step 14), {247} blocked by 13(2)n8 (step 15c))
16a. -> 13(2)n8 and 13(3)n8 form killer pair on {46} within n8
16b. -> no 4,6 elsewhere in n8
16c. cleanup: no 7 in r8c6, no 2 in r9c46 (step 11)

17. Naked pair (NP) on {89} at r18c6 -> no 8,9 elsewhere in c6

18. r7c5 blocks {129} combo for 12(3)n2
18a. -> no 9 in r3c5
18b. furthermore, 12(3)n2 must have 1 of {12} (hidden killer pair in c5 w/ 13(3)n8)
18c. -> {345} combo blocked
18d. also, {156/246} both blocked by 13(3)n8 (step 16)
18e. -> 12(3)n2 = {138/147/237} (no 5,6)
18f. max. r12c5 = 7
18g. -> no 3,4 in r3c5

19. 9 in c5 locked in n5 -> not elsewhere in n5
19a. -> 20(3)n5 (step 7) = {(38/47/56)9}

20. from step 17, {128} combo now unavailable for 11(3)n5
20a. -> cannot have both of {12}
20b. -> 11(3)n5 and 14(3)n5 form hidden killer pair on {12} in n5
20c. -> 14(3)n5 = {(1/2)..} = {158/167/248/257} (no 3)

21. 5 in r7 locked in n7 -> not elsewhere in n7
21a. -> 8(2)n7 = {17/26} (no 3) = {(1/6)..}, {(2/7)..}
21b. -> {27} combo for 9(2)n7 blocked
21c. -> 9(2)n7 = {18/36} (no 4) = {(1/6)..}
21b. -> 9(2)n7 and 8(2)n7 form killer pair on {16} -> no 1,6 elsewhere in n7

22. Hidden killer triple on {367} in n9, as follows:
22a. only places for {367} in n9 are r7c7, r7c89 and r89c7
22b. r7c89 cannot have 2 of {367} due to 20(3)n69 cage sum
22c. r89c7 cannot have 2 of {367} due to 13(3)n89 cage sum, and because 4 unavailable in r9c6
22d. -> r7c7, r7c89 and r89c7 must each contain exactly 1 of {367}
22e. -> r7c7 = {367}; no 8 in r89c7; no 3,7 in r9c6; no 3 in r9c7
(Note: possibilities for r9c6+r89c7 = [1][39]/[5]{17}/[5]{26})
22f. cleanup: no 1,5 in r9c4 (step 11)

23. 3 in r9 locked in 13(3)n78 = {3..} = {238} (only possible combo)
23a. -> r9c46 = [35] (step 11); r89c3 = {28}, locked for c3 and n7
23b. cleanup: no 4 in r1c4, no 8 in r9c89, no 1 in r8c12, no 6 in r9c12

24. r78c6 = [29]; r1c67 = [89]; r78c4 = [67]; r37c5 = [71]
24a. cleanup: no 4 in r12c5

25. Hidden single (HS) in r2 at r2c2 = 8

26. HS in r9 at r9c7 = 6
26a. -> r8c7 = 2
26b. cleanup: no 4 in r8c89

27. r89c3 = [82], r89c5 = [48]

28. NP on {49} at r9c89 -> no 9 in r7c89

29. Hidden triple (HT) in n7 at r7c123 = {459} (no 3,7)

30. NP on {23} at r12c5 -> no 2,3 elsewhere in c5 and n2
30a. cleanup: no 4 in r1c3

31. NP on {15} at r1c34 -> no 1,5 elsewhere in r1

32. Naked triple (NT) on {569} at r456c5 -> no 5,6 elsewhere in n5

33. 11(3)n5 = {137} (only remaining combo), locked for c6 and n5

34. NP on {46} at r23c6 -> no 4 elsewhere in n2

35. Naked single (NS) at r2c4 = 9

36. NT on {467} at r1c12+r2c3 -> no 4,6 elsewhere in n1

[Note: following step optional and may be skipped!]
37. Unique rectangle (UR) type I on {15} at r13c34
37a. -> no 1,5 in r3c3

38. I/O diff. n1: r4c1 = r13c3
[with step 37:]
38a. -> r3c3 = 3, r4c1 = {48}
[if step 37 skipped:]
38a. -> r3c3 = {135}, r4c1 = {468}

39. Innies r1234: r4c456 = 12(3) = [291/453] (no 6,7,8) = {(4/9)..}

40. Outies n1: r13c4+r4c123 = 25(5)
40a. r13c4 = {15} = 6(2) -> r4c123 = 19(3) = [8]([29]/{47})
(Note: [4]{69} (and [6]{49}, if step 37 skipped) blocked by r4c456 (step 39))
(Note: [8]{56} blocked because it would also require a 5 in r3c4 for 19(4) cage sum)
40b. -> r4c1 = 8
[with step 37:]
40c. -> r1c3 = 5 (step 38)
[if step 37 skipped:]
40c. -> r13c3 = [53] (step 38)

41. r13c4 = [15]
42. I/O diff. n7: r6c1 = r7c3 - 1
42a. -> r6c1 = 3, r7c3 = 4
42b. cleanup: no 5 in r5c12, no 7 in r4c2

43. r8c12 = [63]
43a. cleanup: no 2 in r5c2

44. HS in c1 at r1c1 = 4

45. HS in c1 at r7c1 = 5
45a. -> r7c2 = 9

46. HS in r3/c1/n1 at r3c1 = 9

47. HS in c2/n4 at r4c2 = 4
47a. -> r4c3 = 7 (cage sum)
47b. -> r4c456 = [291] (step 39)
47c. cleanup from step 47a: no 1 in r5c12

48. NS at r2c3 = 6
48a. -> r1c2 = 7

49. NT on {356} at r4c789 -> no 3,5,6 elsewhere in n6
49a. cleanup: no 8 in r5c89

50. I/O diff. n9: r6c9 = r7c7 + 2
50a. -> r6c9 = 9, r7c7 = 7

51. NS at r2c6 = 4
51a. -> split 18(4) at r1c89+r2c78 = {2367} (no 1,5) (step 4)
51b. -> r2c78 = [37]; r1c89 = {26}, locked for r1 and n3

Now it's all naked singles to end
Introduction Walkthrough by Afmob:
Hi!

I think the best way to introduce myself is a walkthrough :) . My first walkthrough had about 60 steps, so I cut out all the moves which led to
nothing. So some clean-up might be missing but it isn't essential for solving this assassin. The important moves are marked with a ! . I'm not sure if step 9 (previous step 15) is too much T&E, so I'm trying to solve it again without this step. But first, here is my walkthrough:

1. R12
a) 17(2) = {89} locked for R1
b) 34(5) = {46789} locked -> R2C1 <> 4,6,7,8,9
c) 8,9 locked in R2C234 for R2
d) 22(5): 3,6,7 locked (no 8,9 because of 1a, 1b) -> R2C9 <> 3,6,7
e) R12C5 <> 6,7 since 6,7 is locked in 34(5) and 22(5)

2. N8
a) Innies = 8(2) -> R9C46 = {17/26/35}

3. R89
a) Innies = 28(4) -> no 1,2,3 in R8C456+R9C5
b) 8,9 locked for N8 (in Innies)
c) R8C4 <> 4,5
d) Outies = 9(3) -> no 7 in R7C456
-> R8C4 <> 6
-> R8C6 <> 4
e) R7C56 <> 4,5,6 since outies of R89 would be
larger than 9 because of R7C7 (456)
f) R8C6 <> 5,6,7

4. C6
a) Naked pair in R18C6 (8,9) locked

5. R8
a) Killer pair (23,39) in 11(2) blocks following combinations
of 13(3): {139,238} -> 13(3) <> 9
b) 9 locked in R8C46

6. R9
a) 8(2) + Innies of N8 = 8(2) -> 13(2) <> 6,7 since both 8(2) would be {35}

7. N9
a) Killer pair (45) in 6(2) + 13(2) -> no 4,5 elsewhere
b) Killer pair (45) in 6(2) blocks (45) in 9(2) in N7 -> R8C12 <> 4,5
c) 20(3): R6C9 <> 6,7 since they would lead to a conflicting
cage sum
d) R6C9 <> 3 since R7C89 would be 89 and that's not possible
because of killer pair (89) of 13(2)
e) 13(3): R9C6 <> 6,7 since they need a 5 which is not possible in R89C7

8. N8
a) Innies = 8(2): R9C4 <> 1,2

9. N7 !
a) 13(3) <> 9 because of following conflict:
13(3) = {139} -> 8(2) = {26} -> 9(2) has no possible combination

10. R7
a) 9 locked in R7C123

11. N9
a) 20(3): R6C9 <> 4,8 would lead to conflicting cage sum
b) 20(3) <> 6 because R6C9 is the only position where
5 and 9 is possible
c) 8 locked in R7C89 for R7 and N9
d) 13(2) = {49} -> locked for N9 and R9
e) 6(2) = {15} -> locked for N9 and R8
f) 5 locked in 13(3) -> R9C6 = 5

12. N89
a) Innies = 8(2): R9C4 = 3
b) R89C7 = {26} locked for C7
c) R7C6 = 2, R8C6 = 9, R7C5 = 1
d) 13(2) = {67} -> R7C4 = 6, R8C4 = 7, R9C5 = 8, R8C5 = 4

13. R1
a) R1C6 = 8, R1C7 = 9

14. N2
a) R3C5 = 7 since 12(3) = {237}
b) R12C5 = {23} -> locked for N2 and C5

15. N5
a) 20(3) = {569} locked
b) 11(3) = {137} locked for N5 and C6
c) 14(3) = {248} locked for C4

16. N2
a) R2C4 = 9
b) 6(2) = {15} since R1C4 = (15) -> locked for R1

17. R7
a) R7C789 = {378} locked

18. N7
a) 3 locked in 9(2) = {36} locked for N7 and R8
b) 8(2) = {17} locked
c) R9C3 = 2, R8C3 = 8
d) 17(3) = 5{39/48} -> R6C1 = (38) because of R7C12 = 5(4/9)
e) 5 locked in R7C12

19. N9
a) R8C7 = 2, R9C7 = 6

20. R2
a) Hidden Single: R2C2 = 8

21. R1234
a) Innies = 12(3) -> R4C4 <> 8 (R4C5 is at least 5)
b) R4C6 <> 7 (same reason)
c) R4C5 <> 6 (conflict in cage sum)

22. N69 !
a) Innies = 14(3) -> R4C789 <> 8 because:
-> 248 blocked by R4C4 = (24)
-> 158 blocked by killer pair (15) of Innies 12(3)
of R1234 = {129/345}

23. R4
a) Hidden Single: R4C1 = 8
b) R6C1 = 3, R8C1 = 6, R8C2 = 3
c) Hidden Single: R3C3 = 3 @ N1

24. N7
a) 17(3) = {359} <> 4
b) Hidden Single: R7C3 = 4, R4C2 = 4 @ N4
c) R4C4 = 2

25. N1
a) 19(4) = {3457} -> no 1,6,9
b) R3C4 = 5, R4C3 = 7, R2C3 = 6, R1C2 = 7, R1C1 = 4
c) R9C2 = 1, R9C1 = 7, R1C4 = 1, R1C3 = 5, R2C6 = 4, R3C6 = 6

26. R456
a) Hidden Single: R4C9 = 6 @ R4, R1C8 = 6 @ N3
b) 8(2) = {26} -> R4C2 = 6, R4C1 = 2
c) R2C1 = 1, R3C1 = 9, R3C2 = 2, R7C1 = 5, R7C2 = 9, R6C2 = 5
d) R6C9 = 9, R6C5 = 6, R6C3 = 1, R5C3 = 9, R9C9 = 4, R9C8 = 9
e) R6C6 = 7, R5C5 = 5, R4C5 = 9
f) Hidden Single: R6C8 = 2 @ R6

27. N3
a) 22(5) = {23467} -> R1C9 = 2

28. Rest is clean-up and singles.

Edit: Had a look at Cathy's walkthrough and step 4 also eliminates 9 from R9C3 which is one of my two main steps
Walkthrough by Andrew:
A bit late with this one. I only started it earlier in the week after posting some other walkthroughs and only finished going through the posted walkthroughs today.

It was interesting to see what I missed, some of it obvious and some definitely not obvious like Mike's excellent hidden killer triple in N9. As it happened the obvious things, like the grouped X-Wing in R12 and those interactions in R9 that I missed, only seemed to speed up the solutions by a limited amount. Still I ought to have spotted them; for some other puzzles they might have been critical.
gary w wrote:
..I guess this means the path is quite tightly constrained.

My solving path was quite a lot different than the others. My breakthrough came from combination work in C456 including innies in C4 and C6 that nobody else used.

I'll agree with Mike's estimated rating of 1.25. My combination work probably isn't any more difficult than his hidden killer triple.


Here is my walkthrough for A69.

1. R1C34 = {15/24}

2. R1C67 = {89}, locked for R1

3. R5C12 = {17/26/35}, no 4,8,9

4. R5C89 = {29/38/47/56}, no 1

5. R78C4 = {49/58/67}, no 1,2,3

6. R78C6 = {29/38/47/56}, no 1

7. R8C12 = {18/27/36/45}, no 9

8. R8C89 = {15/24}

9. R9C12 = {17/26/35}, no 4,8,9

10. R9C89 = {49/58/67}, no 1,2,3

11. R456C5 = {389/479/569/578}, no 1,2

12. R456C6 = {128/137/146/236/245}, no 9

13. 20(3) cage at R6C9 = {389/479/569/578}, no 1,2

14. 34(5) cage at R1C1 = {46789} (only possible combination)
14a. CPE R2C1 = {1235}
14b. 8,9 in R2 locked in R2C234, locked for R2

15. 22(5) cage at R1C8 = {13567/23467} = 367{15/24}
15a. CPE no 3,6,7 in R2C9

16. 45 rule on R12 1 outie R3C5 – 1 = 1 innies R2C19, min R2C19 = 3 -> min R3C5 = 4

17. 45 rule on R89 4 innies R8C456 + R9C5 = 28 = {4789/5689}, 8,9 locked for N8, clean-up: no 4,5 in R8C4

18. 45 rule on N8 2 innies R9C46 = 8 = {17/26/35}, no 4

19. Hidden killer triple 1,2,3 in R7C56 + R9C46 -> R7C5 = {123}, R7C6 = {23}, clean-up: R8C6 = {89}

20. Killer pair {89} in R18C6, locked for C6

21. R789C5 = {148/247/256} (cannot be {139/238} because 1,2,3 only in R7C5, cannot be {157/346} which clash with R8C456 + R9C5), no 3,9

22. 9 in N8 locked in R8C46, locked for R8
22a. R7C4 = 4 or R7C6 = 2 -> R789C5 (step 21) = {148/256} (cannot be {247}), no 7

23. 45 rule on R89 3 outies R7C456 = 9 = {126/234} (cannot be {135} which clashes with R789C5), no 5,7, clean-up: no 6,8 in R8C4
23a. 2 locked in R7C56, locked for R7 and N8, clean-up: no 6 in R9C46
[Alternatively Killer Pair 4,6 in R7C4 and R8C456 + R9C5, locked for N8, clean-up: no 2 in R9C46]

24. R7C5 = {12} -> R123C5 must contain 1/2 = {138/147/156/237/246}, no 9

25. 9 in C5 locked in R456C5, locked for N5
25a. R456C5 (step 11) = {389/479/569}

26. R456C6 = {137/146/245} (cannot be {236} which clashes with R7C6)
26a. R456C5 (step 25a) = {389/569} (cannot be {479} which clashes with R456C6), no 4,7

27. Killer Pair 6,8 in R456C5 and R89C5, locked for C5

28. 7 in C5 locked in R123C5, locked for N2
28a. R123C5 (step 24) = 7{14/23), no 5
28b. 7 of 34(5) cage at R1C1 locked in R1C12 + R2C23, locked for N1

29. R456C6 contains 1/2 -> R456C4 must contain 1/2
29a. R456C4 = {167/248/257} (cannot be {158} which clashes with R456C5), no 3

30. 3 in C4 locked in R39C4
30a. R456C4 contains 1/2 -> R1239C4 must contain 1/2
30b. R1C6 = {89} -> R123C4 must contain 8,9
30c. 45 rule on C4 4 innies R1239C4 = 18 = 3{159/168/249/258}, no 7, clean-up: no 1 in R9C6 (step 18)

31. R1C6 = {89}, no other 8,9 in R1239C6
31a. 45 rule on C6 4 innies R1239C6 = 23 = {1679/2579/3578/4568} (cannot be {2678/3479/3569} which clash with R456C6)
31b. 7 of {3578} must be in R9C6 -> no 3 in R9C6, clean-up: no 5 in R9C4 (step 18)
31c. R1239C6 = {169}7/{259}7/{358}7/{468}5

32. R1239C4 (step 30c) = 3{159/168} (cannot be {2349} which clashes with R12C5 because the 3 must be in R9C4, cannot be {258}3 => R9C6 = 5 when R123C4 clashes with R123C6 = {458}) = 13{59/68}, no 2,4, 1 locked for C4, clean-up: no 2,4 in R1C3
32a. {1359} must be {159}3 because {359}1 => R9C6 = 7 when R123C4 clashes with all remaining combinations for R123C6 in step 31c
32b. 1 of {1368} must be in R1C4
32c. R1239C4 = {159}3/{168}3 -> R9C4 = 3, R9C6 = 5 (step 18), R7C56 = [12], R7C4 = 6 (step 23), R8C46 = [79], R1C67 = [89], R2C4 = 9, clean-up: no 2 in R8C12, no 8 in R9C89
32d. Naked pair {15} in R13C4, locked for C4 and N2
32e. 8 in N3 locked in R3C789, locked for R3
[At this stage there was an UR on R13C34 which I never spotted. That’s not surprising since it’s a technique that I refuse to use. I only saw it after working through posted walkthroughs from others that do use it.]

33. R9C89 = {49} (cannot be {67} which clashes with R9C12)
33a. Naked pair {49} in R9C89, locked for R9 and N9 -> R89C5 = [48], R3C5 = 7, clean-up: no 5 in R8C12, no 2 in R8C89

34. Naked pair {23} in R12C5, locked for C5 and N2
34a. Naked pair {46} in R23C6, locked for C6

35. Naked pair {15} in R8C89, locked for R8 and N9, clean-up: no 8 in R8C12

36. Naked triple {378} in R7C789, locked for R7 and N9

37. Naked pair {36} in R8C12, locked for R8 and N7 -> R89C7 = [26], R8C3 = 8, R9C3 = 2

38. R2C2 = 8 (hidden single in R2)
38a. Naked triple {467} in R1C12 + R2C3, locked for N1

39. Naked pair {15} in R1C34, locked for R1

40. 45 rule on R1234 3 innies R4C456 = 12 = [291/453] (only remaining permutations), no 6,7,8

41. 45 rule on N7 1 innie R7C3 – 1 = 1 remaining outie R6C1 -> R6C1 = {348}
41a. 17(3) cage at R6C1 = {359/458}, 5 locked in R7C12, locked for R7, clean-up: no 4 in R6C1 (step 41)

42. 45 rule on N9 1 remaining outie R6C9 – 2 = 1 innie R7C7 -> R6C9 = {59}, no 8 in R7C7

43. 45 rule on N1 2 remaining outies R1C4 + R4C1 – 6 = 1 innie R3C3
43a. Min R1C4+R4C1 = 7 -> no 1 in R4C1
43b. Max R1C4+R4C1 = 14 -> no 9 in R3C3
43c. 2,9 in N1 locked in R2C1 + R3C12 for 20(4) cage -> no 2,9 in R4C1
43d. 20(4) cage at R2C1 = 29{18/36/45}, no 7
43e. 4,6,8 only in R4C1 -> R4C1 = {468}

44. 45 rule on N4 3 innies R4C123 – 1 = 3 outies R7C123
44a. R7C123 = {459} = 18 -> R4C123 = 19 = {289/478/568} (cannot be {379} because R4C1 only contains 4,6,8, cannot be {469} which clashes with R4C456) = 8{29/47/56}, no 1,3 -> R4C1 = 8 -> R6C1 = 3, R7C3 = 4 (step 41), R8C12 = [63], clean-up: no 5 in R2C1 + R3C12 (step 43d), no 5 in R5C1, no 2,5 in R5C2
44b. R4C2 = 4 (hidden single in N4), R4C4 = 2

45. R1C1 = 4 (hidden single in C1)

46. Naked triple {129} in R2C1 + R3C23, locked for N1 -> R1C34 = [51], R3C34 = [35], R4C3 = 7 (cage sum), R2C3 = 6, R1C2 = 7, clean-up: no 1 in R5C12 -> R5C12 = [26], R2C1 = 1, R3C12 = [92], R7C12 = [59], R9C12 = [71], R6C2= 5, R6C9 = 9, R9C89 = [94], R23C6 = [46], clean-up: no 5 in 22(5) cage at R1C8 (step 15), no 7 in R5C8

47. R2C9 = 5 (hidden single in R2), R8C89 = [51]

48. R6C9 = 9 -> R7C89 = 11 = {38}, no 7

49. R7C7 = 7 (hidden single in R7)

50. R5C9 = 7 (hidden single in C9), R5C8 = 4

51. R1C9 = 2 (hidden single in C9), R12C5 = [32]

and the rest is naked singles

The final stages could have been a bit quicker if I'd used some of the naked singles instead of looking for hidden singles.

In case anyone still wonders why, the reason I refuse to use UR is because I don't consider it to be solving the complete puzzle. I know that all Ruud's Assassins, all forum puzzles and all puzzles on the other website I use will have unique solutions. However I prefer to work right through a puzzle to reach the only solution rather than to rely on uniqueness. To me UR is a shortcut that bypasses part of the puzzle. Therefore I feel that using it must lessen the satisfaction of solving the puzzle although I don't doubt that those who spot an UR probably get some satisfaction from doing so. I suppose one could say that I'm philosophically opposed to using it. Others take the opposite view and find it acceptable. I certainly wouldn't try to stop them using UR.

Another point is that UR can be considered unnecessary. As far as I can see any Sudoku that requires UR as the only technique to solve it cannot be unique. If I'm wrong on that then maybe at some stage in the future I might reluctantly be forced to use it.


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PostPosted: Sun Jul 06, 2008 11:12 pm 
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Posts: 1040
Location: Sydney, Australia
Discussion on use of UR (Sept 07)
Post 1: garyw:
Looked up URs on another site...of course!!..I'm really rather appalled that I didn't see the implications of the two 1/5 combos.I don't think it's come up in any other sudoku I've done but no excuses!.I'll know the next time.
Post 2: Andrew:
In case anyone still wonders why, the reason I refuse to use UR is because I don't consider it to be solving the complete puzzle. I know that all Ruud's Assassins, all forum puzzles and all puzzles on the other website I use will have unique solutions. However I prefer to work right through a puzzle to reach the only solution rather than to rely on uniqueness. To me UR is a shortcut that bypasses part of the puzzle. Therefore I feel that using it must lessen the satisfaction of solving the puzzle although I don't doubt that those who spot an UR probably get some satisfaction from doing so. I suppose one could say that I'm philosophically opposed to using it. Others take the opposite view and find it acceptable. I certainly wouldn't try to stop them using UR.

Another point is that UR can be considered unnecessary. As far as I can see any Sudoku that requires UR as the only technique to solve it cannot be unique. If I'm wrong on that then maybe at some stage in the future I might reluctantly be forced to use it.
Post 3: Ruud:
Andrew,
Please do not take the following argument too seriously, it's just an illustration.

Suppose I make a mistake and create a puzzle based on a solution with a duplicate 3 in row 5. This is not a valid puzzle.

When you are attempting to solve this puzzle, you encounter a chain that gives you the choices:
A: There are two numbers 3 in row 5;
B: Both these candidates are false.

As an experienced solver you will no doubt choose option B, which happens to be the wrong option for this puzzle, leading you to a partial solution which is not even close to the one I set.

When the puzzle is invalid, all bets are off. A puzzle which does not have a unique solution is not a valid puzzle. As a player, you ONLY want to solve valid puzzles. So you go to a source where you expect ALL puzzles to be valid.
Many players use software which can validate a puzzle without revealing the solution to the user. Consider this a double check that you are indeed working on a valid puzzle.

Now here comes the core argument:

A valid puzzle has certain properties. It can be completed in such a way than each cell contains a single value and it has one of each number in each row, column and nonet. For killers, the caged numbers add up to the sum given for the cage and no number may be repeated in a cage. And it does not contain a pattern which can be altered in such a way that another valid solution is formed.

When you rinse the candidates after a placement, you are using one of these properties. When you eliminate a candidate with an XY-Wing, you are using a combination of properties. When you avoid a UR you are using yet another property.

The whole exercise of solving a puzzle is recognizing and avoiding patterns with are known not to be present in a valid puzzle.

The only logical reason to avoid uniqueness-based techniques is that the puzzle may have more than one solution and therefore could contain these patterns. Such a source may also provide you with a puzzle that has duplicate digits in a unit, so you should also avoid all solving techniques which are based on the premise that each unit contains only one of each digit.

I'm right in the middle of a Uniqueness discussion on another forum, so I'm just test firing my arguments here. Feel free to rip them to pieces.

Ruud
Post 4: Para:
Hi Ruud

I think this whole UR-based techniques will never be solved. I like to use UR-based techniques in regular sudoku. There are some nice ways to use UR's in regular sudoku. Lately had a UR-xy-wing. That was kinda fun, used a UR as a pivot cell. Bu this might also be because my vanilla sudoku arsenal is kinda limited to single digit techniques and wings(and maybe occasionally an ALS-xz)
I always try to avoid using them when writing a walk-through for a Killer Sudoku, but when i am solving it for fun or on speed i don't mind using it. For Killer Sudoku i like to avoid UR's, because i rather find another way around it. I guess i just like to stick to killer techniques in Killer Sudoku. I have seen some older forum messages where you used ALS-techniques to solve Killer Sudoku and also seen Udosuk use y-wings to shorten the solving path. But maybe my level of Killer sudoku solving is higher than my regular sudoku techniques. But i guess it is mostly that i like killer techniques over UR-techniques in killer, but i don't prefer regular sudoku techniques over Ur-techniques in vanilla sudokus.
But i never really opposed UR-techniques. I solve a lot of other logic puzzles and one puzzle type where uniqueness-based are almost inevitable are domino puzzles(you get a grid with numbers(grid with dominoes with the edges taken away) and have to locate the original placement of the dominoes). Never really tried to even solve one without assuming it was unique. They can be hard enough without this assumption. Sometimes i think these sudoku guys take it all too serious.

Yeah i am not really taking a position on all this (and i never really will). You should just solve puzzles the way you want. I don't like people saying that you shouldn't do something, just as long as you know why you are solving the way you are. Sudoku is a form of puzzling, puzzling is essentially a form of relaxation, ergo sudoku is a form of relaxation. So just have fun :wink:

greetings

Para, puzzle fan :bounce: :thumright:
Post 5: CathyW:
I'm with you Ruud! Using unique rectangles is perfectly logical and acceptable to me, including in killers.

I understand that others prefer to "prove" that a puzzle is unique by solving without them but that's all it amounts to - a preference for which weapon to use from the arsenal of many techniques.

It's part of the fascination of this forum in particular to see how different people tackle the same puzzle.
Post 6: mhparker:
Ruud wrote:
I'm right in the middle of a Uniqueness discussion on another forum, so I'm just test firing my arguments here.
You've got full support from me on this one, Ruud! Not only do we share the same opinion on this topic, but the way you presented your arguments is very similar to the way I would have presented them.

Good luck with the debate on the other forum!
Post 7: Andrew:
The only reason why I posted my message about why I won't use UR was as a follow-up to my earlier reply to Gary's messages.

It stirred up some interesting replies. It's good that we can express our different views and still work well together on and off the forum.

Even though I'm opposed to UR, I'll still wish Ruud good luck in his debate on the other forum!


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PostPosted: Sun Jul 06, 2008 11:13 pm 
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Assassin 69 V1.5 by mhparker (Sept 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:5888:5888:3074:3074:3332:2821:2821:6919:6919:5385:5888:5888:5888:3332:6919:6919:6919:3857:5385:5385:5908:5908:3332:4887:4887:3857:3857:5385:5908:5908:3614:4383:3616:4887:4887:3857:2852:2852:6438:3614:4383:3616:4650:3371:3371:2861:6438:6438:3614:4383:3616:4650:4650:5429:2861:2861:6438:1337:3898:2875:4650:5429:5429:2623:2623:4161:1337:3898:2875:4165:2630:2630:2376:2376:4161:4161:3898:4165:4165:1359:1359:
Solution:
+-------+-------+-------+
| 3 1 7 | 5 4 2 | 9 6 8 |
| 2 8 4 | 7 6 9 | 3 1 5 |
| 9 6 5 | 8 3 1 | 4 7 2 |
+-------+-------+-------+
| 4 7 3 | 9 2 5 | 6 8 1 |
| 6 5 8 | 1 7 3 | 2 9 4 |
| 1 9 2 | 4 8 6 | 5 3 7 |
+-------+-------+-------+
| 7 3 6 | 2 1 4 | 8 5 9 |
| 8 2 9 | 3 5 7 | 1 4 6 |
| 5 4 1 | 6 9 8 | 7 2 3 |
+-------+-------+-------+
Quote:
mhparker, lead-in: Est. rating: 1.5
Para: a typical Mike Killer.The rating seems to fit. One really tricky move
Afmob: V1.5 was definitely harder than the original because it didn't fall that fast and I needed several contradiction moves
Andrew in 2011: There was one thing which Para, Afmob and I had in common. Our solving paths all used ....as the key breakthrough but we used different ways to achieve ..I'll rate my walkthrough...at Easy 1.5. .... short and localised.... analysis
A forum Revisit to this puzzle in 2022 here
Walkthrough by Para:
Hi all

Here's the walk-through for Assassin 69V1.5. Again a typical Mike Killer.
The rating seems to fit. One really tricky move, which probably isn't necessary if i had seen the things i did after this step before this step.

Walk-through Assassin 69V1.5

1. R1C34 = {39/48/57}: no 1,2,6

2. R1C67, R5C12 and R78C6 = {29/38/47/56}: no 1

3. R5C89 = {49/58/67}: no 1,2,3

4. 11(3) at R6C1 = {128/137/146/236/245}: no 9

5. 21(3) at R6C9 = {489/579/678}: no 1,2,3

6. R78C4 and R9C89 = {14/23}: no 5,6,7,8,9

7. R8C12 and R8C89 = {19/28/37/46}: no 5

8. R9C12 = {18/27/36/45}: no 9

9. 45 on N8: 2 innies: R9C46 = 14 = {59/68} = {5|6..}: no 1,2,3,4,7
9a. R78C6 = {29/38/47}: {56} blocked by R9C46: no 5,6

10. 45 on R89: R7C456 = 7 = {124} -->> locked for R7 and N8
10a. R8C4 = 3; R7C456 = [214]; R8C6 = 7
10b. Clean up: R1C3: no 9; R1C7: no 4,7
10c. 15(3) at R7C5 = 1{59/68} = {5|8..},{6|9..},{5|6..}

11. 11(3) at R6C1 = [2]{36}/[1]{37} -->> R6C1 = {12}; R7C12 = {36/37} -->> 3 locked for R7 and N7
11a. Clean up: R9C12: no 6

12. 14(3) at R4C4 = {149/158/167}: 1 locked for C4 and N5

13. 14(3) at R4C6 = {239/356}={6|9..}: no 8; 3 locked for C6 and N5
13a. Clean up: R1C7: no 8

14. 17(3) at R4C5 = {278/467}: {269/458} blocked by 15(3) at R7C5: no 5,9; 7 locked for C5 and N5
14a. Clean up: R456C4: no 6(step 12)

15. 13(3) at R1C5 = {238/346}= {6|8..}: {256} blocked by 15(3) at R7C5: no 5,9

16. Hidden Pair: R89C5 = {59} -->> locked for N8

17. Naked Pair: R9C46 = {68} -->> locked for R9
17a. Clean up: R9C12: no 1

18. Killer Pair {24} in R9C12 + R9C89 -->> locked for R9

19. 45 on N9: 1 outie and 3 innies: R9C6 + 14 = R7C789
19a. R9C6 = 6 -->> R7C789 = {569/578}
19b. R9C6 = 8 -->> R7C789 = {589}: {679} blocked by R7C12}
19c. 5 locked for R7 and N9

20. 16(3) at R8C7(R89C7 + R9C6] = {19}[6]/[17][8]: R89C7 = {19}/[17]: 1 locked for C7 and N9
20a. R9C89 = {23}(last combo) -->> locked for R9 and N9
20b. R8C89 = {46}(last combo) -->> locked for R8 and N9
20c. R9C12 = {45} (last combo) -->> locked for R9 and N7
20d. R89C5 = [59]

21. 16(3) at R8C4 = [178/916]: R89C3 = [17/91]: 1 locked for C3 and N7
21a. R8C12 = {28} (last combo) -->> locked for N7
21b. R5C4 = 1(hidden)

22. 9 in N7 locked for C3

23. 45 on R1234: 3 innies: R4C456 = 16 = [862/925]: [826/475] blocked by 17(3) at R4C5, [943/583] blocked by 14(3) at R4C4, [529] blocked by 14(3) at R4C6; 2 locked for R4 and N5
[Moderator edit: See the introduction to Andrew's 2011 walkthrough for comments on this step and step 24.]
23a. Clean up: R6C4: no 8,9; R56C5: no 6; R56C6: no 5
Just noticed this, been there for a while. Kinda cool.

24. Killer XY-Wing: R4C4 = {89}; R9C4 = {68}; 14(3) at R4C6 = {6|9..}: eliminate 6 from Common Peers of R9C4 and 14(3): R9C6: no 6
24a. R9C46 = [68]; R89C3 = [91](last combo); R89C7 = [17]
24b. Clean up: R1C7: no 3

25. 21(3) at R6C9 = [4]{89}/[7]{59}: R6C9 = {47}; R7C89 = {59/89}: 9 locked for R7

26. 18(4) at R5C7 needs one of {58} in R7C7, can’t have both {47} because of R6C9 or both {17} because R6C8 only cell with {17}
26a. 18(4) = {1359/1368/1458/2358/3456}: no 7; R6C8: no 9(only place for 1)

27. R5C89 = {4|5|7..},{4|7|8..}; R6C9 = {47} -->> 18(4) at R5C7 can’t have both {45} or {48} in N6 -->> 18(4): {1458} blocked

28. 45 on N9: 1 innie and 1 outie: R6C9 + 1 = R7C7: R6C9 + R7C7 = [45/78]

29. 18(4) = {1359/1368/2358}: {3456} is blocked by step 28: no 4, R6C8: no 6(only place for 1; 3 locked for N6

30. 3 in R4 locked for N4
30a. Clean up: R5C12: no 8

31. R5C5 = {478}; R5C89 = {4|7|8..} -->> R5C12: {47} combo blocked: no 4,7
31a. R5C12 = {29/56} = {5|9..}

32. 25(4) at R5C3 = {928}[6]/{478}[6]/{468}[7]: {4579} blocked by R5C12: no 1,5; R6C2: no 2(only place for 9); 8 locked for N4
32a. {478}[6] blocked by R6C5 + R6C9(can’t have 2 of {478} in R6C23) -->> 25(4) = {289}[6]/{468}[7]: R5C3 + R6C23: no 7

33. 7 in N4 locked for R4

34. 45 on N47: 3 innies: R4C123 = 14 = {347}(only option because both 3 and 7 locked within these cells) -->> locked for R4 and N4

35. 25(4) at R5C3 = {289}[6] -->> R7C3 = 6; R6C2 = 9(only place in cage); R56C3 = {28} -->> locked for C3 and N4
35a. R6C1 = 1

36. Naked Pair: R5C12 = {56} -->> locked for R5
36a. R5C89 = {49}(last combo) -->> locked for N6
36b. R6C9 = 7; R56C6 = [36]; R4C56 = [25]; R6C4 = 4; R4C4 = 9; R56C5 = [78]
36c. R56C3 = [82]; R5C7 = 2; R7C7 = 8(step 28); R4C7 = 6
36d. R1C67 = [29](last combo)
36e. Clean up: R1C3: no 3

37. 23(4) at R3C3 = {3578}(last combo): no 4; R3C4 = 8(only place in cage); R3C3 = 5(only place in cage)
37a. R4C1 = 4(hidden); R9C12 = [54]; R5C12 = [65]

38. R1C34 = [75](last combo)
38a. R2C4 = 7; R4C23 = [73]; R7C12 = [73]; R2C3 = 4
38b. R3C7 = 4(hidden); R1C5 = 4(hidden); R3C8 = 7(hidden)

39. 19(4) at R3C6 = 46[18](last combo) -->> R3C6 = 1; R4C8 = 8
39a. R2C6 = 9; R4C9 = 1; R3C1 = 9(hidden)

40. 15(4) at R2C9 = 71[52](last combo) -->> R23C9 = [52]

And the rest is all singles.


greetings

Para
Walkthrough by Afmob:
V1.5 was definitely harder than the original because it didn't fall that fast and I needed several contradiction moves like step 15 from my first walkthrough to solve this one.

Walkthrough for A69 V1.5:

1. R89
a) Outies = 7(3) = {124} locked for R7 + N8
b) R8C4 = 3, R7C4 = 2, R7C6 = 4, R8C5 = 7, R7C5 = 1

2. N5
a) 14(3) @ R4C4 = 1{49/58/67} -> 1 locked for C4 + N5
b) 14(3) @ R4C6 = 3{29/56} -> 3 locked for C6 + N5

3. N2
a) 3 locked in 13(3) = 3{28/46} -> no 5,7,9

4. C5
a) 7 locked in 17(3) = 7{28/46} -> no 5,9
b) 14(3) @ R4C4 <> 7 -> no 6 since {167} is not possible
c) 5 locked in 15(3) -> R89C5 = {59}
d) Innies 14(2) of N8 = {68} locked for R9
e) Clean up: 9(2) in R9 = {27/45}

5. N7
a) 11(3): R7C12 <> 5,8 (conflict with cage sum)
b) 11(3) = {137/236} -> 3 locked in R7C12 for R7 + N7
c) R6C1 = (12)
d) 16(3) <> 4,5 because: {259},{457} not possible because R9C4 = (68)
f) ! 9(2) = {45} locked for R9 + N9 since {27} leads to a conflict:
-> 9(2) = {27} -> 16(3) = {169} -> 10(2) = {46} -> R7C1 = 3 = R7C2

6. N8
a) R9C5 = 9, R8C5 = 5

7. N7
a) Clean up: 10(2) = {19/28}

8. N9
a) Clean up: 5(2) = {23} locked in R9 + N9
b) 16(3) = 1{69/78} -> 1 locked in R89C7 for C7 + N9
c) 10(2) = {46} -> locked for R8 + N9
d) R8C7 <> 8 because of R9C6 = (68) since 16(3) cannot have 6 and 8

9. R1
a) 12(2): R1C3 <> 9
b) 11(2): R1C7 <> 4,7,8

10. N7
a) 10(2) = {28} locked for N8 since {19} is blocked by R8C7 = (19)
b) 16(3) = 1{69/78} -> 1 locked in R89C3 for C3
c) 9 locked in R78C3 for C3

11. R5
a) Hidden Single: R5C4 = 1

12. N7
a) 25(4) <> 1 since R567C3 would be {789} which is blocked by
Killer pair (79) of 16(3)

13. R1234
a) Innies 16(3) <> 3 because:
-> no {367} since R4C5 would be 7 and R4C4 has neither 3 nor 6
-> no {349} since R4C46 = {49} -> no 3 possible in R4C56
-> no {358} since R4C46 = {58} -> no 5,8 possible in R4C56
-> 16(3) = {259/268/457}
b) Innies 16(3): R4C5 <> 8 because R4C4 has no 2 or 6
c) R4C5 <> 4 since 7 is only possible in R4C5

14. N9
a) 5,8 locked in R7C789:
-> R7C7 = (58) since 21(3) cannot have both 58
-> 21(3): R6C9 <> 8 because R7C89 must have a 5 which
is not possible (21(3) = 5 + 8 + ?)

15. C6 !
a) R123C6 <> 8 because:
-> Innies 20(4) = 1{289/568}, because of R9C6 = (68)
-> (R1+R2+R3)C6 = 12/14 = 1{29/56/58}
-> 158 not possible because 13(3) in N2 would be {346} -> N2 <> 2
b) Hidden Single: R9C6 = 8, R9C4 = 6
c) Clean up: R1C7 <> 3

16. N9
a) 16(3) = {178} -> R9C7 = 7, R8C7 = 1
b) 21(3) = 9{48/57} -> no 6
c) R6C9 = (47)

17. N7
a) 16(3) = {169} -> R9C4 = 6, R9C3 = 1, R8C3 = 9

18. N6 !
a) 18(4) <> 7 because:
-> if R7C7 = 5 -> R6C9 = 4 -> 18(4) = {1359/2358}
-> if R7C7 = 8 -> R6C9 = 7 -> 18(4) = {1368/1458/2358}
-> 18(4) = {1359/1368/1458/2358}
b) 18(4) <> 4 because if 18(4) = {1458} -> 13(3) = 67 -> R6C9 = ?
c) 3 locked in 18(4)

19. N4
a) 3 locked in R4C123
b) 11(2) <> 8

20. N47
a) Innies 14(3) = 3{29/47/56} -> no 1,8
b) Hidden Single: R6C1 = 1 @ N4
c) 11(3) = {137} -> R7C3 <> 7
d) R7C3 = 6
e) 25(4) = 68{29/47} -> no 5

21. N6
a) 18(4) = {2358} -> no 6,9

22. N5
a) 6 locked in R6C56

23. N6
a) 18(4) = {2358} -> 2 locked for N6
b) R4C7 <> 5,8
c) 13(2) <> 5,8 since 5 or 8 is locked in 18(4)
d) Killer pair (47) of 13(2) blocks {47} of 11(2) @ N4
e) Killer pair (47) of 13(2) + R6C9 = (47) -> locked in N6

24. N3
a) 4 locked in R23C7
b) 15(4) = {1239/1257/1356} -> no 8

25. N69
a) Innies 15(3) = 1{59/68} -> Killer pair (56) blocks {356}
of Innies N47 -> R4C123 <> 5,6

26. N4
a) 5 locked in 11(2) = {56} locked for R5

27. N6
a) 13(2) = {49} locked for R5+N6
b) R4C7 = 6, R6C9 = 7
c) 21(3) = {579} -> {59} locked in R7C89 for N9
d) R7C7 = 8
e) 18(4): 2,3,5 locked in N6
f) R4C9 = 1, R4C8 = 8

28. N3
a) Hidden Single: R1C9 = 8

29. N5
a) 17(3) = {278} locked for N5+C5 because R4C5 = (27) and R5C5 = (278)
b) R5C6 = 3, R5C7 = 2
c) 14(3) @ R4C4 = {149} locked for R5+N6
d) R4C6 = 5 -> R6C6 = 6

30. N2
a) 11(2) = {29} -> R1C7 = 9, R1C6 = 2
b) 19(4) = {1468} -> R3C6 = 1, R3C7 = 4, R2C6 = 9

31. N3
a) 27(5) = 389{16/25} because {12789} not possible because of
R2C7 = (35) -> 27(5) <> 7

32. C8
a) Hidden Single: R3C8 = 7
b) 15(4) = {1257} -> 2,5 locked for C9+N3
c) R2C7 = 3, R6C7 = 5, R6C8 = 3, R9C8 = 2, R9C9 = 3, R7C9 = 9
e) R7C8 = 5, R5C9 = 4, R5C8 = 9, R8C9 = 6, R8C8 = 4

33. R3
a) 9 locked in 21(4) = 9{237/246/345} -> no 8
b) R4C1 <> 9
c) Hidden Single: R8C1 = 8 @ C1 -> R8C2 = 2

34. N1
a) Hidden Single: R3C1 = 9 @ C1
b) 5 locked in R123C3
c) 12(2) = {57} locked for R1
d) R3C2 = {36} -> no 3,6 in R24C1
e) 3 locked in 23(4) = 37{49/58} -> no 2
f) Hidden Single: R3C9 = 2 @ R3 -> R2C9 = 5
g) 23(5) = 348{17/26} -> 3 locked in R1C12 for R1+N1
h) R3C2 = 6

35. Rest is clean-up and singles
2011 Walkthrough by Andrew:
Thanks Mike for another nice variant; as Para said "a typical Mike Killer".

Para's walkthrough has a neat breakthrough in step 24. However I was concerned about step 23. I don't think that [475] is blocked by 17(3) at R4C5; it's easy to make mistakes when analysing overlapping cages. It should start
45 on R1234: 3 innies: R4C456 = 16 = [475/862/925]: ...
When I showed this to Ed, before asking him to add this post to the archive, he came up with the alternative start to step 24
Quote:
14(3)r4c4 and r9c4 cannot both have 8 -> must have 9 in 14(3)r4c4 (-> 6 in 14(3)r4c6): or must have 6 in r9c4: either way, no 6 in r9c6.
Many thanks Ed for "saving" Para's walkthrough with a slightly different way of expressing his step.

There was one thing which Para, Afmob and I had in common. Our solving paths all used a hidden single in either R9C4 or R9C6 as the key breakthrough but we used different ways to achieve the hidden single; therefore that was a narrow point for this puzzle.

Here is my walkthrough for A69 V1.5.

Prelims

a) R1C34 = {39/48/57}, no 1,2,6
b) R1C67 = {29/38/47/56}, no 1
c) R5C12 = {29/38/47/56}, no 1
d) R5C89 = {49/58/67}, no 1,2,3
e) R78C4 = {14/23}
f) R78C6 = {29/38/47/56}, no 1
g) R8C12 = {19/28/37/46}, no 5
h) R8C89 = {19/28/37/46}, no 5
i) R9C12 = {18/27/36/45}, no 9
j) R9C89 = {14/23}
k) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
l) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3

1. 45 rule on N8 2 innies R9C46 = 14 = {59/68}
1a. R78C6 = {29/38/47} (cannot be {56} which clashes with R9C46), no 5,6
1b. Hidden killer pair 5,6 in 15(3) cage and R9C46 for N8, R9C46 contains one of 5,6 -> 15(3) cage must contain one of 5,6 = {159/168/267/357} (cannot be {249/348} which don’t contain 5 or 6, cannot be {258/456} which clash with R9C46), no 4

2. 45 rule on C123 4 outies R1239C4 = 26 = {2789/3689/4589/4679/5678}, no 1

3. 45 rule on N7 2(1+1) outies R6C1 + R9C4 = 1 innie R7C3 + 1
3a. Min R6C1 + R9C4 = 6 -> min R7C3 = 5
3b. Max R6C1 + R9C4 = 10 -> max R6C1 = 5

4. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 1 innie R7C7 + 7
4a. Min R6C9 + R9C6 = 9 -> min R7C7 = 2

5. 45 rule on R89 3 outies R7C456 = 7 = {124}, locked for R7 and N8 -> R8C4 = 3, R7C4 = 2, R7C5 = 1, R7C6 = 4, R8C6 = 7, clean-up: no 9 in R1C3, no 4,7 in R1C7
5a. 1 in N2 only in R23C6, locked for C6

6. 11(3) cage at R6C1 = {137/236} (cannot be {128/146/245} because 1,2,4 only in R6C1), no 5,8
6a. 1,2 only in R6C1 -> R6C1 = {12}
6b. 11(3) cage at R6C1 = {137/236}, 3 locked for R7 and N7, clean-up: no 6 in R9C12

7. 14(3) cage at R4C6 = {239/356}, no 8, 3 locked for C6 and N5, clean-up: no 8 in R1C7

8. 3 in N2 only in 13(3) cage = {238/346}, no 5,7,9
8a. 7 in N2 only in R123C4, locked for C4

9. 15(3) cage in N8 (step 1b) = {159} (only remaining combination, cannot be {168} which clashes with 13(3) cage in N2), locked for C5 and N8

10. Naked pair {68} in R9C46, locked for R9, clean-up: no 1 in R9C12

11. Killer pair 2,4 in R9C12 and R9C89, locked for R9

12. 45 rule on R9 3 remaining innies R9C357 = 17 = {179/359}
12a. 3 of {359} must be in R9C7 -> no 5 in R9C7

13. 1 in N5 only in 14(3) cage at R4C4 = {149/158}, no 6

14. 16(3) cage at R8C3 = {169/178} (cannot be {259/457} because R9C4 only contains 6,8, cannot be {268} because no 2,6,8 in R9C3), no 2,4,5, 1 locked for C3 and N7, clean-up: no 9 in R8C12
14a. R9C4 = {68} -> no 6,8 in R8C3
14b. 9 in N7 only in R789C3, locked for C3

15. 16(3) cage at R8C7 = {169/178/358} (cannot be {259/349/457} because R9C6 only contains 6,8, cannot be {268} because no 2,6,8 in R9C7, cannot be {367} because 3,7 only in R9C7), no 2,4
15a. R9C6 = {68} -> no 6,8 in R8C7

16. Naked triple {159} in R8C357, locked for R8

17. R6C1 + R9C4 = R7C3 + 1 (step 3)
17a. Min R6C1 + R9C4 = 7 -> min R7C3 = 6

18. 5 in N7 only in R9C12 = {45}, locked for R9 and N7 -> R89C5 = [59], clean-up: no 6 in R8C12, no 1 in R9C89
18a. Naked pair {28} in R8C12, locked for R8 and N7
18b. Naked pair {46} in R8C89, locked for N9
18c. Naked pair {23} in R9C89, locked for N9

19. 1 in N9 only in R89C7, locked for C7
19a. R5C4 = 1 (hidden single in R5)

20. 21(3) cage at R6C9 = {489/579/678}
20a. 4,6 of {489/678} must be in R6C9 -> no 8 in R6C9
20b. 5 of {579} must be in R7C89 (R7C89 cannot be {79} which clashes with R89C7, ALS block) -> no 5 in R6C9
20c. Hidden killer pair 5,8 in R7C7 and 21(3) cage at R6C9 for R7, 21(3) cage must contain one of 5,8 in R7C89 -> R7C7 = {58}

21. 45 rule on R1234 2 innies R4C56 = 1 remaining outie R6C4 + 3, IOU no 3 in R4C6
21a. 45 rule on R1234 3 innies R4C456 = 16 = {259/268/457}
21b. 8 of {268} must be in R4C4 -> no 8 in R4C5
21c. 7 of {457} must be in R4C5 -> no 4 in R4C5
[Can probably make further candidate eliminations within N5, looking at the interactions between R4C456 and the three vertical cages but I’ll leave that until later when I hope that I’ll have been able to reduce the number of combinations in the cages in N5.]

22. 13(3) cage in N2 (step 8) = {238/346}
22a. 7 in C4 only in R123C4 -> R1239C4 (step 2) = {4679/5678}
22b. Consider placements for 4 in N2
4 in R123C4 => R1239C4 = {4679} = {479}6, no 6 in R23C4
or 4 in 13(3) cage = {346}, 6 locked for N2
-> no 6 in R23C4

23. R9C4 = 6 (hidden single in C4) -> R89C3 (step 14) = [91], R9C6 = 8, R89C7 = [17], clean-up: no 3 in R1C7

24. 21(3) cage at R6C9 = {489/579} (cannot be {678} because 6,7 only in R6C9), no 6
24a. 4,7 only in R6C9 -> R6C9 = {47}

25. 25(4) cage at R5C3 = {2689/3679/4579/4678} (cannot be {1789} because 1,9 only in R6C2, cannot be {3589} because R7C3 only contains 6,7), no 1
25a. 9 of {2689/3579} must be in R6C2 -> no 2,3 in R6C2

26. 45 rule on N47 3 remaining innies R4C123 = 14 = {149/167/239/347} (cannot be {158/248/257/356} which clash with R4C456), no 5

27. 45 rule on N69 3 remaining innies R4C789 = 15 = {159/168/348/357} (cannot be {249/258/267/456} which clash with R4C456), no 2

28. 2 in N6 only in 18(4) cage at R5C7 = {2358/2457} (cannot be {1269/2349/2367} because R7C7 only contains one of 5,8, cannot be {1278} because 1,7 only in R6C8, cannot be {2457} which clashes with R6C9), no 1,6,9, CPE no 5 in R4C7

29. R6C1 = 1 (hidden single in R6), 11(3) cage at R6C1 (step 6) = {137} (only remaining combination) -> R7C12 = {37}, locked for R7, R7C3 = 6

30. R7C3 = 6 -> 25(4) cage at R5C3 (step 25) = {2689/3679/4678}, no 5

31. 5 in N4 only in R5C12 = {56}, locked for R5 and N4, clean-up: no 7,8 in R5C89
31a. Naked pair {49} in R5C89, locked for R5 and N6 -> R6C9 = 7
31b. 21(3) cage at R6C9 (step 24) = {579} (only remaining combination), 5,9 locked for R7 -> R7C7 = 8

32. Naked triple {235} in R5C7 + R6C78, locked for N6, 5 also locked for R6 -> R4C7 = 6, clean-up: no 5 in R1C6
32a. Naked pair {18} in R4C89, locked for R4, CPE no 1,8 in R3C8
32b. Naked pair {23} in R5C67, locked for R5

33. 17(3) cage at R4C5 = {278} (only remaining combination, cannot be {467} because 4,6 only in R6C5), locked for C5 and N5 -> R5C6 = 3, R5C7 = 2, clean-up: no 9 in R1C6, no 5 in R4C4 (step 13)
33a. Naked pair {49} in R46C4, locked for C4 and N5 -> R4C6 = 5, R6C6 = 6, R1C6 = 2, R1C7 = 9, clean-up: no 3,8 in R1C3
33b. Naked pair {35} in R6C78, locked for R6

34. 4 in C7 only in R23C7, locked for N3

35. 19(4) cage at R3C6 contains 6 = {1369/1468}, no 5

36. 5 in C3 only in R123C3, locked for N1

37. 15(4) cage at R2C9 = {1257/1356}, no 8 -> R4C9 = 1, R4C8 = 8
37a. 7 of {1257} must be in R3C8 -> no 2 in R3C8

38. 19(4) cage at R3C6 (step 35) contains 6,8 = {1468} (only remaining combination) -> R3C67 = [14], R2C6 = 9

39. 15(4) cage at R2C9 (step 37) = {1257} (only remaining combination, cannot be {1356} which clashes with R2C7) -> R3C8 = 7, R23C9 = {25}, locked for C9 and N3 -> R2C7 = 3, R6C78 = [53], R7C89 = [59], R5C89 = [94], R8C89 = [46], R9C89 = [23], R1C9 = 8, clean-up: no 4 in R1C3
39a. Naked pair {57} in R1C34, locked for R1

40. 9 in R3 only in R3C12, locked for 21(4) cage at R2C1, no 9 in R4C1
40a. 9 in N4 only in R46C2, locked for C2

41. R3C1 = 9 (hidden single in R3) -> 21(4) cage at R2C1 = {2379/2469}, no 8
41a. 3 of {2379} must be in R3C2 (R24C1 cannot be [73] which clashes with R7C1) -> no 3 in R4C1

42. 3 in N4 only in R4C23, locked for 23(4) cage at R3C3, no 3 in R3C3
42a. 23(4) cage = {3578} (only remaining combination, cannot be {3479} because R3C4 only contains 5,8), no 2,4,9
42b. Naked pair {37} in R4C23, locked for R4 and N4 -> R5C3 = 8

and the rest is naked singles.


Rating Comment. I'll rate my walkthrough for A69 V1.5 at Easy 1.5. The forcing chain in step 22 is short and localised, almost combination analysis.


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