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Andrew
PostPosted: Thu Jul 14, 2011 5:11 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 123 "Eight T's" V2 by udosuk (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:5121:5121:4354:4611:4611:4611:5380:5125:5125:5121:4354:4358:4358:4611:4103:4103:5380:5125:4354:5640:4354:4358:4611:4103:5380:5385:5380:6666:5640:5640:4354:6411:5380:5385:5385:6156:6666:6666:6666:6411:6411:6411:6156:6156:6156:6666:2317:2317:7438:6411:8463:2064:2064:6156:7438:2317:7438:3601:8210:3347:8463:2064:8463:2580:7438:3601:3601:8210:3347:3347:8463:2581:2580:2580:7438:8210:8210:8210:8463:2581:2581:
Solution: 
+-------+-------+-------+
| 6 5 4 | 1 3 2 | 9 7 8 |
| 9 1 2 | 6 8 7 | 4 3 5 |
| 3 8 7 | 9 4 5 | 1 6 2 |
+-------+-------+-------+
| 4 9 5 | 2 1 6 | 7 8 3 |
| 1 6 8 | 7 5 3 | 2 4 9 |
| 7 2 3 | 4 9 8 | 5 1 6 |
+-------+-------+-------+
| 8 4 9 | 5 6 1 | 3 2 7 |
| 5 7 6 | 3 2 4 | 8 9 1 |
| 2 3 1 | 8 7 9 | 6 5 4 |
+-------+-------+-------+
Quote:
udosuk: Can't get to sleep so will post the v2 now (note it is a diagonal (X) puzzle):
SSR: 2.69. "True" SSR: 1.94.
The property as observed by Para still stands in this one, but I think the big guns don't really need it. So fire away guys! :alien:

Afmob: Wow! V2 was really strange since like Messy One 6 it opened up itself without much trouble and all of a sudden I couldn't progress without using ...
Rating: 2.0.

Andrew (in 2012): A123 V2 is another of the very hardest puzzles that I've tried to solve.
Thanks udosuk for an interesting puzzle, as well as a very challenging one! I loved the cage pattern.
As Afmob said, it starts fairly easily before becoming very difficult.
Congratulations Afmob for an excellent solving path and for managing to post your optimised walkthrough only a day and a half after the puzzle was posted!
I got stuck after my step 39 and only managed to make further progress after I had another look at this puzzle fairly recently.
Rating for my walkthrough: at least 2.5.

Walkthrough by Afmob:
Wow! V2 was really strange since like Messy One 6 it opened up itself without much trouble and all of a sudden I couldn't progress without using forcing chains.

It's also interesting that all of my chains didn't use the Killer property so they could have been applied to an X Sudoku with the same candidates left. The most difficult moves were step 10i and its symmetrical twin step 10j which were forcing chains that used forcing chains. :pallid:
JSudoku used a different move to get the same result: :bigoops: It used two Finned Swordfishs. One was 3 in C278 for R159 with fins being R28C28+R37C37 (all those cells belong to the diagonals). I think with 6 fins it's probably as hard as my move.

I could have used the symmetry technique here but since it requires uniqueness I refused to do so. :brickwall:

One thing about the rating, despite my rating of 2.0 it felt a lot easier than A74 Brickwall. If I had used Gurth's symmetrical present then I would have rated it my wt around 1.75.

A123 V2 Walkthrough:

1. N2468
a) Innies N2 = 27(4) <> 1,2
b) Outies N2 = 6(2) = {15/24}
c) Outies N4 = 12(2) = [75/84/93]
d) Outies N6 = 8(2) = [53/62/71]
e) Innies N8 = 13(4) <> 8,9
f) Outies N8 = 14(2) = {59/68}

2. R456
a) Innies R1234 = 8(3) = 1{25/34} -> 1 locked for R4
b) Outies R6789 = 22(3) = 9{58/67} -> 9 locked for R6
c) 17(5) must have 1 -> 1 locked for N1
d) 33(5) must have 9 -> 9 locked for N9

3. N28
a) Outies N2 = 6(2): R2C7 <> 5
b) 16(3): R23C6 <> 4 since R2C7 = (124)
c) Outies N8 = 14(2): R8C3 <> 5
d) 14(3): R78C4 <> 6 since R8C3 = (689)

4. N1379
a) Innies+Outies N7: -6 = R6C4 - (R7C2+R8C3): R6C4 <> 1,2 since R7C2+R8C3 >= 9
b) Innies+Outies N9: -2 = R6C6 - (R7C8+R8C7): R6C6 <> 3 since R7C8+R8C7 >= 6
c) Innies+Outies N1: -8 = R4C4 - (R2C3+R3C2): R4C4 <> 7 since R2C3+R3C2 <= 14
d) Innies+Outies N3: -4 = R4C6 - (R2C7+R3C8): R4C6 <> 8,9 since R2C7+R3C8 <= 11

5. R456
a) 1,2 in R6 locked in 9(3) and 8(3) and none of them can have two of (12) @ R6
-> Both cages must have one of (12) @ R6
b) 8(3): R6C78 <> 3 because 4 only possible @ R6
c) 8,9 in R4 locked in 22(3) + 21(3) and none of them can have two of (89) @ R4
-> Both cages must have one of (89) @ R4
d) 22(3): R4C23 <> 7 because 6 only possible there
e) Outies N89 = 20(3+1): R6C6 <> 5 since R6C78 @ 8(3) cannot be {24}
f) Outies N12 = 20(3+1): R4C4 <> 5 since R4C23 @ 22(3) cannot be {68}

6. N1 + R456 !
a) ! 20(3) = 9{38/47/56} because {578} blocked by Killer pair (57) of 17(5)
-> 9 locked for N1
b) Outies N4 = 12(2) <> 3
c) 22(3) = 9{58/67} -> 9 locked for R4+N4
d) 9(3) = 3{15/24} -> 3 locked for R6+N4; R6C23 <> 4,5
e) 26(5) = 478{16/25} -> 8 locked for N4
f) 21(3) = {678} -> 8 locked for N6
g) Outies N6 = 8(2) <> 3
h) 8(3) = {125} -> 5 locked for R6+N6
i) Outies R6789 = 22(3) = {679} locked for R6
j) Naked pair (48) locked in R6C46 for N5

7. N9 + R456 !
a) 33(5) must have 7 -> 7 locked for N9
b) ! 10(3) = 1{36/45} because {35} is a Killer pair of 33(5) -> 1 locked for N9
c) R7C8 = 2
d) Outie N6 = R3C8 = 6
e) 8(3) = {125} -> 1 locked for R6+N6
f) 9(3) = {234} -> R7C2 = 4; 2 locked for N4
g) Outie N4 = R3C2 = 8
h) 22(3) = {589} -> 5 locked for R4+N4
i) Outies R1234 = 8(3) = {134} locked for R4

8. R789
a) Killer pair (56) locked in 33(5) + 10(3) for N9
b) R8C7 = 8, R4C7 = 7, R4C8 = 8
c) Outie N8 = R8C3 = 6
d) Outie N7 = R6C4 = 4
e) R6C6 = 8
f) 13(3) = [14/32]8
g) 14(3) = 6{17/35}
h) Killer pair (13) locked in 14(3) + R7C6 for N8

9. R123
a) Innies+Outies N3: 2 = R4C6 - R2C7 -> R4C6 = 6, R2C7 = 4
b) Outie N2 = R2C3 = 2
c) 17(3) = 2[69/87]
d) R4C4 = 2, R6C3 = 3, R6C2 = 2
e) 17(5) must have 3 -> 3 locked for N1
f) Killer pair (79) locked in R3C4 + 16(3) for N2

10. D\/ !
a) 1 in R2 locked in R2C258 -> CPE: R5C5 <> 1
b) 9 in R8 locked in R8C258 -> CPE: R5C5 <> 9
c) 17(5): R2C2+R3C1 <> 5 since (36) only possible there
d) 33(5): R7C9+R8C8 <> 5 since (47) only possible there
e) 20(3) @ N1: R1C1 <> 7 because 4 only possible there
f) 10(3) @ N9: R9C9 <> 3 because 6 only possible there
g) Consider placement of 7 in D\ -> R2C8 <> 7
- i) 7 in R2C2, R5C5 or R8C8 -> CPE: R2C8 <> 7
- ii) 7 in R3C3 -> R2C6 = 7 (HS @ N2)
h) Consider placement of 3 in D\ -> R8C2 <> 3
- i) 3 in R2C2, R5C5 or R8C8 -> CPE: R8C2 <> 3
- ii) 3 in R7C7 -> R8C4 = 3 (HS @ N8)

i) ! Consider placement of 3 in C2 -> R5C5 <> 3
- i) 3 in R2C2
- ii) 3 in R9C2 -> 3 in N9:
- ii) a) Either 3 in R7C7+R8C8 locked for D\ or
- ii) b) 3 in R78C9 locked for C9 -> 3 locked R5C78 @ N6 for R5
j) ! Consider placement of 7 in C8 -> R5C5 <> 7
- i) 7 in R8C8
- ii) 7 in R1C8 -> 7 in N1:
- ii) a) Either 7 in R2C2+R3C3 locked for D\ or
- ii) b) 7 in R23C1 locked for C1 -> 7 locked in R5C23 @ N4 for R5

11. D\/ !
a) R5C5 = 5
b) 21(5): R3C9 <> 5 since 7 only possible there
c) 29(5): R7C1 <> 5 since 3 only possible there
d) Hidden Single: R3C6 = 5 @ R3 -> R2C6 = 7
e) R3C4 = 9 -> R2C4 = 6
f) Hidden Single: R7C4 = 5 @ R7 -> R8C4 = 3
g) R7C6 = 1 -> R8C4 = 4
h) 33(5) = {36789} -> 3,6 locked for N9
i) 17(5) = {12347} -> 4,7 locked for N1

12. R456 + D\/
a) 24(5): R5C79 <> 3 since R5C789 = {239} blocked by R5C6 = (39)
b) ! Consider placement of 6 in R6 -> R5C7 <> 6
- i) R6C1 = 6 -> R7C7 = 6 (HS @ D\)
- ii) R6C9 = 6
c) ! Consider placement of 2 in D/ -> R5C6 <> 9
- i) R3C7 = 2 -> R5C7 = 9
- ii) R9C1 = 2 -> R9C6 = 9
d) R5C6 = 3, R4C5 = 1, R5C4 = 7, R6C5 = 9
e) R4C1 = 4, R4C9 = 3, R6C9 = 6, R6C1 = 7
f) R1C6 = 2, R9C6 = 9, R9C4 = 8, R1C4 = 1
g) 33(5) = {36789} -> 3 locked for C7

13. R123
a) Naked triple (569) locked in R1C127 for R1
b) 20(3) @ N3 = 8{39/57} -> R2C9 = (59)
c) 7 locked in R13C3 for C3
d) Hidden Single: R1C9 = 8 @ N3

14. Rest is singles without considering diagonals.

Rating: 2.0. I used lots of forcing chains and two were chains in chains.
Walkthrough by Andrew (finished in 2012):
A123 V2 is another of the very hardest puzzles that I've tried to solve.

Thanks udosuk for an interesting puzzle, as well as a very challenging one! I loved the cage pattern.

As Afmob said, it starts fairly easily before becoming very difficult.

Congratulations Afmob for an excellent solving path and for managing to post your optimised walkthrough only a day and a half after the puzzle was posted!

I got stuck after my step 39 and only managed to make further progress after I had another look at this puzzle fairly recently.

Afmob wrote:
The most difficult moves were ... and its symmetrical twin ... which were forcing chains that used forcing chains.
I wish I'd spotted those particular forcing chains; in that case I might not be posting my walkthrough, although our earlier steps were also different. After reaching the position where I got stuck, I spent a long time after returning to this puzzle looking at forcing chains based on the values in R4C19 and R6C19 but was unable to make any progress. Then I changed to multi-level forcing chains based on combinations in the four corner cages and their effect on other corner cages. These were successful although the first and hardest of these can be considered to be 4 levels deep! My walkthrough has lots of forcing chains but only a small amount of combination analysis and I was able to avoid using any contradiction moves.

Here is my walkthrough for A123 V2

The note about rotational symmetry for A123 also applies for this puzzle. Again I’ll try to solve it without using that feature, which I consider to be something like UR and not solving the whole puzzle.

However knowledge of the features of rotational symmetry was helpful in the later steps. Once I’d achieved something in one part of the grid I knew that I needed to look for a complementary step on the other side of the grid, although I made a point of never using rotational symmetry to achieve this.

Prelims

a) 20(3) cage in N1 = {389/479/569/578}, no 1,2
b) 20(3) cage in N3 = {389/479/569/578}, no 1,2
c) 22(3) cage at R3C2 = {589/679}
d) 21(3) cage at R3C8 = {489/579/678}, no 1,2,3
e) 9(3) cage at R6C2 = {126/135/234}, no 7,8,9
f) 8(3) cage at R6C7 = {125/134}
g) 10(3) cage in N7 = {127/136/145/235}, no 8,9
h) 10(3) cage in N9 = {127/136/145/235}, no 8,9
i) 17(5) cage at R1C3 = {12347/12356}, no 8,9
j) 18(5) cage in N2 = {12348/12357/12456}, no 9
k) 33(5) cage at R6C6 = {36789/45789}, no 1,2
l) 32(5) cage in N8 = {26789/35789/45689}, no 1

Steps resulting from Prelims
1a. 22(3) cage at R3C2 = {589/679}, CPE no 9 in R5C2
1b. 8(3) cage at R6C7 = {125/134}, CPE no 1 in R5C8
1c. 18(5) cage in N2 = {12348/12357/12456}, 1,2 locked for N2
1d. 32(5) cage in N8 = {26789/35789/45689}, 8,9 locked for N8

2. 45 rule on R1234 3 innies R4C159 = 8 = {125/134}, 1 locked for R4
2a. 17(5) cage at R1C3 = {12347/12356}, 1 locked for N1

3. 45 rule on R6789 3 innies R6C159 = 22 = {589/679}, 9 locked for R6
3a. 33(5) cage at R6C6 = {36789/45789}, 9 locked for N9

4. 45 rule on N2 2 outies R2C37 = 6 = [24/42/51]

5. 45 rule on N4 2 outies R37C2 = 12 = [75/84/93]
5a. 9(3) cage at R6C2 = {135/234} (cannot be {126} because R7C2 only contains 3,4,5), no 6, CPE no 3 in R5C2

6. 45 rule on N6 2 outies R37C8 = 8 = [53/62/71]
6a. 21(3) cage at R3C8 = {579/678} (cannot be {489} because R3C8 only contains 5,6,7), no 4, CPE no 7 in R5C8

7. 45 rule on N8 2 outies R8C37 = 14 = [68/86/95]

8. 45 rule on R123 2 innies R3C28 = 2 outies R4C46 + 6
8a. Max R3C28 = 16 -> max R4C46 = 10, no 9 in R4C6

9. 45 rule on R789 2 outies R6C46 = 2 innies R7C28 + 6
9a. Min R7C28 = 4 -> min R6C46 = 10, no 1 in R6C4

10. 16(3) cage at R2C6 = {169/178/259/268/349/457} (cannot be {358/367} because R2C7 only contains 1,2,4)
10a. 4 of {349/457} must be in R2C7 -> no 4 in R23C6

11. 45 rule on N1 2 innies R2C3 + R3C2 = 1 outie R4C4 + 8
11a. Max R2C3 + R3C2 = 13 (cannot be [59] which clashes with 20(3) cage) -> max R4C4 = 5

12. 45 rule on N3 2 innies R2C7 + R3C8 = 1 outie R4C6 + 4
12a. Max R2C7 + R3C8 = 11 -> max R4C6 = 7

13. 45 rule on N7 2 innies R7C2 + R8C3 = 1 outie R6C4 + 6
13a. Min R7C2 + R8C3 = 9 -> min R6C4 = 3

14. 45 rule on N9 2 innies R7C8 + R8C7 = 1 outie R6C6 + 2
14a. Min R7C8 + R8C7 = 7 (cannot be [15] which clashes with 10(3) cage) -> min R6C6 = 5

15. 22(3) cage at R3C2 = {589/679}
15a. 7 of {679} must be in R3C2 -> no 7 in R4C23
15b. 21(3) cage at R3C8 (step 6a) = {579/678}
15c. 5 of {579} must be in R3C8 (R4C78 cannot be {59} which clashes with 22(3) cage at R3C2), no 5 in R4C78

16. 45 rule on N2 4 innies R23C46 = 27 = {3789/4689/5679}
16a. 17(3) cage at R2C3 = {269/278/359/458/467} (cannot be {368} because R2C3 only contains 2,4,5)
16b. 5 of {359} must be in R2C3, 5 of {458} must be in R2C3 (R23C4 cannot be {58} because R23C46 only contains one of 5,8) -> no 5 in R23C4

17. 14(3) cage at R7C4 = {149/158/167/239/248/356} (cannot be {257/347} because R8C3 only contains 6,8,9)
17a. 6 of {167/356} must be in R8C3 -> no 6 in R78C4

18. 45 rule on N8 4 innies R78C46 = 13 = {1237/1246/1345}
18a. 13(3) cage at R7C6 = {148/157/238/256/346} (cannot be {247} because R8C7 only contains 5,6,8)
18b. 5 of {157} must be in R8C7, 5 of {256} must be in R8C7 (R78C6 cannot be {25} because R78C46 only contains one of 2,5), no 5 in R78C6

19. R2C3 + R3C2 = R4C4 + 8 (step 11)
19a. Consider placements for R4C4 = {2345}
R4C4 = 2 => R2C3 = 2 (hidden single in N1) => R3C2 = 8 => no 8 in R4C23
or R4C4 = 3 => 3 in N1 only in 20(3) cage = {389}, locked for N1 => R3C2 = 7, R4C23 = {69} => no 8 in R4C23
or R4C4 = 4 => R2C3 + R3C2 = 12 = [48/57] => R4C23 = {59/69} => no 8 in R4C23
or R4C4 = 5 => 22(3) cage at R3C2 cannot be {589} => no 8 in R4C23
-> no 8 in R4C23
[An alternative way looks like 45 rule on N1 3 outies R4C234 = 1 innie R2C3 + 14, but that still needs interactions with the cages in N1.]

20. 8 in R4 only in R4C78, locked for N6
20a. 21(3) cage at R3C8 (step 6a) contains 8 = {678} (only remaining combination), no 5,9, CPE no 6 in R5C8, clean-up: no 3 in R7C8 (step 6)

21. 9 in R4 only in R4C23, locked for N4 and 22(3) cage at R3C2, no 9 in R3C2, clean-up: no 3 in R7C2 (step 5)

22. 9(3) cage at R6C2 (step 5a) = {135/234}, 3 locked for R6 and N4
22a. R7C2 = {45} -> no 4,5 in R7C23

23. 8(3) cage at R6C7 = {125} (only remaining combination), no 4, 5 locked for R6 and N6, CPE no 2 in R5C8

24. R6C159 (step 3) = {679} (only remaining combination), locked for R6 -> R6C6 = 8, placed for D\, R6C4 = 4, placed for D/

25. 45 rule on N5 2 remaining innies R4C46 = 8 = {26} (cannot be {35} which clashes with R4C159) -> R4C4 = 2, placed for D\, R4C6 = 6, placed for D/

26. Naked pair {59} in R4C23, locked for R4 and N4, R3C2 = 8 (cage sum), R7C2 = 4 (step 5)

27. Naked pair {78} in R4C78, locked for N6 and 21(3) cage at R3C8 -> R3C8 = 6, R7C8 = 2 (step 6)
27a. Naked pair {15} in R6C78, locked for R6 and N6

28. R2C3 = 2 (hidden single in N1), R2C7 = 4 (step 4), R6C23 = [23]
28a. R2C3 = 2 -> R23C4 = 15 = [69/87]
28b. R2C7 = 4 -> R23C6 = 12 = {39/57}
28c. Killer pair 7,9 in R3C4 and R23C6, locked for N2

29. R8C7 = 8 (hidden single in N9), R4C78 = [78], R8C3 = 6 (step 7)
29a. R8C3 = 6 -> R78C4 = 8 = {17/35}
29b. R8C7 = 8 -> R78C6 = 5 = [14/32]
29c. Killer pair 1,3 in R78C4 and R7C6, locked for N8

30. 20(3) cage in N1 = {479/569}, no 3
30a. 4 of {479} must be in R1C1 -> no 7 in R1C1

31. 10(3) cage in N9 = {136/145}, no 7
31a. 6 of {136} must be in R9C9 -> no 3 in R9C9

32. 17(5) cage at R1C3 = {12347/12356}
32a. 1,5 of {12356} must be in R13C3 -> no 5 in R2C2 + R3C1

33. 33(5) cage at R6C6 = {36789/45789}
33a. 5,9 of {45789} must be in R79C7 -> no 5 in R7C9 + R8C8

34. 20(3) cage in N1 = {479/569}
34a. 4 of {479} must be in R1C1 => R2C2 = 6 (hidden single in N1), R4C1 = 1 => R5C2 = 7 -> no 7 in R1C2

35. 10(3) cage in N9 (step 31) = {136/145}
35a. 6 of {136} must be in R9C9 => R8C8 = 4 (hidden single in N9), R6C9 = 9 => R5C8 = 3 -> no 3 in R9C8

36. 20(3) cage in N3 = {389/578}
36a. 20(3) cage = {578}, locked for N3 or {389} => R1C3 = 7 (hidden single in R1) => 7 on D\ only in R5C5 + R8C8, CPE no 7 in R2C8
-> no 7 in R2C8
36b. 20(3) cage = {578}, locked for N3 or {389} => R1C3 = 7 (hidden single in R1) => R2C6 = 7 (hidden single in R2) => R3C6 = 5 (step 28b) => no 5 in R3C79
-> no 5 in R3C79

37. 10(3) cage in N7 = {127/235}
37a. 10(3) cage = {235}, locked for N7 or 10(3) cage = {127} => R9C7 = 3 (hidden single in R9) => 3 on D\ only in R2C2 + R5C5, CPE no 3 in R8C2
-> no 3 in R8C2
37b. 10(3) cage = {235}, locked for N7 or 10(3) cage = {127} => R9C7 = 3 (hidden single in R9) => R8C4 = 3 (hidden single in R8), R7C4 = 5 (step 29a)
-> no 5 in R7C13

38. R23C4 (step 28a) = [69/87]
38a. 45 rule on C1234 3 innies R159C4 = 16 = {169/178/358} (cannot be {367} which clashes with R23C4)
38b. R23C46 (step 16) = {3789/5679}
38c. 5 of {358} must be in R59C4 (R159C4 cannot be [538] which clashes with R23C46), no 5 in R1C4

39. R78C6 (step 29b) = [14/32]
39a. 45 rule on C6789 3 innies R159C6 = 14 = {149/239/257} (cannot be {347} which clashes with R78C6)
39b. R78C46 (step 18) = {1237/1345}
39c. 5 of {257} must be in R15C6 (R159C6 cannot be [275] which clashes with R78C46), no 5 in R9C6

[I was stuck at this position for a very, very long time, even after I came back to this puzzle again this month. I then found a long contradiction move
10(3) cage in N9 (step 31) = {136/145} cannot be {136}, here’s how
10(3) cage = {136} = [316] => R8C8 = 4 (hidden single in N9), placed for D\, R8C6 = 2, R9C1 = 2 (hidden single in C1), R4C9 = 4, R4C1 = 1, R6C1 = 6 (hidden single in R6), 20(3) cage in N1 = {569} (only remaining combination), R12C1 = {59}, locked for C1 => no remaining combinations for 10(3) cage in N7 because {127} is blocked by R4C1 = 1, R9C8 = 1 and {235} is blocked by R12C1 = {59}, R8C9 = 3 and R9C1 = 2
-> 10(3) cage in N9 = {145}, locked for N9
However I wasn’t satisfied with using this since it felt too much like bifurcation.

Then after persevering with multi-level forcing chains, at first trying chains based on the values in R4C1 and R6C1, I then looked at chains based on the corner cages and eventually found…]
40. Consider combinations for 20(3) cage in N1 = {479/569} and their effect on 10(3) cage in N9 = [316]/{145}
20(3) cage = {479} => R2C2 = 6 (hidden single in N1), locked for D\ blocks 10(3) cage in N9 = [316] => 10(3) cage = {145}
or 20(3) cage = {569} with 6 in R12C1 => R6C9 = 6 (hidden single in R6) blocks 10(3) cage in N9 = [316] => 10(3) cage in N9 = {145}
or 20(3) cage = {569} with 6 in R1C2, R12C1 = {59}, locked for C1,
then R4C1 = 4, R4C9 = 3 and/or R6C1 = 7, R6C9 = 6 (hidden single in R6) both block 10(3) cage in N9 = [316] => 10(3) cage = {145}
or R46C1 = [16] after which
either 10(3) cage in N7 = {127} => R9C2 = 1 blocks 10(3) cage in N9 = [316] => 10(3) cage = {145}
or 10(3) cage in N7 = {235} = {23}5
now R8C1 = 2 => R8C6 = 4 => 4 in N9 only in 10(3) cage = {145}
or R9C1 = 2, R8C1 = 3 blocks 10(3) cage in N9 = [316] => 10(3) cage = {145}
-> 10(3) cage in N9 = {145}, locked for N9

41. 5 in R7 only in R7C45, clean-up: no 3 in R7C4 (step 29a)

42. Consider placements for 10(3) cage in N9 = {145} and their effect on 20(3) cage in N1 = [497]/{569}
4 in 10(3) cage in R89C9 => R4C9 = 3, R4C1 = 4 (hidden single in R4) blocks 20(3) cage = [497] => 20(3) cage = {569}
or 4 in 10(3) cage in R9C8, R89C9 = {15}, locked for C9
then
either 20(3) cage in N3 = {389} => R1C3 = 7 (hidden single in R1) blocks 20(3) cage in N1 = [497] => 20(3) cage = {569}
or 20(3) cage in N3 = {578} = 5{78}
now R1C9 = 7, R2C9 = 8, R2C4 = 6 => 6 in N1 only in 20(3) cage = {569}
or R2C9 = 7 blocks 20(3) cage in N1 = [497] => 20(3) cage = {569}
-> 20(3) cage in N1 = {569}, locked for N1

43. 5 in R3 only in R3C56, locked for N2, clean-up: no 7 in R3C6 (step 28b)

44. Consider placements for 6 on D\
R1C1 = 6 => R6C1 = 7, R6C9 = 6 (hidden single in R6), no 6 in R7C9
or R7C7 = 6, no 6 in R7C9
-> no 6 in R7C9
44a. 6 in N9 only in R79C7, locked for C7

45. Consider placements for 4 on D\
R3C3 = 4, no 4 in R3C1
or R9C9 = 4, R4C9 = 3, R4C1 = 4 (hidden single in R4), no 4 in R3C1
-> no 4 in R3C1
45a. 4 in N1 only in R13C3, locked for C3

46. 5 of D\ only in R1C1 + R5C5 + R9C9, CPE no 5 in R1C9 + R9C1

47. R159C4 (step 38a) = {169/178/358}
47a. 5 of {358} must be in R5C4 -> no 3 in R5C4

48. R159C6 (step 39a) = {149/239/257}
48a. 5 of {257} must be in R5C6 -> no 7 in R5C6

49. Deleted.

50. Deleted.

[Steps 51 and 53 were partly re-worked after deleting incorrect steps 49 and 50.]
51. Consider combinations for 20(3) cage in N3 = {389/578} and their effect on 10(3) cage in N7 = {127/235}
20(3) cage = 3{89}, 9 locked for C9, R8C8 = 7 (hidden single in C8), R7C9 = 3 => 3 in N7 only in 10(3) cage = {235}
or 20(3) cage = 9{38}, 3 locked for C9, R4C9 = 4, R4C1 = 1, R1C3 = 7 (hidden single in R1), R3C1 = 3, R9C2 = 3 (hidden single in N7) => 10(3) cage in N7 = {235}
or 20(3) cage = [578], R8C8 = 7 (hidden single in C8), placed for D\, R3C1 = 7 (hidden single in N1), R6C1 = 6, R6C9 = 9, R7C9 = 3 => 3 in N7 only in 10(3) cage = {235}
or 20(3) cage = [587], R8C8 = 7 (hidden single in C8), R9C6 = 7 (hidden single in C6) => 10(3) cage in N7 = {235} (only remaining combination)
or 20(3) cage = [785], R9C8 = 5 (hidden single in N9), R6C78 = [51], R28C8 = {39}, CPE no 3 in R2C2 using D\, R9C2 = 3 (hidden single in C2) => 10(3) cage in N7 = {235}
-> 10(3) cage in N7 = {235}, locked for N7

52. R4C3 = 5 (hidden single in C3), R4C2 = 9
52a. 9 in N1 only in R12C1, locked for C1

53. Consider permutations for 10(3) cage in N7 = {235} and their effect on 20(3) cage in N3 = {389/578}
10(3) cage = [235], R2C2 = 3 (hidden single in C2), placed for D\, R7C9 = 3 (hidden single in N9), R4C9 = 4, R4C1 = 1, R3C1 = 7 => 7 in N3 only in 20(3) cage = {578}
or 10(3) cage = [325], R2C2 = 3 (hidden single in C2), R1C4 = 3 (hidden single in C4) => 20(3) cage = {578} (only remaining combination)
or 10(3) cage = [523], naked pair {17} in R28C2, CPE no 7 in R8C8 using D\ => R1C8 = 7 (hidden single in C8) => 20(3) cage = {578}
-> 20(3) cage in N3 = {578}, locked for N3

54. R6C7 = 5 (hidden single in C7), R6C8 = 1
54a. 1 in N9 only in R89C9, locked for C9

55. 1,7 in C2 only in R258C2, CPE no 1,7 in R5C5 using both diagonals
55a. 3,9 in C8 only in R258C8, CPE no 3,9 in R5C5 using both diagonals -> R5C5 = 5, placed for D\

56. R7C4 = 5 (hidden single in C4), R8C4 = 3 (step 29a), R7C6 = 1, R8C6 = 4 (step 29b)

57. R3C6 = 5 (hidden single in C6), R2C6 = 7 (step 28b), R3C4 = 9, R2C4 = 6 (step 28a)

58. 7 in N3 only in R1C89, locked for R1
58a. 3 in N7 only in R9C12, locked for R9

59. Hidden killer pair 7,8 in R1C9 and R7C3 + R8C2 for D/, R1C9 = {78} -> R7C3 + R8C2 must contain one of 7,8
59a. Killer pair 7,8 in R7C1 and R7C3 + R8C2, locked for N7
59b. 7 in R9 only in R9C45, locked for N8
59c. Naked pair {29} in R8C5 + R9C6, locked for N8

60. Hidden killer pair 2,3 in R2C8 + R3C7 and R9C1 for D/, R9C1 = {23} -> R2C8 + R3C7 must contain one of 2,3
60a. Killer pair 2,3 in R2C8 + R3C7 and R3C9, locked for N3
60b. 3 in R1 only in R1C56, locked for N2
60c. Naked pair {18} in R1C4 + R2C5, locked for N2

61. 9 in R1 only in R1C17, CPE no 9 in R7C7 using D\
61a. 1 in R9 only in R9C39, CPE no 1 in R3C3 using D\

62. Hidden killer pair 6,9 in R1C1 and R7C7 + R8C8 for D\, R1C1 = {69} -> R7C7 + R8C8 must contain one of 6,9
62a. Killer pair 6,9 in R7C7 + R8C8 and R9C7, locked for N9

[And for completeness before looking at hidden singles …]
63. Hidden killer pair 1,4 in R2C2 + R3C3 and R9C9 for D\, R9C9 = {14} -> R2C2 + R3C3 must contain one of 1,4
63a. Killer pair 1,4 in R1C3 and R2C2 + R3C3, locked for N1

64. R7C3 = 9 (hidden single in R7)

and the rest is naked singles, without using the diagonals.

Rating Comment: The highest rating comment I've ever given is "at least 2.5". The way I solved A123 V2 was at a similar level of difficulty, although it didn't include any contradiction moves, so I'll rate my walkthrough for A123 V2 at least 2.5. Using rotational symmetry to fix R5C5 = 5 wouldn't have changed my rating, since my solving path was based on reducing the four corner cages to single combinations. However after reducing those cages in N9 and N7 to single combinations, rotational symmetry could have been used as "shortcuts" to reduce the complementary cages in N1 and N3 to single combinations.


Last edited by Andrew on Thu Jan 26, 2012 4:52 am, edited 1 time in total.

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Andrew
PostPosted: Thu Jul 14, 2011 5:29 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Old SSv3.2.1 scores: 
Score = SudokuSolver v3.2.1 Score, rounded to nearest 0.05
E = Easy
H = Hard
In these tables, Rating is the lowest of the ratings given by Afmob,
Andrew and estimates for puzzles by Ed, Frank and Para

+--------------------------+--------------------------+--------------------------+
| Puzzle    Rating   Score | Puzzle    Rating   Score | Puzzle    Rating   Score |
+--------------------------+--------------------------+--------------------------+
| MO#6      H1.00     0.95 | A124V2     1.75     2.60 | A125V0     0.75     0.70 |
| A124       1.00     1.40 | A125      H1.00     1.10 | A125V2     1.25     1.60 |
+--------------------------+--------------------------+--------------------------+

Puzzle rating table, with links to archive entries; each of these has a link to the puzzle thread.

Abbreviations used in Rating Table on this page:
Est = Estimated rating by puzzle maker
E = Easy
H = Hard
Score = SudokuSolver v3.3.0 score, rounded to nearest 0.05
! indicates that the Score has changed at least 0.10 from the SS v3.2.1 score
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Puzzle                 |  Made By  |  Est  | Afmob | Andrew|     Other Raters      | Score |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Messy One #6X          |    Ed     |       | H1.00 |       |                       |  1.00 |
| Assassin 124           |   Frank   |       |  1.00 | H1.00 |                       | !1.10 |
| Assassin 124V2         |   Frank   |       | H1.75 |  1.75 |                       | !2.95 |
| Assassin 125           |   Para    |  1.00 | H1.00 | E1.25 |                       |  1.05 |
| Assassin 125V0         |   Para    |  0.75 |       |  0.75 |                       |  0.70 |
| Assassin 125V2         |   Para    |  1.50 |  1.25 |  1.50 |                       |  1.55 |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+


Some of the selected quotes in the puzzle entries have been edited to remove "spoilers"; the full rating comments are included with the walkthroughs. In some cases the puzzle makers gave hints; these are included in tiny text in the selected quotes.


Last edited by Andrew on Thu Aug 04, 2011 1:21 am, edited 2 times in total.

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Andrew
PostPosted: Sat Jul 16, 2011 3:17 am 
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Posts: 1895
Location: Lethbridge, Alberta, Canada
Messy One #6X by Ed (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:5889:3586:3586:3843:3843:6660:6660:3077:5638:5889:5127:3586:5127:3843:6660:3077:3077:5638:5889:5127:5127:5127:6660:6660:4872:4872:5638:5385:5385:5385:5130:5130:4872:4872:3339:5638:3084:5385:5130:5130:2829:5390:5391:3339:3339:3084:3084:2320:2829:2829:5390:5391:5391:5391:3089:3602:2320:10515:5908:5390:5908:4629:4629:3089:3602:4630:10515:5908:5908:5908:4629:4629:4630:4630:4630:10515:10515:10515:10515:10515:10515:
Solution: 
+-------+-------+-------+
| 6 4 7 | 5 9 3 | 8 1 2 |
| 9 2 3 | 8 1 6 | 4 7 5 |
| 8 1 5 | 4 7 2 | 9 3 6 |
+-------+-------+-------+
| 3 7 6 | 1 8 5 | 2 4 9 |
| 2 5 4 | 7 3 9 | 6 8 1 |
| 1 9 8 | 6 2 4 | 3 5 7 |
+-------+-------+-------+
| 5 6 1 | 9 4 8 | 7 2 3 |
| 7 8 2 | 3 6 1 | 5 9 4 |
| 4 3 9 | 2 5 7 | 1 6 8 |
+-------+-------+-------+
Quote:
Ed: This Messy One gets an SSscore of only 0.96. Sure felt a lot harder than that..but...some early placements make it irresistible and very enjoyable.
Just a reminder that Messy One's series 2 finishes at the end of November, so, any more killer creators want to get their entry in before the next poll? About 1 month to go. Just a reminder, must get a SudokuSolver score from 0.95 - 1.05 and be messy, ie, NOT symmetrical. A thread showing how to make a killer is here.

Afmob: Nice Killer, Ed! It starts really easy but suddenly it gets tougher. It took me some time to find ...
Rating: 1.0 - (Hard) 1.0.

Walkthrough by Afmob:
Nice Killer, Ed! It starts really easy but suddenly it gets tougher.

It took me some time to find step 4i which I probably didn't see because you had to use the placements. After that, step 5a followed easily.

Messy One #6 Walkthrough:

1. First placements
a) Innie N7 = R7C3 = 1
b) Outie N7 = R6C3 = 8
c) Innie N4 = R5C3 = 4
d) Outie N12 = R1C7 = 8
e) Innie N89 = R7C6 = 8
f) Innie N5 = R4C6 = 5

2. C123
a) 23(3) = {689} locked for C1+N1
b) 12(2) = {57} locked for C1+N7
c) 14(2) = {68} -> R7C2 = 6, R8C2 = 8
d) Hidden Single: R9C1 = 4 @ C1, R4C3 = 6 @ C3
e) 21(4) = 56{19/37} -> R5C2 = 5; R4C2 = (79)
f) Hidden pair (79) in R46C2 @ N4 locked for C2
g) Innies N1 = 8(3) = 1{25/34} -> 1 locked for N1+20(5)
h) 14(3): R1C2 <> 3 because 4 only possible there
i) Innies N1 = 8(3): R3C3 <> 2 since R23C2 <> 5
j) Innies N1 = 8(3): R23C2 <> 3 since R3C3 <> 1,4
k) Hidden Single: R9C2 = 3 @ C2

3. C456
a) 21(3) = {489} because {678} blocked by Killer pair (67) of 11(3)
-> R5C6 = 9, R6C6 = 4
b) 41(8) must have 3 -> 3 locked for C4+N8
c) 11(3) = 3{17/26} -> 3 locked for C5
d) Outies N1 = 12(2) = {48/57}
e) 20(5) = 145{28/37} -> CPE: R3C5 <> 4,5
f) 26(5) = 268{19/37} -> 2,6 locked for N2
g) 15(3) = {159} locked for N2
h) Outies N1 = 12(2) = {48} locked for C4+20(5)
i) 20(5) = {12458} -> R3C3 = 5
j) Hidden Single: R4C5 = 8 @ C5

4. R456 + D\ !
a) Hidden Single: R9C9 = 8 @ D\, R5C8 = 8 @ N6
b) 13(3) = 8{14/23}: R4C8 <> 1
c) Innies N6 = 11(2) = {29/47}
d) Killer pair (79) locked in R4C2 + Innies N6 for R4
e) 20(4) = 48{17/26} -> R5C4 = (67)
f) Naked pair (12) locked in R2C2+R4C4 for D\
g) 11(3): R56C5 <> 7 because R6C4 <> 1,3
h) 5,6 locked in 21(4) @ N6 = 56{19/37}
i) ! Outies R6 = 11(3) = 3{17/26}: R5C1 <> 3 since R5C7 <> 2
-> 3 locked for R5; CPE: R37C7 <> 3
j) 19(4) = 5{149/167/239/347} -> R3C8 = (13)

5. R456 + C7 !
a) ! Killer pair (79) locked in 19(4) + R7C7 for C7
b) Outies R6 = 11(3) = {236} -> R5C1 = 2; 6 locked for R5, CPE: R3C7 <> 6
c) R5C4 = 7 -> R4C4 = 1, R5C9 = 1 -> R4C8 = 4, R4C1 = 3 -> R4C2 = 7
d) 19(4) = {2359} -> R3C8 = 3; {29} locked for C7
e) 22(4) = 29{47/56} because R4C9 = (29) -> 2,9 locked for C9
f) Killer pair (29) locked in 22(4)+R3C7 for N3
g) Killer pair (67) locked in 22(4)+R2C8 for N3

6. R1+N9
a) Naked triple (159) locked in R1C458 for R1
b) R1C1 = 6, R7C7 = 7, R8C8 = 9, R8C3 = 2
c) 18(4) = {2349} -> R7C8 = 2; {34} locked for C9+N9
d) 23(5) = {14567} -> CPE: R8C4 <> 5,6

7. Rest is singles without considering diagonals.

Rating: 1.0 - (Hard) 1.0. I used CPE and a Killer pair.
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Andrew
PostPosted: Sat Jul 16, 2011 3:46 am 
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Location: Lethbridge, Alberta, Canada
Assassin 124 by Frank (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3329:2051:2051:4612:4612:4612:2823:2823:2565:3329:4362:4620:4620:4612:3598:3598:5648:2565:3329:4362:4620:4885:11542:3351:3598:5648:2565:4362:4362:4885:4885:11542:3351:3351:5648:5648:3620:3620:11542:11542:11542:11542:11542:3115:3115:4672:4672:3631:3631:11542:3899:3899:4422:4422:4159:4672:5432:3631:11542:3899:4420:4422:3655:4159:4672:5432:5432:4419:4420:4420:4422:3655:4159:1792:1792:4419:4419:4419:3842:3842:3655:
Alternative code string
3x3::k:3328:2049:2049:4611:4611:4611:2822:2822:2568:3328:4362:4619:4619:4611:3598:3598:5648:2568:3328:4362:4619:4885:11542:3351:3598:5648:2568:4362:4362:4885:4885:11542:3351:3351:5648:5648:3620:3620:11542:11542:11542:11542:11542:3115:3115:4653:4653:3631:3631:11542:3890:3890:4404:4404:4150:4653:5432:3631:11542:3890:4412:4404:3646:4150:4653:5432:5432:4419:4412:4412:4404:3646:4150:1865:1865:4419:4419:4419:3918:3918:3646:
Solution: 
+-------+-------+-------+
| 6 7 1 | 5 4 8 | 2 9 3 |
| 2 9 8 | 7 1 3 | 4 5 6 |
| 5 4 3 | 6 9 2 | 7 8 1 |
+-------+-------+-------+
| 1 3 4 | 9 8 5 | 6 2 7 |
| 8 6 5 | 4 2 7 | 1 3 9 |
| 7 2 9 | 3 6 1 | 8 4 5 |
+-------+-------+-------+
| 9 8 7 | 2 3 6 | 5 1 4 |
| 4 1 6 | 8 5 9 | 3 7 2 |
| 3 5 2 | 1 7 4 | 9 6 8 |
+-------+-------+-------+
Quote:
Frank: SSR: 1.40.

udosuk: ... nice puzzle. :thumbs: I've solved it but no time to write a walkthrough yet.
Then Here is my complete walkthrough. I don't use numerical ratings but personally I think it ought to be the simplest solving path possible. One big step followed by 3 short steps. :ugeek:

Afmob: Thanks Frank for this fun Killer though I'm a bit puzzled about SudokuSolver's rating. :scratch:
Rating: 1.0.

Andrew: Thanks Frank for an enjoyable puzzle.
Having gone through the posted walkthroughs I can see that my solving path wasn't the most direct but I'll post it anyway; some of my steps may be of interest.
I'll rate my walkthrough as Hard 1.0.
I'm also surprised at the SS score of 1.40. I wonder what SS missed. I can't see anything in Afmob's, udosuk's or my walkthrough that would justify anything like that rating.

Comments about code strings have been omitted from this archive entry.

Walkthrough by udosuk:
Here is my complete walkthrough. I don't use numerical ratings but personally I think it ought to be the simplest solving path possible. One big step followed by 3 short steps. :ugeek:

1.
8/2 @ r1c2={17|26|35}
7/2 @ r9c2={16|25|34}
Innies @ c12: r19c2=12=[75]
=> 8/2 @ r1c2=[71], 7/2 @ r9c2=[52]
=> 14/2 @ r5c1=[59|68|86]
=> r5c1 from {568}

Innie-outies @ c1: r46c1=r5c2+2
IOU: r46c1 can't have 2
=> 2 @ c1,n1 locked @ 13/3 @ r1c1={238|256} has {56}|8
=> 8 @ c1 locked @ 13/3 @ r1c1 & r5c1

13/3 @ r1c1 has 3|6
=> 16/3 @ r7c1 from {134679} can't be {367}, can't have 7
=> 16/3 @ r7c1 from {13469}={169|349} (9 @ c1,n7 locked)
=> 7 @ c3,n7,21/3 locked @ r78c3
=> 21/3 @ r7c3=[{67}8|{78}6]
=> CPE: r8c12 can't have {68}

13/3 @ r1c1 & 16/3 @ r7c1 form KNP {36} @ c1
=> 14/2 @ r5c1=[59|86]
=> 13/3 @ r1c1 & r5c1 form KNP {58} @ c1
{14} @ n7 locked @ r789c1+r78c2
=> CPE: r7c1 can't have {14}, must be 7

Now r678c2=18-7=11
=> r78c2 can't be {34}
Also 16/3 @ r7c1 has 1|3 & 1|4
=> r78c2 can't be {13|14}
Hence r78c2 can't be from {134}
=> r7c2 can't have {134}, must be from {68}

r7c2+r78c3={678} (NT @ n7)
=> 16/3 @ r7c1 from {1349}={349} (NT @ c1,n7)
=> 13/3 @ r1c1 from {2568}={256} (NT @ c1,n1)
=> r4c1=1, 14/2 @ r5c1=[86]
=> r678c2=11=[281]
=> 21/3 @ r7c3=[{67}8]

2.
Innies @ r1234: r34c5=17={89} (NP @ c5,45/9)
Innies @ r6789: r67c5=9
=> 6 @ 45/9 locked @ r67c5=9={36} (NP @ c5,45/9)
HP @ r5: 12/2 @ r5c8={39}
Innies @ c9: r456c9=21 must be from {456789}
=> 12/2 @ r5c8=[39]
r46c9=21-9=12={48|57} has 4|5
=> 10/3 @ r1c9 can't be {145}, can't have 4
=> 10/3 @ r1c9 must be from {123567}

3.
Innie-outies @ r1: r1c19=r2c5+8
=> Min r1c19=1+8=9
=> r1c19 from {2356} can't have 2, can't be {35}
Also r2c5 can't be 3
=> r1c19 can't be 3+8=11, can't be {56}
=> r1c19=[63], r2c5=6+3-8=1
=> 11/2 @ r1c7 from {24589}={29}
Innies @ c89: r19c8=15=[96]
=> 11/2 @ r1c7=[29], 15/2 @ r9c7=[96]

4.
HS @ r1: r1c6=8
=> r34c5=[98]
HS @ n9: r9c9=8
=> r46c9=12 from {457}=[75]
=> r23c9=[61]
=> r78c9={24} (NP @ n9)
=> r678c8=17-5=12 from {14578}=[417]
=> r456c7=[618]
=> r78c7={35} (NP @ c7)
=> r2c6=14-4-7=3, r8c6=17-3-5=9
=> r67c6=15-8=7=[16]
HS @ n2: r3c4=6
r4c34=19-6=13 from {23459}={49} (NP @ r4)

All naked singles from here.
Walkthrough by Afmob:
Thanks Frank for this fun Killer though I'm a bit puzzled about SudokuSolver's rating. :scratch:

A124 Walkthrough:

1. C123
a) Innies C12 = 12(2) = {57} -> R9C2 = 5, R1C2 = 7
b) Cage sum: R1C3 = 1, R9C3 = 2
c) 13(3) <> 9
d) 14(2): R5C1 <> 9
e) Innies+Outies C1: -2 = R5C2 - R46C1 -> R46C1 <> 2 (IOU @ N4)
f) 2 locked in 13(3) @ C1 for N1 -> 13(3) = 2{38/56}
g) 16(3) <> {367} because (36) is a Killer pair of 13(3)
h) Innies C1 = 16(3) <> 9 because R5C1 = (568) and {169} blocked by Killer pair (19) of 16(3)
i) 9 locked in 16(3) @ C1 for N7 -> 16(3) = 9{16/34}
j) 7 locked in Innies C1 = 7{18/45} for N4 since {367} blocked by Killer pair (36) of 16(3)
k) Innies C1 = 16(3): R46C1 <> 5,8

2. C123 !
a) 7 locked in 21(3) @ N7 = {678} because 5,9 only possible @ R8C4
-> R8C4 <> 7 and CPE: R8C12 <> 6,8
b) 14(2): R5C2 <> 8
c) R6C1 <> 1 since it sees all 1 of N7
d) ! Outies N7 = 17(2+1): R6C2 <> 1,8,9 since 10 <= R6C1+R8C4 <= 15
e) 18(4) = 7{128/146/236} because R6C1 = (47) -> R6C1 = 7
f) 1 locked in 17(4) @ N4 for R4 -> 17(4) = 1{268/349}
g) ! Killer pair (69) locked in 17(4) + R5C2 for C2
h) 18(4) = {1278} -> R6C2 = 2, R8C2 = 1, R7C2 = 8
i) 16(3) = {349} locked for C1
j) 13(3) = {256} locked for C1+N1
k) R5C1 = 8 -> R5C2 = 6
l) 21(3) = {678} -> R8C4 = 8

3. R12345
a) Innies R1234 = {89} locked for C5+45(9)
b) 19(3) = 9{37/46}
c) Grouped X-Wing 9 locked in 19(3) + R34C5 for R34
d) Hidden Single: R2C2 = 9 @ C2
e) 8 locked in 18(3) @ C3 = 8{37/46} -> R2C4 = (67)
f) 9 locked in 12(2) @ R5 = {39} locked for R5+N6
g) 9 locked in 11(2) @ N3 = {29} locked for R1+N3
h) 10(3) = 1{36/45} -> 1 locked for C9+N3

4. C789
a) Innies C89 = 15(2) = {69} -> R1C8 = 9, R9C8 = 6
b) Cage sum: R1C7 = 2, R9C7 = 9
c) R5C8 = 3, R5C9 = 9
d) Innies C9 = 12(2) = [48/75/84]
e) 17(4) = {1457} -> 7 locked for C8+N9; CPE: R4C8 <> 4,5
f) 14(3) @ N9 = {248} -> R9C9 = 8; 4 locked for C9+N9
g) 17(3) = {359} since R78C7 = (135) -> R8C6 = 9; {35} locked for C7+N9
h) 22(4) = {2578} -> R4C9 = 7, R4C8 = 2; {58} locked for C8+N3
i) 10(3) = {136} locked for N3
j) 14(3) = {347} because R23C7 = (47) -> R2C6 = 3; 4 locked for C7

5. R6789
a) R5C7 = 1, R6C8 = 4, R6C9 = 5
b) Innies R6789 = 9(2) = {36} locked for C5
c) 15(3) = {168} because R6C67 = (168) -> 1 locked for C6

6. N2
a) 18(4) = {1458} since R1C456 = (4568) -> R2C5 = 1, R1C6 = 8; {45} locked for R1+N2
b) R2C9 = 6, R2C4 = 7
c) 18(3) = {378} -> R2C3 = 8, R3C3 = 3

7. Rest is singles.

Rating: 1.0. I used Killer pairs and some Outies analysis.
Walkthrough by Andrew:
Thanks Frank for an enjoyable puzzle.

Having gone through the posted walkthroughs I can see that my solving path wasn't the most direct but I'll post it anyway; some of my steps may be of interest.

I'll rate my walkthrough as Hard 1.0 although I'm not really sure how step 10a, my hardest one, should be rated.

I'm also surprised at the SS score of 1.40. I wonder what SS missed. I can't see anything in Afmob's, udosuk's or my walkthrough that would justify anything like that rating.

Here is my walkthrough

Prelims

a) R1C23 = {17/26/35}, no 4,8,9
b) R1C78 = {29/38/47/56}, no 1
c) R5C12 = {59/68}
d) R5C89 = {39/48/57}, no 1,2,6
e) R9C23 = {16/25/34}, no 7,8,9
f) R9C78 = {69/78}
g) R123C9 = {127/136/145/235}, no 8,9
h) 19(3) cage at R3C4 = {289/379/469/478/568}, no 1
i) 21(3) cage at R7C3 = {489/579/678}, no 1,2,3

1. 1,2 in R5 locked in R5C34567 for 45(9) cage -> no 1,2 in R3467C5

2. 45 rule on R1234 2 innies R34C5 = 17 = {89}, locked for C5 and 45(9) cage

3. 45 rule on R6789 2 innies R67C5 = 9 = {36/45}, no 7
3a. 7 in 45(9) cage locked in R5C34567, locked for R5, clean-up: no 5 in R5C89

4. 45 rule on C12 2 outies R19C3 = 3 = {12}, locked for C3, R1C2 = {67}, R9C2 = {56}
4a. 45 rule on C12 2 innies R19C2 = 12 = [75], R19C3 = [12], clean-up: no 4 in R1C78, no 9 in R5C1

5. 45 rule on C89 2 innies R19C8 = 15 = [69/87/96], clean-up: no 6,8,9 in R1C7, no 7 in R9C7

6. R123C9 = {127/136/145} (cannot be {235} which clashes with R1C7), 1 locked for C9 and N3

7. 45 rule on C9 3 innies R456C9 = 21 = {489/579/678}, no 2,3, clean-up: no 9 in R5C8
7a. 9 of {489} must be in R5C9 (R5C9 of {489} cannot be 4 or 8 because R5C89 = {48} clashes with R456C9)
7b. 8,9 of {579/678} must be in R5C9
7c. -> R5C9 = {89}, no 9 in R46C9, clean-up: no 8 in R5C8

8. 45 rule on C1 3 innies R456C1 = 16 = {169/178/358/367/457} (cannot be {259/268} which clash with R5C12 because no 2 in R5C1, cannot be {349} because R5C1 only contains 5,6,8), no 2
8a. 6 of {169/367} must be in R5C1 -> no 6 in R46C1

9. 2 in C1 locked in R123C1, locked for N1
9a. R123C1 = {238/256}, no 4,9
9b. R456C1 (step 8) = {169/178/457} (cannot be {358/367} which clash with R123C1), no 3
9c. 5,8 of {178/457} must be in R5C1 -> no 5,8 in R46C1

10. 2,5 in C1 locked in R123C1 + R5C1 -> either 5 in R5C1 or R123C1 = {256}
10a. R456C1 (step 9b) = {178/457} (cannot be {169} which clashes with R123C1 = {256}), no 6,9, 7 locked for C1 and N4, clean-up: no 8 in R5C2
10b. 9 in C1 locked in R789C1, locked for N7
10c. R789C1 = {169/349}, no 8

11. 7 in C3 locked in R78C3, no 7 in R8C4
11a. 21(3) cage at R7C3 = {678} (cannot be {579} because 5,9 only in R8C4), no 4,5,9, CPE no 6 in R8C1, no 6,8 in R8C2

12. 45 rule on N1 3 outies R2C4 + R4C12 = 11
12a. Min R4C12 = 3 -> max R2C4 = 8

13. 45 rule on N3 3 outies R2C6 + R4C89 = 12
13a. Min R4C9 = 4 -> max R2C6 + R4C8 = 8, no 8,9

14. 45 rule on N7 3 outies R6C12 + R8C4 = 17
14a. R8C4 = {68} -> R6C12 = 9,11 = [18/72/74], no 4 in R6C1, no 1,3,6,9 in R6C2
14b. R78C2 = 7,9 = [34/43/61/63/81], no 1 in R7C2
14c. 18(4) cage at R6C1 = [1278/1368/1467/2367]
14d. 6,8 must be in R7C2 (2,7 of {1278} must be in R6C12 so 8 must be in R7C2) -> R7C2 = {68}

15. Naked triple {678} in R7C23 + R8C3, locked for N7, clean-up: no 1 in R789C1 (step 10c)
15a. Naked triple {349} in R789C1, locked for C1 and N7 -> R8C2 = 1, R46C1 = [17], R5C1 = 8 (step 10a), R5C2 = 6, R5C9 = 9, R5C8 = 3, R7C2 = 8, R6C2 = 2 (step 14c)
15b. Naked pair {67} in R78C3, locked for C3 and 21(3) cage -> R8C4 = 8
15c. Naked triple {256} in R123C1, locked for N1
15d. R456C9 (step 7) = {489/579}, no 6
15e. R46C9 = [48/75/84], no 5 in R4C9

16. 8 in N1 locked in R23C3 -> 18(3) cage at R2C3 = {189/378/468}, no 2,5
16a. 1,6,7 only in R2C4 -> R2C4 = {167}

17. R67C5 = {36} (hidden pair in 45(9) cage at R3C5), locked for C5

18. R789C9 = {248/257/356} (cannot be {347} which clashes with R46C9)
18a. 7 of {257} must be in R9C9 -> no 7 in R78C9
18b. 8 of {248} must be in R9C9 -> no 4 in R9C9
18c. R123C9 (step 6) = {127/136} (cannot be {145} which clashes with R789C9), no 4,5

19. 1 in R9 locked in R9C456, locked for N8
19a. 17(4) cage in N8 = {1349/1457} (cannot be {1259} because 2,5 only in R8C5, cannot be {1367} which clashes with R7C5), no 2,6, 4 locked for N8
19b. 6 in R9 locked in R9C789, locked for N9

20. 45 rule on N9 3 outies R6C89 + R8C6 = 18
20a. Max R6C89 = 14 -> no 2,3 in R8C6
20b. 2 in N8 locked in R7C46, locked for R7
20c. R789C9 (step 18) = {248/257/356}
20d. 2 of {248} must be in R8C9 -> no 4 in R8C9
20e. 6,7,8 only in R9C9 -> R9C9 = {678}

21. 19(3) cage at R3C4 = {379/469}, no 2,5
21a. Grouped X-Wing for 9 in 19(3) cage at R3C4 and R34C5, no other 9 in R34
21b. R2C2 = 9 (hidden single in C2)

22. 45 rule on R9 2 innies R9C19 = 1 outie R8C5 + 6
22a. R9C19 cannot total 13 -> no 7 in R8C5
22b. R8C5 = {45} -> R9C19 = 10,11 = [37/46/38/47], no 9 in R9C1

23. R2C6 + R4C89 = 12 (step 13)
23a. R4C89 cannot total 5,7 or 8 -> no 4,5,7 in R2C6

24. 45 rule on R1 2 innies R1C19 = 1 outie R2C5 + 8
24a. Max R1C19 = 11 -> no 4,5,7 in R2C5
24b. R1C19 cannot total 10 -> R2C5 = 1, R1C19 = 9 = [63], R23C9 = {16} (step 18c) = [61], R2C4 = 7, clean-up: no 5 in R1C7, no 8 in R1C8

26. R1C78 = [29], R9C8 = 6 (step 5), R9C7 = 9

27. R1C6 = 8 (hidden single in R1), R1C45 = {45}, locked for N2, R34C5 = [98]

28. 17(4) cage in N8 (step 19a) = {1457} (only remaining combination) -> R8C5 = 5, R9C456 = {147}, locked for R9 and N8 -> R9C1 = 3, R9C9 = 8, R8C9 = 2, R7C9 = 4 (step 18), R78C1 = [94], R8C78 = [37], R78C3 = [76], R8C6 = 9, R46C9 = [75], R7C8 = 1, R6C8 = 4 (cage sum), R4C78 = [62], R567C7 = [185], R23C7 = [47]

29. R4C6 = 5 (hidden single in R4), R3C6 = 2 (cage sum)

and the rest is naked singles and a cage sum.
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Andrew
PostPosted: Sat Jul 16, 2011 4:22 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 124 V2 by Frank (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3073:3331:3331:3588:3588:3588:2311:2311:4613:3073:6154:4108:4108:3588:2574:2574:4880:4613:3073:6154:4108:4885:11542:5143:2574:4880:4613:6154:6154:4885:4885:11542:5143:5143:4880:4880:2340:2340:11542:11542:11542:11542:11542:2603:2603:4928:4928:4143:4143:11542:4667:4667:4422:4422:4159:4928:3640:4143:11542:4667:3908:4422:4423:4159:4928:3640:3640:5187:3908:3908:4422:4423:4159:1024:1024:5187:5187:5187:2818:2818:4423:
Solution: 
+-------+-------+-------+
| 3 7 6 | 2 8 1 | 5 4 9 |
| 4 1 9 | 5 3 7 | 2 8 6 |
| 5 8 2 | 6 4 9 | 1 7 3 |
+-------+-------+-------+
| 9 6 5 | 8 2 4 | 7 3 1 |
| 7 2 3 | 9 1 5 | 8 6 4 |
| 1 4 8 | 7 6 3 | 9 2 5 |
+-------+-------+-------+
| 2 5 4 | 1 7 6 | 3 9 8 |
| 6 9 7 | 3 5 8 | 4 1 2 |
| 8 3 1 | 4 9 2 | 6 5 7 |
+-------+-------+-------+
Quote:
Frank: This version is courtesy of JSudoku.
SSR: 2.60.

udosuk: Nice v2 Frank! :thumbs:
No time for a full walkthrough, just a brief walkin including the trickiest steps for me:
I'll be quite busy from now on, and will be participating less actively. :salute:
Then I've written a complete walkthrough for A124v2:
Except for the extra-tricky step 1 the rest should be quite short and elegant enough. :geek:

Afmob: That was a hard Killer! If I had spotted udosuk's step 1 then my wt would have been quite shorter and would have had a lower difficulty (around 1.5). Despite the length of my wt it was quite fun and I could make use of ...
Rating: 1.75 - (Hard) 1.75.
How did you solve A124 V2, Frank?

Andrew (in 2010): Another challenging puzzle from my Unfinished folder.
udosuk wrote:
Except for the extra-tricky step 1 ...
That was an ingenious step! I'll agree with "extra-tricky" so I don't go along with Afmob's suggestion about the rating using that step; I'd put it in the same rating range as Afmob's walkthrough and my one.
My solving path was fairly similar to Afmob's but the way I saw some of the key steps was different so I'm belatedly posting my walkthrough.
I'll rate my walkthrough for A124 V2 at 1.75.

Walkthrough by udosuk:
Nice v2 Frank! :thumbs:
No time for a full walkthrough, just a brief walkin including the trickiest steps for me:
I'll be quite busy from now on, and will be participating less actively. :salute:

Then I've written a complete walkthrough for A124v2:

1.
4/2 @ r9c2={13} (NP @ r9,n7)
Innies @ c12: r19c2=10=[73|91]
=> 13/2 @ r1c2=[76|94]
=> r19c23=[7631|9413]
=> Either r1c3=6 or r9c2=1
=> Either r789c1+r78c2 have 6 or r9c2=1
=> Outies @ n7: r6c12+r8c4=8 can't be [611] (CPE)
=> r6c1 can't be 6, must be from {12345}

2.
Innies @ r1234: r34c5=6={15|24}
Innies @ r6789: r67c5=13={49|58|67}
3 @ r5,45/9 locked @ r5c34567
=> 9/2 @ r5c1={18|27|45}
=> 10/2 @ r5c8={19|28|46}
LOL @ r5,45/9: r5c1289=r3467c5
=> r5c1289 can't be [{18}{46}]
=> 9/2 @ r5c1 can't be {18}, must be {27|45}

3.
16/3 @ r7c1 from {2456789}={259|268|457} has 2|{457}
=> 2 @ c1 locked @ r5c1+16/3 @ r7c1
=> 12/3 @ r1c1 can't be {129}, can't have 9
13/2 @ r1c2=[76|94] has 4|7
=> 12/3 @ r1c1 can't be {147|237}, can't have 7
16/3 @ r7c1={259|268|457} has 5|6
=> 12/3 @ r1c1 can't be {156|246}, can't have 6
=> 12/3 @ r1c1 from {13458}={138|345} (3 @ c1,n1 locked)
=> 12/3 @ r1c1 & 16/3 @ r7c1 form KNP {58} @ c1

4.
Innies @ c1: r456c1=17=[674|971]
=> 9/2 @ r5c1=[72]
LOL @ r5,45/9: {27} locked @ r3467c5
=> r34c5=6={24}, r67c5=13={67} (NPs @ c5,45/9)
HP @ r5: 10/2 @ r5c8={46} (NP @ n6)
Innies @ c9: r456c9=10={145|136}
=> r46c9={13|15} (1 @ c9,n6 locked)

5.
11/2 @ r9c7 from {2456789}={29|38|56}
Innies @ c89: r19c8=9={27|[36]|45}
=> 9/2 @ r1c7={27|[63]|45} has 4|6|7
13/2 @ r1c2=[76|94] has 4|{67}
=> 4 @ r1 locked @ 13/2 @ r1c2 & 9/2 @ r1c7
=> 14/4 @ r1c4 from {1235678}={1238|1256}
=> 1 @ n2, 2 @ r1,n2 locked @ 14/4 @ r1c4
=> r34c5=[42], 9/2 @ r1c7={[63]|45}
=> r1c3 & 9/2 @ r1c7 form KNP {46} @ r1
=> 14/4 @ r1c4 from {12358}={1238} (NQ @ n2)

6.
Outies @ n3: r2c6+r4c89=11
Min r2c6=5
=> Max r4c89=11-5=6
=> r4c89=[31|51]
r19c8=9={[36]|45}
=> r1459c8={3456} (NQ @ c8)
=> r234c8=19-1=18=[{78}3]
=> r456c9=10=[145], 10/2 @ r5c8=[64]
=> r2c6+r4c89=11=[731]
=> r23c8=[87]
9/2 @ r1c7={45} (NP @ r1,n3)
=> r19c23=[7631]

7.
r23c7=10-7=3={12} (NP @ c7,n3)
=> r123c9={369} (NT @ c9), r456c7={789} (NT @ c7,n6)
=> r6c8=2, 11/2 @ r9c7 from {456}=[65]
=> 9/2 @ r1c7=[54], r78c7={34}
=> r8c6=15-3-4=8
=> r9c5=9
=> r8c5+r9c46=20-9=11={245}
=> r8c5=5, r9c46={24} (NP @ r9,n8)
=> r9c19=[87]
=> 17/3 @ r7c9=[827]
=> 16/3 @ r7c1=[268]
=> r456c1=17=[971]
=> 12/3 @ r1c1=[345]
=> r234c2=24-9=15=[186]
=> r6c12+r8c4=8=[143]
=> 20/3 @ r3c6=[947]

All naked singles from here.

Except for the extra-tricky step 1 the rest should be quite short and elegant enough. :geek:
Walkthrough by Afmob:
That was a hard Killer! If I had spotted udosuk's step 1 then my wt would have been quite shorter and would have had a lower difficulty (around 1.5). Despite the length of my wt it was quite fun and I could make use of some nice Killer subsets.

How did you solve A124 V2, Frank?

A124 V2 Walkthrough:

1. C123
a) 4(2) = {13} locked for R9+N7
b) Innies C12 = 10(2) = [73/91]
c) Outies C12 = 7(2) = [43/61]
d) Outies N7 = 8(2+1) <> 7,8,9; R8C4 <> 6
e) 19(4): R78C2 <> 2 because R6C12 @ Outies N7 <= 7
f) 19(4) <> {1279} since it's blocked by R1C2 = (79)
g) Outies N7 = 8(2+1): R8C4 <> 5 because R6C12 >= 4
h) 12(3) <> 6,7 since 4{17/26} blocked by Killer pairs (46,47) of 13(2)
and {156/237} blocked by Killer pairs (27,56) of 16(3) @ C1
i) Outies N1 = 20(2+1) <> 1; R2C4 <> 2
j) Innies+Outies C1: -8 = R5C2 - R46C1 -> R4C1 <> 8 (IOU @ N4)

2. 45(9) + R5 !
a) Innies R1234 = 6(2) = {15/24}
b) Outies R1234 = 13(2) <> 1,2,3
c) 45(9) must have 3 -> 3 locked for R5
d) 9(2) <> 6
e) 10(2) <> 7
f) Killer triple (456) in each of Innies R1234 and Innies R6789 for 45(9) + C5
-> R5C34567 can only have one of (456)
g) ! Two of (456) in R5 must be in 9(2) and 10(2) -> Either 9(2) = {45} or 10(2) = {46}
-> 4 locked for R5
h) 4 locked in Outies R5 @ 45(9) for C5 = 19(4) = 9(2) + 10(2) = 4{159/258/267}
i) Hidden Killer pair (57) in R5C34567 for 45(9)+R5
j) Hidden Killer pair (57) in 9(2) for R5 -> 9(2) <> 1,8

3. C789
a) Innies C9 = 10(3) <> 8,9
b) 10(2): R5C8 <> 1,2
c) 11(2) <> 8
d) Outies C89 = 11(2) <> 1,3,8; R9C7 <> 2
e) Innies C89 = 9(2) <> 9; R1C8 <> 1,6,8

4. R1234
a) 13(2) + 9(2) = 4{279/369/567} -> 4 locked for R1
b) 14(4) = 12{38/56} -> 1,2 locked for N2
c) Innies R1234 = 6(2): R4C5 <> 4,5

5. C123
a) Innies C1 = 17(3) <> 3,5 since {359} blocked by Killer pair (39) of 12(3)
b) Innies C1 = 17(3): R46C1 <> 2 because R5C1 <> 6,9
c) 9(2): R5C2 <> 4
d) 3 locked in 12(3) @ C1 for N1 -> 12(3) = 3{18/45}
e) Outies N7 = 8(2+1): R6C2 <> 2 because (24) is a Killer pair of 9(2)

6. C123 !
a) 19(4) <> {1369} since (39) is a Killer pair of Innies C12
b) 14(3) <> 5{18/36} because (56,58) are Killer pairs of 16(3) @ N7
c) Killer triple (789) in each of 16(3) and 14(3) for N7
-> 19(4) can only have one of (789)
-> 19(4) = {1459/1468/1567/3457}
d) Outies N7 = 8(2+1): R8C4 <> 4 since R6C12 @ 19(4) >= 5
e) 24(4) <> 58{29/47} because (58) is a Killer pair of 19(4)
f) ! Hidden Killer triple (258) in R5C2 for C2 since each of 24(4) and 19(4)
can only have one of (258) -> R5C2 <> 7
g) 9(2): R5C1 <> 2
h) 2 locked in 16(3) @ C1 for N7 -> 16(3) = 2{59/68}
i) 7 locked in R45C1 @ C1 for N4

7. C123 !
a) ! Killer quad (1235) in 24(4) + R59C2 for C2
-> 19(4) can only have one of (1235) @ C2
-> 19(4) <> {3457} since (35) would be @ C2
b) 19(4) = 1{459/468/567} <> 3 -> 1 locked for R6+N4
c) Killer triple (789) in R1C2 + 19(4) for C2 -> 24(4) can only have one of (789) @ C2
d) ! 24(4): R23C2 <> 2 because only possible placements are {26}[79] / {27}[96] / {29}[76]
- {26}[79] -> 13(2) = [94] -> R1C2 = R4C2 = 9
- {27}[96] -> Innies C1 = [971] -> 9(2) = [72] -> Two 2s in C2
- {29}[76] blocked by Killer pair (69) of 19(4)
e) 2 locked in 16(3) @ N1 for C3 -> 16(3) = 2{59/68}
f) Hidden Single: R9C3 = 1 @ C3
g) Outie C12 = R1C3 = 6
h) Cage sum: R1C2 = 7, R9C2 = 3
i) 24(4) = 69{18/45} because (67) only possible @ R4C12 -> 6 locked for R4+N6

8. 45(9) + R5
a) Hidden Single: R5C1 = 7 @ C1, R5C2 = 2 @ C2
b) 2 in 45(9) locked in Innies R1234 = 6(2) = {24} -> R4C5 = 2, R3C5 = 4
c) 7 in 45(9) locked in Innies R6789 = 13(2) = {67} locked for C5+45(9)
d) 6 in R5 locked in 10(2) = {46} locked for N6

9. C789
a) 9(2) = {45} locked for R1+N3
b) Innies C89 = 9(2) = {45} locked for C2
c) R5C8 = 6, R5C9 = 4
d) 17(4) = 5{129/138/237} -> R6C9 = 5
e) Innie C1 = R4C9 = 1
f) Outies N3 = 10(1+1) = {37}

10. N2+C789
a) 14(4) = {1238} locked for N2 -> 2 also locked for R1
b) R2C6 = 7 -> R23C7 = 3(2) = {12} locked for C7+N3
c) Hidden Single: R1C9 = 9 @ R1
d) 18(3) @ N1 = {369} -> 3,6 locked for C9+N3
e) Hidden Single: R6C8 = 2 @ N6
f) Outie N9 = R8C6 = 8
g) 15(3) = {348} -> 3,4 locked for C7+N9
h) R9C8 = 5 -> R9C7 = 6, R9C5 = 9

11. N578
a) 20(4) = {2459} since (13) only possible @ R8C7 -> R8C7 = 5, {24} locked for R9+N8
b) 14(3) = 4{19/37} -> 4 locked for C3+N7
c) R9C1 = 8 -> R78C1 = 8(2) = {26} locked for C1+N7
d) 19(4) = {1459} -> R8C2 = 9, R7C2 = 5, {14} locked for R6+N4
e) 14(3) = {347} -> R8C4 = 3
f) R4C1 = 9
g) 16(3) = 7{18/36} because R6C3 = (38) -> 7 locked for C4; R6C4 <> 8
h) 19(3) = 6{49/58} -> R3C4 = 6

12. N1
a) 24(4) = {1689} -> R4C2 = 6, {18} locked for C2+N1

13. Rest is singles.

Rating: 1.75 - (Hard) 1.75. I used Killer triples, Killer quads and a (small?) contradiction move.
Walkthrough by Andrew:
My solving path was fairly similar to Afmob's but the way I saw some of the key steps was different so I'm belatedly posting my walkthrough.

Rating Comment. I'll rate my walkthrough for A124 V2 at 1.75. I used Law of Leftovers, combination analysis and some contradiction moves. I don't want to imply that my solving path was easier than Afmob's, it wasn't. However I felt that Afmob's walkthrough was 1.75 rather than Hard 1.75. Although I used several contradiction moves, some of which might not have been necessary, my largest contradiction move was effectively the same as Afmob's one.

Here is my walkthrough for A124 V2.

Prelims

a) R1C23 = {49/58/67}, no 1,2,3
b) R1C78 = {18/27/36/45}, no 9
c) R5C12 = {18/27/36/45}, no 9
d) R5C89 = {19/28/37/46}, no 5
e) R9C23 = {13}
f) R9C78 = {29/38/47/56}
g) 10(3) cage at R2C6 = {127/136/145/235}, no 8,9
h) 19(3) cage at R3C4 = {289/379/469/478/568}, no 1
i) 20(3) cage at R3C6 = {389/479/569/578}, no 1,2
j) 14(4) cage in N2 = {1238/1247/1256/1346/2345}, no 9

1. Naked pair {13} in R9C23, locked for R9 and N7, clean-up: no 8 in R9C78

2. 45 rule on R1234 2 innies R34C5 = 6 = {15/24}

3. 45 rule on R6789 2 innies R67C5 = 13 = {49/58/67}, no 1,2,3

4. 3 in 45(9) cage at R3C5 only in R5C34567, locked for R5, clean-up: no 6 in R5C12, no 7 in R5C89

5. 45 rule on C12 2 innies R19C2 = 10 = [73/91], R1C23 = [76/94]

6. 45 rule on C89 2 outies R19C7 = 11 = [29/47/56/65/74], no 1,3,8 in R1C7, no 2 in R9C7, clean-up: no 1,6,8 in R1C8, no 9 in R9C8

7. 45 rule on C9 3 innies R456C9 = 10 = {127/136/145/235}, no 8,9, clean-up: no 1,2 in R5C8

8. 45 rule on C1 2 innies R46C1 = 1 outie R5C2 + 8, IOU no 8 in R46C1

9. 45 rule on R9 2 innies R9C19 = 1 outie R8C5 + 10, max R9C19 = 17 -> max R8C5 = 7

10. 45 rule on N1 3 outies R2C4 + R4C12 = 20
10a. Min R4C12 = 11, no 1
10b. Max R4C12 = 17 -> min R2C4 = 3

11. 45 rule on N7 3 outies R6C12 + R8C4 = 8
11a. Min R6C12 = 4 (cannot be 3 because R78C2 cannot be 16 = {79} which clashes with R1C2) -> max R8C4 = 4
11b. Max R6C12 = 7, no 7,8,9
11c. Max R6C12 = 7 -> min R78C2 = 12, no 2

12. R46C1 = R5C2 + 8 (step 8)
12a. Min R46C1 = 9, no 2 in R4C1

13. 45 rule on C1 3 innies R456C1 = 17
13a. Max R46C1 = 15 -> min R5C1 = 2, clean-up: no 8 in R5C2

14. 45 rule on R5 4 outies R3467C5 = 4 innies R5C1289 must contain the same 4 numbers because of the 45(9) cage (Law of Leftovers)
14a. 8 in R67C5 must be in {58} (step 3) -> 8 in R5C1289 must be in R5C8 because 5 can only be in R5C12, otherwise 8 in 45(9) cage at R3C5 must be in R5 -> no 8 in R5C1, clean-up: no 1 in R5C2
14b. R67C5 = {49/58} would lock 4,5 in R3467C5 and therefore in R5C1289
14c. R67C5 = {67} => R5C1289 = {27}+{46}
14d. -> 4 locked in R5C1289, locked for R5
14e. 4 in 45(9) cage at R3C5 only in R3467C5, locked for C5

15. R1C23 = [76/94], R1C78 = {27/45}/[63] -> combined cage R1C2378 = [76]{45}/[94]{27}/[94][63], 4 locked for R1

16. 14(4) cage in N2 = {1238/1256}, no 7, 1,2 locked for N2, clean-up: no 4,5 in R4C5

17. R456C1 (step 13) = 17 = {179/269/359/467}
17a. 9 of {359} must be in R4C1 -> no 3,5 in R4C1
17b. 2,5 of {269/359} must be in R5C1 -> no 2,5 in R6C1

18. R789C1 = {259/268/457}
18a. R123C1 = {129/138/345} (cannot be {147} which clashes with R1C23, cannot be {156/237/246} which clash with R789C1), no 6,7
18b. 8 in C1 must be in R123C1 = {138} or R789C1 = {268} -> R123C1 cannot be {129} (locking-out cages)
18c. R123C1 = {138/345}, no 2,9, 3 locked for C1 and N1, clean-up: no 5 in R5C1 (step 17), no 4 in R5C2

19. Max R6C12 = 7 (step 11c)
19a. R6C12 cannot be [12] (step 11a), cannot be {23/25} because 2,3,5 only in R6C2, cannot be [42] which clashes with R5C12 -> no 2 in R6C2

20. 10(3) cage at R2C6 = {127/136/145/235}
20a. 7 of {127} must be in R2C6 -> no 7 in R23C7

21. 45 rule on N3 3 outies R2C6 + R4C89 = 11, min R2C6 = 3 -> max R4C89 = 8, no 8,9
21a. Max R4C89 = 8 -> min R23C8 = 11, no 1

[I originally got as far as here when I first tried this puzzle. I’ve done some minor editing of earlier steps, for example in step 18b I didn’t know the term locking-out cages when I did that step.]

[Just spotted some IOUs and a CPE which I ought to have seen earlier.]

22. 45 rule on N1 2 innies R23C2 = 1 outie R2C4 + 4, IOU no 4 in R3C2

23. 45 rule on N3 2 innies R23C8 = 1 outie R2C6 + 8, IOU no 8 in R3C8

24. 45 rule on N9 2 innies R78C8 = 1 outie R8C6 + 2, IOU no 2 in R7C8

25. 9 in N2 only in R2C4 + R3C46, CPE no 9 in R3C3

26. R456C1 (step 17) = {179/269/467}, R789C1 (step 18) = {259/268/457}
26a. 19(4) cage at R6C1 = {1378/1459/1468/1567/3457} (cannot be {1369} which clashes with R19C2)
Cannot be {1378} = [13]{78} which clashes with R789C1 which must be {268} when R6C1 = 1
Cannot be {3457} = [43]{57} which clashes with R789C1 which must be {259} when R6C1 = 4
26b. -> 19(4) cage at R6C1 = {1459/1468/1567}, no 3, 1 locked for R6 and N4
26c. 1 in N5 locked in R4C5 + R5C456, locked for 45(9) cage, no 1 in R5C7
26d. Min R6C12 = 5 -> max R8C4 = 3 (step 11)

[At this stage I made the mistake of thinking that [45] can be eliminated from R5C12 because of clashes with the 19(4) cage at R6C1; I hadn’t looked carefully enough at what happens when {1468} has 4 in R78C2. Therefore I’ve had to re-work from here.]

27. 24(4) cage at R2C2 = {1689/2679/3489/3579/3678/4569} (cannot be {2589/4578} which clash with 16(4) cage at R6C1)
27a. Hidden killer triple 2,5,8 in 24(4) cage at R2C2, R5C2 and 19(4) cage at R6C1 for C2, 24(4) cage contains one of 2,5,8, 19(4) cage contains one of 5,8 -> R5C2 must contain one of 2,5 -> R5C2 = {25}, no 7, clean-up: no 2 in R5C1
27b. 2 in C1 only in R789C1, locked for N7
27c. R789C1 (step 18) = {259/268}, no 4,7
27d. 7 in C1 only in R45C1, locked for N4

28. Hidden killer triple 7,8,9 in R1C2, R234C2 and R78C2 for C2, R1C2 = {79}, R78C2 contains one of 7,8,9 -> R234C2 must contain one of 7,8,9
28a. 24(4) cage at R2C2 (step 27) = {1689/2679/3489/3579/3678/4569}
28b. {1689/2679/3489/3579/3678} must have one of 7,9 in R4C1 because R234C2 only contains one of 7,8,9
28c. 9 of {1689} must be in R4C1 with 6 in R4C2 (otherwise {1689} clashes with R1C23 = [76]), 3 of {3489/3678} must be in R4C2, other combinations don’t contain 8 -> no 8 in R4C2
28d. 8 in N4 only in R456C3, locked for C3
28e. Max R4C12 = 16 -> min R2C4 = 4 (step 10)

29. 24(4) cage at R2C2 (step 27) cannot be {3678}, here’s how
{3678} => R5C2 = 2 (hidden single in C2), R5C1 = 7 clashes with {3678} which must have 7 in R4C1 (step 28b)
-> 24(4) cage at R2C2 (step 27) = {1689/2679/3489/3579/4569}
29a. 24(4) cage at R2C2 = {1689/2679/3579/4569} (cannot be {3489} because 9 must be in R4C1 (step 28b), 3 in R4C2 and {48} clashes with R123C1)

30. 24(4) cage at R2C2 (step 29a) = {1689/2679/3579/4569}
30a. 2 of {2679} cannot be in R23C2, here’s how
R23C2 = {26} => R4C12 = [79] => R1C2 = 7, R1C3 = 6 clashes with R23C2
R23C2 = {27} => 9 must be in R4C1 (step 28b), R4C2 = 6, R5C2 = 5, R45C2 = [65] clash with all combinations for 19(4) cage at R6C1
R23C2 = {29} => 7 must be in R4C1 (step 28b), R4C2 = 6, R5C2 = 5, R45C2 = [65] clash with all combinations for 19(4) cage at R6C1
-> no 2 in R23C2

31. 2 in N1 only in R23C3, locked for C3
31a. 16(3) cage at R2C3 must contain 2 = {259/268}, no 1,4,7
31b. 8 of {268} must be in R2C4 -> no 6 in R2C4

32. R9C3 = 1 (hidden single in C3), R9C2 = 3, R1C2 = 7 (step 5), R1C3 = 6, clean-up: no 2,3 in R1C78, no 4,5,9 in R9C7 (step 6), no 2,6,7 in R9C8
32a. Naked pair {45} in R1C78, locked for R1 and N3
32b. Naked pair {45} in R19C8, locked for C8, clean-up: no 6 in R5C9

33. R1C9 = 9 (hidden single in R1)

34. 2 in R1 only in R1C456, locked for N2
34a. 14(4) cage in N2 (step 16) = {1238} (only remaining combination, cannot be {1256} because 5,6 only in R2C5), locked for N2

35. 7 in C3 only in R78C3 -> 14(3) cage at R7C3 = {257/347}, no 1,9

36. Naked triple {259} in 16(3) cage at R2C3, CPE no 5,9 in R2C12

37. 24(4) cage at R2C2 (step 29a) = {1689/4569} (cannot be {2679} because 2,6,7 only in R4C12), no 2,7, 6 locked for R4 and N4
37a. 4 of {4569} must be in R2C2 -> no 4 in R4C12

38. R5C1 = 7, R5C2 = 2 (hidden singles in C1 and C2), R5C9 = 4 (hidden single in R5), R5C8 = 6

39. R67C5 = {67} (hidden pair in 45(9) cage at R3C5), locked for C5, R34C5 = [42] (hidden pair in 45(9) cage at R3C5)
[Alternatively R3467C5 = R5C1289 (step 14) = {2467} -> R4C5 = 2, R67C5 = {67}, locked for C5, R3C5 = 4
Or naked quint {13589} in R5C34567, locked for 45(9) cage at R3C5 ...]

40. R456C9 (step 7) = 10
40a. R5C9 = 4 -> R46C9 = 6 = [15]

41. Naked pair {14} in R6C12, locked for R6, N4 and 19(4) cage at R6C1

42. R78C3 = {47} (hidden pair in C3), R8C4 = 3 (step 35)

43. 19(4) cage at R2C8 = {1378} (only remaining combination) -> R2C8 = 8, R34C8 = {37}, locked for C8

44. R123C9 = {279/369}
44a. Killer pair 3,7 in R23C9 and R3C8, locked for N3

45. 10(3) cage at R2C6 = {127} (only remaining combination) -> R2C6 = 7, R23C7 = {12}, locked for C7 and N3, clean-up: no 7 in R3C9 (step 44)
45a. Naked pair {36} in R23C9, locked for C9 and N3 -> R34C8 = [73]

46. Naked triple {278} in R789C9, locked for N9 -> R9C7 = 6, R9C8 = 5, R1C78 = [54]

47. R7C7 = 3, R8C7 = 4 (hidden singles in C7), R8C6 = 8 (cage sum), R9C5 = 9, R78C3 = [47], R8C9 = 2

48. 20(4) cage in N8 = {2459} (only remaining combination) -> R8C5 = 5, R9C46 = {24}, locked for R9 and N8 -> R9C1 = 8, R9C9 = 7, R7C9 = 8
48a. Naked triple {167} in R7C456, locked for R7 -> R7C8 = 9, R7C2 = 5, R7C1 = 2, R8C1 = 6 (cage sum)

49. R123C1 (step 18c) = {345} (only remaining combination) -> R1C1 = 3, R23C1 = [45], R4C12 = [96], R23C2 = [18]

50. 20(3) cage at R3C6 = {479} (only remaining combination, cannot be {569} because 6,9 only in R3C6, cannot be {578} because 7,8 only in R4C7) -> R4C6 = 4, R3C6 = 9, R4C7 = 7

and the rest is naked singles.
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Andrew
PostPosted: Sat Jul 16, 2011 4:41 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 125 by Para (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4096:6145:6145:6145:2820:2820:3590:3590:3590:4096:2314:2314:6145:4877:2820:5135:5135:4881:4096:2314:3860:4877:4877:4375:4375:5135:4881:1819:1819:3860:3860:6687:4375:2849:4881:4881:1819:5157:5157:6687:6687:6687:2849:2849:4140:4909:4909:5157:3888:6687:5426:5426:4140:4140:4909:3895:3888:3888:2106:2106:5426:3389:4670:4909:3895:3895:4930:2106:3908:3389:3389:4670:4680:4680:4680:4930:4930:3908:3908:3908:4670:
Solution: 
+-------+-------+-------+
| 3 9 4 | 6 1 8 | 5 2 7 |
| 8 6 1 | 5 7 2 | 9 3 4 |
| 5 2 7 | 3 9 4 | 6 8 1 |
+-------+-------+-------+
| 4 1 6 | 2 8 7 | 3 9 5 |
| 2 3 9 | 4 6 5 | 7 1 8 |
| 7 5 8 | 1 3 9 | 4 6 2 |
+-------+-------+-------+
| 6 4 5 | 9 2 1 | 8 7 3 |
| 1 8 3 | 7 5 6 | 2 4 9 |
| 9 7 2 | 8 4 3 | 1 5 6 |
+-------+-------+-------+
Quote:
Para: Here's another puzzle for which Ruud tended to be (in)famous for: killer puzzles without 2 cell cages. One 5-cell cage, four 4-cell cages and 20 3-cell cages to keep you busy. I know udosuk just did one, but this was already made before i knew that.
SS-Score: 1.09. Estimated rating: 1.0.
Enjoy.

Afmob: Thanks for this week's Assassin! And thanks for posting it so soon since else Andrew would have posted the first walkthrough. :cheesey:
I probably took a technical harder way than necessary but I couldn't resist using ...
Rating: (Hard) 1.0.

Andrew: That wasn't going to happen this time. I found A125 hard to get into until I realised that I'd been looking at ... wrong way round. :oops: Once I spotted my mistake the rest of this puzzle wasn't too difficult.
I'll rate A125 at Easy 1.25 the way that I solved it.

Walkthrough by Afmob:
Thanks for this week's Assassin! And thanks for posting it so soon since else Andrew would have posted the first walkthrough. :cheesey:

I probably took a technical harder way than necessary but I couldn't resist using a Killer triple which I disguised as a Killer pair (step 1d). :whistle:

A125 Walkthrough:

1. C123 !
a) 7(3) = {124} locked for N4
b) Innies+Outies C1: -3 = R46C2 - R9C1 -> R6C2 = (35) and R4C2 <> 4; R9C1 = (789)
c) 7(3) = {124} -> 4 locked for C1
d) ! Killer pair (12) locked in 19(4) + R45C1 for C1
e) 16(3) = 3{58/67} -> 3 locked for C1+N1
f) 9(3) = {126} locked for N1
g) 16(3) = {358} locked for C1+N1
h) Innies+Outies C1: -3 = R46C2 - R9C1; R9C1 = (79) -> R46C2 = 4/6(2) = 1{3/5} -> R4C2 = 1
i) Hidden Single: R2C3 = 1 @ N1

2. C123
a) 9(3) = {126} -> 2,6 locked for C2
b) 7(3) = {124} -> 2 locked for C1
c) 1 locked in Innies N7 = 12(3) = 1{29/47/56} -> R7C3 = (245)
d) Innies N47 = 11(2) = [65/92] since {47} blocked by R13C3 = (479)
e) 15(3) @ N4 must have one of (235) -> R4C4 = (235)
f) 15(3) @ N4 = {249/267/456} <> 3 because R4C3 = (69); R3C3 <> 9

3. N12 !
a) Innies N12 = 11(2) = {47} locked for R3
b) 9 locked in 24(4) @ N1 for R1 -> 24(4) = 9{267/348/357/456} <> 1 since R1C23 = (479); R12C4 <> 4,7,9
c) 9 locked in 19(3) @ N2 = 9{28/37/46} <> 5
d) 1 locked in 11(3) @ N2 for R1; 11(3) = 1{28/37/46} <> 5
e) ! 5 locked in 24(4) @ N2 for C4 -> 24(4) = 59{37/46}
f) R4C4 = 2, R4C1 = 4, R5C1 = 2

4. N3689
a) 2 locked in 16(3) @ R6 = 2{59/68}
b) 1 locked in 11(3) @ N6 for R5 = 1{37/46}
c) Innies N89 = 17(2) = {89} locked for R7
d) Hidden Single: R6C4 = 1 @ C4
e) 15(3) = {159} -> R7C3 = 5, R7C4 = 9
f) R7C7 = 8
g) 18(3) <> 1
h) Hidden Single: R3C9 = 1 @ C9
i) 17(3) = {278/458/467} <> 3,9 because R3C6 = (47)

5. N1346
a) Innie N47 = R4C3 = 6
b) Cage sum: R3C3 = 7
c) 24(4) = {4569} -> 4 locked for R1, 6 locked for C4+N2
d) 14(3) = 5{27/36} -> 5 locked for N3
e) R3C6 = 4
f) 17(3) = {467} because (58) only possible @ R4C6 -> R3C7 = 6, R4C6 = 7
g) R4C7 = 3
h) 19(4) = {1459} since R4C89 = (589) -> R2C9 = 4, {59} locked for R4+N6
i) 11(3) = {137} -> 7 locked for R5+N6
j) R6C7 = 4 -> R6C6 = 9
k) 20(3) @ N3 = {389} -> R2C7 = 9, {38} locked for C8+N3

6. R789
a) R6C1 = 7, R9C1 = 9
b) 18(3) = {279} -> R9C2 = 7, R9C3 = 2
c) 19(3) = {478} because (56) only possible @ R9C5 -> R8C4 = 7; {48} locked for R9+N8
d) 8(3) = {125} -> R8C5 = 5; {12} locked for R7+N8
e) 13(3) = {247} -> R8C7 = 2, R8C8 = 4, R7C8 = 7

7. Rest is singles.

Rating: (Hard) 1.0. I nearly used a Killer triple. :)
Walkthrough by Andrew:
Afmob wrote:
Thanks for this week's Assassin! And thanks for posting it so soon since else Andrew would have posted the first walkthrough.
That wasn't going to happen this time. I found A125 hard to get into until I realised that I'd been looking at the innie-outie difference in step 9 the wrong way round. :oops: Once I spotted my mistake the rest of this puzzle wasn't too difficult.

I'll rate A125 at Easy 1.25 the way that I solved it.

Here is my walkthrough.

Prelims

a) 9(3) cage in N1 = {126/135/234}, no 7,8,9
b) 11(3) cage in N2 = {128/137/146/236/245}, no 9
c) 19(3) cage in N2 = {289/379/469/478/568}, no 1
d) 20(3) cage in N3 = {389/479/569/578}, no 1,2
e) 7(3) cage in N4 = {124}, locked for N4
f) 11(3) cage in N6 = {128/137/146/236/245}, no 9
g) 21(3) cage at R6C6 = {489/579/678}, no 1,2,3
h) 8(3) cage in N8 = {125/134}, 1 locked for N8

1. 45 rule on N1 3 innies R1C23 + R3C3 = 20 = {389/479/569/578}, no 1,2

2. 45 rule on N3 3 innies R23C9 + R3C7 = 11 = {128/137/146/236/245}, no 9

3. 45 rule on N89 2 innies R7C47 = 17 = {89}, locked for R7
3a. 19(3) cage in N8 = {379/469/478/568} (cannot be {289} which clashes with R7C4), no 2
3b. Killer pair 8,9 in R7C4 and 19(3) cage, locked for N8
3c. R789C9 cannot be {189} which clashes with R7C7, no 1
3d. Min R7C4 = 8 -> max R6C4 + R7C3 = 7, no 7,8,9

4. 45 rule on N36 2 innies R36C7 = 10 = [19/28/37/46/64], no 5, no 7,8 in R3C7

5. 45 rule on N12 2 innies R3C36 = 11 = {38/47/56}/[92], no 1,9 in R3C6
5a. R3C36 = 11 -> R3C67 cannot be 11 because of overlap -> no 6 in R4C6
5b. Max R3C67 = 14 -> min R4C6 = 3

6. 45 rule on N47 2 innies R47C3 = 11 = [56/65/74/83/92], no 3 in R4C3, no 1 in R7C3
6a. R47C3 = 11 -> R34C3 cannot be 11 because of overlap -> no 4 in R4C4
6b. Min R34C3 = 8 -> max R4C4 = 7
6c. Min R7C34 = 10 -> max R6C4 = 5

7. 45 rule on N9 3 innies R7C7 + R9C78 = 14, min R7C7 = 8 -> max R9C78 = 6, no 6,7,8,9

8. 45 rule on R1234 4 innies R4C1257 = 16 = {1249/1258/1267/1348/1456/2347} (cannot be {1357/2356} because R4C12 must have two of 1,2,4)
8a. 9 of {1249} must be in R4C5, 1,2,4 of the other combinations must be in R4C12 -> no 1,2,4 in R4C5

9. 45 rule on C1 1 innie R9C1 = 2 outies R46C2 + 3
9a. Min R46C2 = 4 -> min R9C1 = 7
9b. Max R9C1 = 9 -> max R46C2 = 6, no 4 in R4C2, no 6,7,8,9 in R6C2
9c. 4 in N4 locked in R45C1, locked for C1
9d. Killer pair 3,5 in 20(3) cage and R6C2 for N4, locked for N4, clean-up: no 6 in R7C3 (step 6)
9e. Min R34C3 = 9 -> max R4C4 = 5 (because 15(3) cage cannot be [366])
9f. Min R4C34 = 7 -> max R3C3 = 8, clean-up: no 2 in R3C6 (step 5)

10. 19(4) cage at R6C1 = {1369/1378/1567/2359/2368} (cannot be {1279} because R6C2 only contains 3,5)
10a. Killer pair 1,2 in R45C1 and R78C1, locked for C1

11. 1,2 in N1 locked in 9(3) cage = {126}, locked for N1, clean-up: no 5 in R3C6 (step 5)
11a. Naked triple {126} in R234C2, locked for C2
11b. 6 in C2 locked in R23C2, locked for N1

12. R123C1 = {358} (only remaining combination), locked for C1 and N1, clean-up: no 3,6,8 in R3C6 (step 5)
12a. Naked pair {47} in R3C36, locked for R3, clean-up: no 6 in R6C7 (step 4)
12b. R3C3 + R4C4 can only total 8 as [71] -> no 7 in R4C3, clean-up: no 4 in R7C3 (step 6)
12c. 9 in N1 locked in R1C23, locked for R1 and 24(4) cage

13. 6 in C1 locked in R678C1
13a. 19(4) cage at R6C1 (step 10) = {1369/1567}, no 2, 1 locked in R78C1, locked for C1 and N7
13b. Grouped X-Wing for 1 in R78C1 and 8(3) cage in N8, no other 1 in R78
13c. 2 in N7 locked in R789C3, locked for C3 -> R2C3 = 1
13d. R4C2 = 1 (hidden single in C2)

14. Max R3C67 = 13 -> min R4C6 = 4
14a. R34C6 cannot total 14 -> no 3 in R3C7, clean-up: no 7 in R6C7 (step 4)
14b. R3C67 cannot total 12 -> no 5 in R4C6

15. 21(3) cage at R6C6 = {489} (only remaining combination), 4 locked in R6C67, locked for R6

16. 45 rule on N7 3 innies R78C1 + R7C3 = 12 = {129/156} (cannot be {237} because 2,3 only in R7C3), no 3,7, clean-up: no 8 in R4C3 (step 6)

17. 9 locked in 24(4) cage at R1C2 = {2679/3489/3579/4569} (cannot be {1689/2589} because R1C23 must contain two of 4,7,9), no 1
17a. R1C23 must contain two of 4,7,9 -> no 4,7 in R12C4
17b. 1 in N2 locked in R1C56, locked for R1
17c. 11(3) cage = {128/137/146}, no 5

18. 1 in N3 locked in R3C79
18a. R23C9 + R3C7 (step 2) = {128/137/146}, no 5
18b. 4,7 of {137/146} must be in R2C9 -> no 3,6 in R2C9

19. 45 rule on N4 3 innies R4C3 + R6C12 = 18 = {369/567}, 6 locked for N4

20. 9 in N2 locked in 19(3) cage = {289/379/469}, no 5
20a. 5 in N2 locked in R12C4, locked for C4
20b. 24(4) cage at R1C2 (step 17) = {3579/4569}, no 2,8

21. R34C3 cannot total 12 -> R4C4 = 2, R45C1 = [42]
21a. R7C34 cannot total 12 -> R6C4 = 1, R7C34 = 14 = [59], R4C3 = 6 (step 6), R3C3 = 7 (cage sum), R3C6 = 4, R7C7 = 8, R6C67 = [94], R6C1 = 7, R6C2 = 5 (step 19), R9C1 = 9, R9C23 = 9 = [72]
21b. Naked pair {49} in R1C23, locked for R1, R12C4 (step 17) = {56}, locked for C4 and N2
21c. 8 in N7 locked in R8C23, locked for R8

22. 45 rule on N5 1 remaining innie R4C6 = 7, R3C7 = 6 (cage sum), R23C2 = [62], R12C4 = [65]
22a. R3C7 = 6 -> R23C9 (step 2) = [23/41]
22b. R23C9 = 5 -> R4C89 = 14 = {59}, locked for R4 and N6 -> R4C7 = 3, R4C5 = 8
22c. R4C7 = 3 -> R5C78 = 8 = {17}, locked for N6

23. R9C4 = 8 (hidden single in R9), R3C45 = [39], R2C5 = 7 (step 20), R2C7 = 9, R3C9 = 1, R2C9 = 4 (step 22a), R58C4 = [47], R9C5 = 4 (cage sum), clean-up: no 3 in 8(3) cage in N8 (prelim h)
23a. R2C7 = 9 -> R23C8 = 11 = [38], R123C1 = [385], R2C6 = 2, R1C56 = [18], R7C56 = [21], R8C5 = 5, R78C1 = [61], R8C7 = 2
23b. R5C6 = 5 (hidden single in C6)

24. R8C7 = 2 -> R78C8 = 11 = [74]

and the rest is naked singles.
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Andrew
PostPosted: Sat Jul 16, 2011 4:54 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 125 V0 "Newbie Killer" by Para (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5376:5121:5121:5121:3844:3844:1798:1798:1798:5376:2570:2570:5121:4109:3844:5903:5903:5649:5376:2570:1556:4109:4109:2839:2839:5903:5649:3611:3611:1556:1556:7967:2839:3873:5649:5649:3611:5925:5925:7967:7967:7967:3873:3873:3372:3373:3373:5925:3888:7967:5426:5426:3372:3372:3373:6199:3888:3888:2874:2874:5426:2621:4670:3373:6199:6199:4162:2874:5444:2621:2621:4670:2376:2376:2376:4162:4162:5444:5444:5444:4670:
Solution: 
+-------+-------+-------+
| 9 6 7 | 3 5 8 | 4 2 1 |
| 8 3 5 | 4 1 2 | 6 9 7 |
| 4 2 1 | 9 6 7 | 3 8 5 |
+-------+-------+-------+
| 7 5 3 | 2 8 1 | 9 6 4 |
| 2 9 6 | 7 3 4 | 1 5 8 |
| 1 4 8 | 5 9 6 | 7 3 2 |
+-------+-------+-------+
| 3 7 4 | 6 2 5 | 8 1 9 |
| 5 8 9 | 1 4 3 | 2 7 6 |
| 6 1 2 | 8 7 9 | 5 4 3 |
+-------+-------+-------+
Quote:
Para: I'm posting a second puzzle to encourage some other players on the forum to try a killer too. It's an easy killer, prob like a times killer although i have never done one of those. There will also be a V2 out later this week.
SS-score: 0.69. Estimated rating: 0.5-0.75.
For everyone new trying this, I hope you enjoyed it. Don't shy away from trying the V1.

Ronnie: Thanks Para
I think your V0 is a first, but welcome addition to the forum. Solved it on my lunch break today, about 20 minutes. Yes, it's about a Times Deadly rating, but a nice quick diversion.
I think it was enxio and Ed (maybe wrong) who first issued a challenge to each other. One would try to solve a killer and the other a samurai. You've given them a golden opportunity! :applause:

Andrew: Since Afmob had already posted a walkthrough for A125 while I was still setting up and colouring my diagrams, I thought I would try A125 V0 first.
Thanks Para for an excellent Newbie Killer! :D
This puzzle can be done by either insertion or elimination solving. Which way did you do it Ronnie? Are you going to post your walkthrough?
Since it was described as being about Times Deadly standard, I decided to do it using insertion solving. I haven't done a Times puzzle but it took me longer than I usually take for the Weekly Extreme Killers on http://www.sudoku.org.uk, which I think are supposed to be around Times Deadly level, so I'll rate A125 V0 at 0.75; it's probably easier and quicker using elimination solving.

Walkthrough by Andrew, using insertion solving:
Thanks Para for an excellent Newbie Killer! :D

This puzzle can be done by either insertion or elimination solving. Which way did you do it Ronnie? Are you going to post your walkthrough?

Since it was described as being about Times Deadly standard, I decided to do it using insertion solving. I haven't done a Times puzzle but it took me longer than I usually take for the Weekly Extreme Killers on http://www.sudoku.org.uk, which I think are supposed to be around Times Deadly level, so I'll rate A125 V0 at 0.75; it's probably easier and quicker using elimination solving.

Here is my walkthrough using insertion solving. I'll leave it hidden even after future puzzles appear (not necessary for the archive).

1. R1C789 = {124}, locked for R1 and N3

2. 23(3) cage in N3 = {689}, locked for N3

3. 6(3) cage at R3C3 = {123}

4. 23(3) cage in N4 = {689}, locked for N4

5. 24(3) cage in N7 = {789}, locked for N7

6. 45 rule on N4 3 innies R4C3 + R6C12 = 8 = {125/134}, 1 locked for N4, R6C12 must contain 4 or 5

7. 14(3) cage in N4 must contain 7 = {257/347}

8. 9 in C1 must be in R123C1, locked for N1, R123C1 = {489/579}

9. 45 rule on R1234 R4C57 15 more than R5C1 -> R5C1 = 2 (step 7), R4C57 = 17 = {89}, locked for R4
[Alternatively can use 45 rule on R1234 4 innies R4C1257 = 29 = {5789} …]
9a. R4C12 = {57} (step 7), locked for R4 and N4

10. R23C9 + R3C7 = {357}
10a. 45 rule on N3 R4C89 7 more than R3C7
10b. Max R4C89 = {46} = 10 -> R3C7 = 3, R4C89 = {46}, locked for R4 and N6, R23C9 = {57}, locked for C9

11. R3C7 = 3 -> R34C6 = 8, R4C6 must be {12} -> R3C6 = {67}

12. 13(4) cage at R6C1 must contain 4 = {1345} (cannot be {1246} because R5C1 = 2) -> R78C1 = {135}, 5 locked for C1 and N7 -> R4C12 = [75]

13. R123C1 (step 8) = {489}, locked for C1 and N1 -> R6C2 = 4 (hidden single in N4)

14. R9C1 = 6 (only place in C1), R9C23 = {12}, locked for R9 and N7 -> R78C1 = {35}, R6C1 = 1, R4C3 = 3
14a. Naked pair {12} in R39C3, locked for C3

15. R7C3 = 4 (only place in N7), R67C4 = 11

16. R5C78 cannot contain 2,4,6
16a. R4C7 = {89} -> R5C78 = 6,7 can only be {15}, locked for R5 and N6, R4C7 = 9, R4C5 = 8

17. 45 rule on N6 1 remaining innie R6C7 = 7
17a. R6C6 + R7C7 = 14 = [68/95]

18. 10(3) cage in N1 = {136/235} (cannot be {127} which clashes with R3C3) -> R2C2 = 3, R2C3 + R3C2 = [52/61], naked pair {12} in R3C23, locked for R3
18a. R1C23 = [67/75/76]

19. 45 rule on N5 2 remaining innies R6C46 = 11 = [56], R7C4 = 6 (step 15), R7C7 = 8 (step 17a), R2C7 = 6, R2C3 = 5, R3C2 = 2, R3C3 = 1, R9C23 = [12], R4C46 = [21], R23C9 = [75], R3C6 = 7 (step 11)

20. R1C23 = {67} = 13 -> R12C4 = 7 = [34]

21. R3C1 = 4 (hidden single in C1), R3C5 = 6 (only place in R3)

22. R2C56 = {12} = [12], R3C4 = 9, R23C8 = [98], R12C1 = [98], R1C56 = [58]

23. 45 rule on N9 2 outies R89C6 = 12 = {39} -> R57C6 = [45] (only places in C6), R78C5 = 6 = [24], R78C1 = [35]

24. R8C4 = 1 (only place in N8), R9C45 = [87], R5C4 = 7, R56C5 = {39}

25. R89C6 = 12 -> R9C78 = 9 = {45}

26. 10(3) cage in N9 = {127/136} -> R7C8 = 1, R5C78 = [15], R9C78 = [54], R1C789 = [421], R8C78 = [27], R4C89= [64]

27. Naked pair {89} in R8C23, locked for R8 -> R7C2 = 7, R7C9 = 9, R1C23 = [67], R5C3 = 6 (hidden single in N4)

28. R89C9 = [63], R89C6 = [39]

29. R6C8 = 3 (remaining single in C8), R56C5 = [39], R6C3 = 8, R5C2 = 9, R8C23 = [89], R56C9 = [82]
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Andrew
PostPosted: Sat Jul 16, 2011 5:18 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 125 V2 by Para (October 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge: 
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3328:3073:3073:3073:6148:6148:3846:3846:3846:3328:4362:4362:3073:3597:6148:4367:4367:4369:3328:4362:4116:3597:3597:3095:3095:4367:4369:4891:4891:4116:4116:4895:3095:4129:4369:4369:4891:4389:4389:4895:4895:4895:4129:4129:4908:4141:4141:4389:3376:4895:4658:4658:4908:4908:4141:4151:3376:3376:4154:4154:4658:4669:2622:4141:4151:4151:4930:4154:3396:4669:4669:2622:4936:4936:4936:4930:4930:3396:3396:3396:2622:
Solution: 
+-------+-------+-------+
| 3 5 4 | 2 7 9 | 1 8 6 |
| 9 2 7 | 1 6 8 | 5 3 4 |
| 1 8 6 | 3 5 4 | 2 9 7 |
+-------+-------+-------+
| 4 7 2 | 8 9 6 | 3 5 1 |
| 8 3 5 | 4 2 1 | 6 7 9 |
| 6 1 9 | 7 3 5 | 4 2 8 |
+-------+-------+-------+
| 2 4 1 | 5 8 7 | 9 6 3 |
| 7 9 3 | 6 1 2 | 8 4 5 |
| 5 6 8 | 9 4 3 | 7 1 2 |
+-------+-------+-------+
Quote:
Para: I have been thinking about which puzzle to post as a V2 for a while. I have 2 fun puzzles and I have finally decided to post this puzzle. I can also post the other one if no-one has a problem with it. I enjoyed solving both puzzles, but decided to post the one that felt more difficult.
SS-Score: 1.62. Estimated Rating: 1.5.
Enjoy.

Afmob: Great Killer, Para! It's quite unusual that you still had to use an advanced move (... in former version of my wt) to finish this Assassin even though almost all placements were made.
Edit: Simplified wt. I thought that the Killer was finished so as always I used SimpleSudoku to verify if only (Naked and Hidden) Singles were needed to solve it but it needed more...
Rating: 1.25.

udosuk: My solving path was quite similar to Afmob's, and I had no time to write the walkthrough anyway, but just like to comment that I did his most critical steps (#5 & #10) quite differently ...
This was how I did the critical steps ...
So in hindsight this is not that worthy as a V2. :ugeek:
Perhaps the other one will be more worthy. Please post it Para! :alien:
(Archive note. The other version wasn't posted.)

Andrew: Thanks Para for a challenging V2. I managed to finish it yesterday but have only gone through Afmob's walkthrough and udosuk's alternative steps today.
This puzzle clearly has a fairly narrow solving path. udosuk commented that his path was similar to Afmob's and so was mine. There are clearly two critical points, one in the middle and the other at the end.
Although they aren't technically difficult the two breakthroughs, whichever way they are done (and I missed the simpler way for the second one), are IMHO difficult to spot so I can't accept udosuk's overall comment; I felt that it was a worthy V2, significantly harder than A125.
I'll rate A125 V2 at 1.5 the way I solved it ... However if I'd found the simpler way ... then it's clearly 1.25.

Walkthrough by Afmob:
Great Killer, Para! It's quite unusual that you still had to use an advanced move (XY-Wing in former version of my wt) to finish this Assassin even though almost all placements were made.

Edit: Simplified wt. I thought that the Killer was finished so as always I used SimpleSudoku to verify if only (Naked and Hidden) Singles were needed to solve it but it needed more. So I immediately looked for cells with two candidates for chains instead of (technical) easier cage blockers to crack it.

A125 V2 Walkthrough:

1. N1247
a) 24(3) = {789} locked for N2
b) 14(3) = {356} locked for N2
c) Innies N12 = 10(2) = [64/82/91]
d) Innies N4 = 9(3) <> 7,8,9
e) Innies N7 = 10(3) <> 8,9
f) Innies N47 = 3(2) = {12} locked for C3
g) 16(3) @ N1 = {169/178/259/268} <> 3,4 because R4C3 = (12); R4C4 <> 1,2

2. N3689
a) Innies N36 = 6(2) = {15/24}
b) Innies N6 = 10(3) <> 8,9
c) Innies N8 = 10(3) <> 8,9
d) Innies N89 = 14(2) = [59/68]
e) 13(3) = 5{17/26} since R7C3 = (12) and R7C4 = (56) -> 5 locked for C4; R6C4 = (567)
f) Innies N8 = 10(3) = {136/145/235} <> 7 since R7C4 = (56); R89C6 <> 5,6
g) 18(3) @ N6: R6C6 <> 1,2,3 because R67C7 <= 14

3. C456
a) 14(3) = {356} -> 5 locked for C5
b) 12(3): R4C6 <> 1,2 since R3C67 <= 9
c) 12(3) = {129/147/156/246/345} <> 8 because R3C67 = (1245); R4C6 = (3679)

4. N1247
a) Outies N47 = 26(3+1): R3C3 <> 9 since R467C4 >= 18
b) Innies N12 = 10(2) <> 1
c) 1 locked in R12C4 @ N2 for C4+12(4)
d) Innies N1 = 15(3): R1C23 <> 6 since R3C3 = (68)
e) 12(4) = {1245} -> 5 locked for R1+N1
f) 17(3) @ N1 <> 3 because {368} blocked by R3C3 = (68)
g) 3 locked in 13(3) @ N1 = 3{19/28/46} <> 7 -> 3 locked for C1
h) Innies N7 = 10(3) = 1{27/45} -> 1 locked for N7

5. R1+C456 !
a) ! Hidden Killer pair (36) in R1C1 + 15(3) for R1 -> R1C1 = (36) and
15(3) = {168/267/348} <> 9
b) 9 locked in R1C56 @ R1 for N2
c) Innies+Outies C6789: -1 = R1C5 - R57C6 -> R57C6 <= 10 must have one of (1234)
d) ! Killer quad (1234) locked in R389C6 + R57C6 for C6
e) 1,2,3,4 locked in 19(5) @ N5 = {12349} locked for N5
f) 12(3) = 4{17/26} because R3C6 = (24) -> 4 locked for R3
g) 5 locked in R6C46 @ N5 for R6

6. N3689
a) Innies N36 = 6(2) = {24} locked for C7
b) 12(3) = {246} -> R4C6 = 6; 2 locked for R3
c) 18(3) @ N6 = 9{27/45} -> R7C7 = 9
d) Innie N89 = R7C4 = 5
e) R6C4 = 7 -> R7C3 = 1, R4C4 = 8, R3C3 = 6, R4C3 = 2
f) R6C6 = 5 -> R6C7 = 4, R3C7 = 2, R1C1 = 3
g) 15(3) = {168} locked for R1+N3
h) 17(3) = {359} locked for N3; 9 also locked for C8
i) R3C9 = 7, R2C9 = 4, R3C4 = 3, R3C5 = 5, R2C5 = 6, R3C8 = 9

7. N469
a) Innies N4 = 7(2) = {16} locked for R6+N4
b) 17(4) = {1457} -> 1,5 locked for R6+N6
c) 19(3) @ N6 = {289} locked for N6
d) 10(3) = 3{16/25} -> 3 locked for N9
e) 18(3) = 6{48/57} -> 6 locked for N9
f) 10(3) = {235} locked for C9+N9
g) 13(4) = {1237} -> 1,7 locked for R9+N9 and R89C6 = {23} locked for C6+N8

8. N478
a) 19(3) @ N8 = {469} locked for N8
b) 8 locked in 19(3) @ R9 for N7 -> 19(3) = 8{29/56}
c) 3 locked in 16(3) @ N7 = 3{49/67}
d) 17(3) = 5{39/48} -> 5 locked for N4
e) 5 locked in R89C1 @ C1 for N7
f) 19(3) @ N7: R9C1 <> 6 because 5 only possible there
g) Hidden Single: R6C1 = 6 @ C1
h) R6C2 = 1, R3C2 = 8, R3C1 = 1 -> R2C1 = 9, R2C3 = 7, R2C2 = 2
i) Naked triple (469) locked in R9C245 for R9
j) R9C3 = 8

9. R789 !
a) 16(3) @ N8 = {178} -> R8C5 = 1; {78} locked for R7
b) 16(4) = 16{27/45} -> R8C1 = (57)
c) ! 16(3) @ N7: R7C2 <> 6 since {367} blocked by R8C169 = (2357)
d) 16(3) @ N7 = {349} because (67) only possible @ R8C2
-> {349} locked for N7 and 9 also locked for R8

10. Rest is singles.

Rating: 1.25. I used a Killer quad.
udosuk's Critical Steps:
My solving path was quite similar to Afmob's, and I had no time to write the walkthrough anyway, but just like to comment that I did his most critical steps (#5 & #10) quite differently, and didn't need any killer pairs/triples/quads, innie-outies or xy-wing.

This was how I did the critical steps

5. Outies @ c789: r34689c6=20 can't contain {1234}
=> r46c6 can't have {34}
=> {1234} @ n5 locked @ 19/5={12349}
...


10. An xy-wing solves it easily, but it feels quite strange to use this (intermediate level) vanilla technique when there are also other simple killer moves available, e.g.

r8c478 from {4689}
=> r8c23 can't be {49}=13
=> r7c2=16-r8c23 can't be 3

-or-

r8c169 from {2357}
=> r8c23 can't be {37}=10
=> r7c2=16-r8c23 can't be 6
...


Any human players with a reasonable IQ shouldn't find these moves any harder than xy-wing.

So in hindsight this is not that worthy as a V2. :ugeek:

Perhaps the other one will be more worthy. Please post it Para! :alien:
Walkthrough by Andrew:
Thanks Para for a challenging V2. I managed to finish it yesterday but have only gone through Afmob's walkthrough and udosuk's alternative steps today.

This puzzle clearly has a fairly narrow solving path. udosuk commented that his path was similar to Afmob's and so was mine. There are clearly two critical points, one in the middle and the other at the end.

Although they aren't technically difficult the two breakthroughs, whichever way they are done (and I missed the simpler way for the second one), are IMHO difficult to spot so I can't accept udosuk's overall comment; I felt that it was a worthy V2, significantly harder than A125.

I'll rate A125 V2 at 1.5 the way I solved it, because my final breakthrough is a very short chain. However if I'd found the simpler way, step 9c in Afmob's edited walkthrough or as given in udosuk's comments, then it's clearly 1.25.

Here is my walkthrough. When I found the first breakthrough at step 17, I removed a few steps that didn't lead to anything helpful but haven't otherwise tried to optimise it. Thanks Afmob for pointing out a couple of typos and a simpler explanation for step 15.

Prelims

a) 24(3) cage in N2 = {789}, locked for N2
b) 19(3) cage in N4 = {289/379/469/478/568}, no 1
c) 19(3) cage in N6 = {289/379/469/478/568}, no 1
d) R9C123 = {289/379/469/478/568}, no 1
e) 19(3) cage in N8 = {289/379/469/478/568}, no 1
f) R789C9 = {127/136/145/235}, no 8,9
g) 12(4) cage at R1C2 = {1236/1245}, no 7,8,9
h) 13(4) cage at R8C6 = {1237/1246/1345}, no 8,9
i) 19(5) cage in N5 must contain 1, locked for N5

1. 45 rule on N2 3 innies R12C4 + R3C6 = 7 = {124}, locked for N2

2. 45 rule on N47 2 innies R47C3 = 3 = {12}, locked for C3
2a. Max R4C3 = 2 -> min R3C3 + R4C4 = 14, no 2,3,4

3. 45 rule on N12 2 innies R3C36 = 10 = [64/82/91], no 5,7

4. 45 rule on N36 2 innies R36C7 = 6 = {15/24}

5. 45 rule on N89 2 innies R7C47 = 14 = {59/68}

6. 45 rule on N4 3 innies R4C3 + R6C12 = 9 = {126/135/234}, no 7,8,9

7. 45 rule on N6 3 innies R4C89 + R6C7 = 10 = {127/136/145/235}, no 8,9

8. 45 rule on N7 3 innies R78C1 + R7C3 = 10 = {127/136/145/235}, no 8,9

9. 45 rule on N9 1 innie R7C7 = 2 outies R89C6 + 4
9a. Min R89C6 = 3 -> no 5,6 in R7C7, clean-up: no 8,9 in R7C4 (step 5)
9b. Max R89C6 = 5, no 5,6,7 in R89C6
9c. Max R67C7 = 14 -> min R6C6 = 4

10. 12(3) cage at R3C6 = {129/147/156/246/345} (cannot be {138/237} because 3,7,8 only in R4C6), no 8
10a. 3,6,7,9 only in R4C6 -> R4C6 = {3679}

11. Max R7C34 = 8 -> min R6C4 = 5
11a. Min R7C34 = 6 -> max R6C4 = 7
11b. 13(3) cage at R6C4 = {157/256}, 5 locked in R67C4, locked for C4
11c. 5 in N2 locked in R23C5, locked for C5
11d. 5 in N8 locked in R7C46, locked for R7

12. 3 in C4 locked in R3589C4
12a. 45 rule on C1234 4 innies R3589C4 = 22 = {2389/3469/3478}, no 1
12a. 1 in C4 locked in R12C4, locked for N2 and 12(4) cage at R1C2, clean-up: no 9 in R3C3 (step 3)

13. 45 rule on N1 3 innies R1C23 + R3C3 = 15
13a. 12(4) cage at R1C2 = {1245} (cannot be {1236} because R1C23 + R3C3 cannot be {36}6}, no 3,6, 5 locked in R1C23, locked for R1 and N1
13b. R1C23 + R3C3 = {258/456}

14. R1C789 = {168/267/348} (cannot be {249} which clashes with R1C23), no 9
14a. Hidden killer pair 3,6 in R1C1 and R1C789 -> R1C1 = {36}
14b. 9 in R1 locked in R1C56, locked for N2

15. R1C123 = {139/238/346} (cannot be {148/247} because R1C1 only contains 3,6), no 7, 3 locked for C1 and N1

16. R78C1 + R7C3 (step 7) = {127/145}, no 6, 1 locked for N7
16a. 5 of {145} must be in R8C1 -> no 4 in R8C1

17. 45 rule on C6789 2 innies R57C6 = 1 outie R1C5 + 1
17a. R1C5 = {789} -> R57C6 = 8,9,10 must contain one of 1,2,3,4
17b. Killer quad 1,2,3,4 in R3C6, R57C6 and R89C6, locked for C6
17c. Min R34C6 = 8 -> max R3C7 = 4, clean-up: no 1 in R6C7 (step 4)

18. 45 rule on N5 4 innies R46C46 = 26 = {5678} (only remaining combination), locked for N5, 5 locked in R6C46, locked for R6, clean-up: no 1 in R3C7 (step 4)
18a. Naked pair {24} in R3C67, locked for R3, R4C6 = 6 (cage sum)
18b. Naked pair {24} in R36C7, locked for C7

19. 18(3) cage at R6C6 = {279/459} -> R7C7 = 9, R7C4 = 5 (step 5), R6C46 = [75], R7C3 = 1 (step 11b), R6C7 = 4, R3C67 = [42], R4C34 = [28], R3C3 = 6, R1C1 = 3, R3C45 = [35], R2C5 = 6
19a. Naked pair {12} in R12C4, locked for C4 and 12(4) cage at R1C2
19b. Naked pair {45} in R1C23, locked for R1 and N1
19c. R1C789 (step 14) = {168} (only remaining combination), locked for R1 and N3 -> R12C4 = [21]
19d. Naked pair {79} in R3C89, locked for R3
19e. Naked pair {18} in R3C12, locked for N1
19f. R2C6 = 8 (hidden single in R2)

20. 17(3) cage in N3 = {359} (only remaining combination) -> R3C8 = 9, R2C78 = {35}, locked for R2 -> R23C9 = [47]
20a. R23C9 = [47] = 11 -> R4C89 = 6 = {15}, locked for R4 and N6

21. R4C3 + R6C12 (step 6) = {126} (only remaining combination), R6C12 = {16}, locked for R6 and N4

22. 6 in C4 locked in 19(3) cage = {469}, locked for N8

23. R7C7 = R89C6 + 4 (step 9)
23a. R7C7 = 9 -> R89C6 = 5 = {23}, locked for C6, N8 and 13(4) cage at R8C6 -> R7C56 = [87], R8C5 = 1, R1C56 = [79], R5C6 = 1
23b. R89C6 = {23} -> R9C78 = {17} (prelim h), locked for R9 and N9

24. R789C9 = {235} (only remaining combination), locked for C9 and N9 -> R4C89 = [51], R2C78 = [53]

25. 19(3) cage in N6 = {289} (only remaining combination) -> R6C8 = 2, R56C9 = {89}, locked for C9 and N6 -> R1C9 = 6
25a. R5C5 = 2 (hidden single in R5)

26. 8 in R9 locked in R9C123, locked for N7
26a. R9C123 = {289/568}, no 3,4
26b. 3 in N7 locked in 16(3) cage = {349/367}, no 2,5
26c. 4 in R9 locked in R9C45, locked for N8

27. 19(3) cage in N4 = {379/478}, no 5, 7 locked for N4
27a. 3 of {379} must be in R4C2 -> no 9 in R4C2
27b. 8 of {478} must be in R5C1 -> no 4 in R5C1
27c. 17(3) cage = {359/458}
27d. 8 of {458} must be in R6C3 -> no 8 in R5C23

28. 5 in C1 locked in R89C1, locked for N7
28a. R9C123 (step 26a) = {289/568}
28b. 5 of {568} must be in R9C1 -> no 6 in R9C1

29. R6C1 = 6 (hidden single in C1), R6C2 = 1, R3C12 = [18], R2C1 = 9 (step 15), R2C23 = [27]
29a. 19(3) cage in N4 (step 27) = {478} (only remaining combination) -> R5C1 = 8, R4C12 = {47}, locked for R4 and N4, R4C7 = 3, R46C5 = [93], R5C4 = 4, R56C9 = [98], R6C3 = 9, R9C3 = 8, R9C5 = 4

30. 16(3) cage in N7 (step 26b) = {349/367}
30a. 7,9 only in R8C2 -> R8C2 = {79}

31. R78C1 + R7C3 (step 16) = {127/145}
31a. R78C1 cannot be [45] => R7C2 + R8C3 = [63] -> R7C12 clashes with R7C8
31b. -> R78C1 = [27]

and the rest is naked singles

In case anyone is wondering about the XY-Wing mentioned by Afmob and udosuk, I didn't spot it either and asked Afmob to show it to me.

In the position after my step 30. R7C1, R7C9 and R8C3 form the XY-Wing. R7C9 and/or R8C3 must be 3, CPE no 3 in R7C2 + R8C9. That then gives a hidden single in R7. IMHO quite a tricky XY-Wing because it uses N7 as one arm of the wing.
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