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
theneither 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.