I’ll start with as many of my previous steps as possible.
Prelims
a) R12C1 = {29/38/47/56}, no 1
b) R12C5 = {29/38/47/56}, no 1
c) 9(3) cage at R3C4 = {126/135/234}, no 7,8,9
d) 9(3) cage at R4C1 = {126/135/234}, no 7,8,9
e) 29(4) cage at R6C1 = {5789}
f) 12(4) cage at R7C3 = {1236/1245}, no 7,8,9
g) 33(5) cage at R1C2 = {36789/45789}, no 1,2
1. 12(4) cage at R7C3 = {1236/1245}, CPE no 1,2 in R7C6
2. 1 in R1 only in R1C6789, CPE no 1 in R23C7
3. 1 in N1 only in R2C3 + R3C123, locked for 19(5) cage at R2C3, no 1 in R4C3
4. 45 rule on N9 3 innies R789C7 = 11 = {128/137/146/236/245}, no 9
5. 34(6) cage at R5C7 must contain 9, locked for N6
6. 45 rule on R789 3 innies R7C127 = 1 outie R6C6 + 21
6a. Max R7C127 = 24 -> max R6C6 = 3
6b. Min R7C127 = 22, no 1,2,3,4 in R7C7
6c. R7C127 = 22,23,24 = {589/679/689/789}, 9 must be in R7C12, locked for R7, N7 and 29(4) cage at R6C1, no 9 in R6C12
7. R789C7 (step 4) = {128/137/146/236/245}
7a. R7C7 = {5678} -> no 5,6,7,8 in R89C7
8. 19(4) cage at R6C6 cannot contain more than two of 1,2,3,4, R6C6 = {123}, R8C7 = {1234} -> no 1,2,3,4 in R78C6
9. Hidden killer quad 1,2,3,4 in 12(4) cage at R7C3 and 17(3) cage at R7C8 for R7, 12(4) cage cannot contain more than three of 1,2,3,4, 17(3) cage only contains one of 1,2,3,4 -> 12(4) cage must contain three of 1,2,3,4 in R7 and 17(3) cage must contain the other one in R7 -> R7C345 = {1234}
9a. 12(4) cage contains one of 5,6 -> R8C5 = {56}
9b. 12(4) cage = {1236/1245}, 1,2 locked for R7
9c. 17(3) cage contains one of 3,4 in R7C89 = {359/368/458/467}, no 1,2
9d. One of 3,4 in R7C89 -> no 3,4 in R8C8
[And for completeness in N9, at this stage …]
9e. 17(3) cage at R8C9 = {179/269/278/359/467} (cannot be {368/458} which clash with 17(3) cage at R7C8)
10. R12C5 = {29/38/47} (cannot be {56} which clashes with R8C5), no 5,6
11. 9(3) cage at R3C4 = {126/135} (cannot be {234} which clashes with R12C5), no 4, 1 locked for R3 and N2
12. R2C3 = 1 (hidden single in N1)
12a. 1 in R7 only in R7C45, locked for N8
[Now a modified version of one of my original steps, to allow me to use some more of my original steps.]
13. 19(5) cage at R2C3 = {12349/12358/12367/12457/13456}
13a. 2 or 3 of {12349/12358} must be in R4C3 (R3C123 cannot contain both of 2,3 which would clash with 9(3) cage at R4C3) -> no 8,9 in R4C3
14. 9 in R4 only in R4C456, locked for N5
14a. 8 in N4 only in R5C23 + R6C123, CPE no 8 in R6C45
15. 9 in N4 only in 33(7) cage at R5C2 = {1234689/1235679}
15a. R6C6 “sees” all of 33(7) cage except for R5C23, R6C6 = {123}, 33(7) cage contains all of 1,2,3 -> one of 1,2,3 in 33(7) cage must be in R5C23
15b. Killer triple 1,2,3 in 9(3) cage at R4C1 and R5C23, locked for N4
15c. Two of 1,2,3 in 33(7) cage must be in N5, killer triple 1,2,3 in 33(7) cage and R6C6, locked for N5
[Or steps 15b and 15c can be considered to be
Double killer triple 1,2,3 in 9(3) cage at R4C1, 33(7) cage at R5C2 and R6C6, locked for N45.]
16. 19(5) cage at R2C3 = {12457/13456} (cannot be {12349/12358/12367} which clash with 9(3) cage at R3C4 now that 2,3 are no longer in R4C3), no 8,9, CPE no 4,5 in R1C3
16a. 7 of {12457} must be in R3C123 (R3C123 cannot be {245} which clashes with 9(3) cage at R3C4) -> no 7 in R4C3
16b. Killer pair 2,3 in 19(5) cage and 9(3) cage, locked for R3
16c. 8,9 in R3 only in R3C789, locked for N3
[Now I’m getting more into new steps.]
16d. 7 in N4 only in R5C23 + R6C123, CPE no 7 in R6C45
17. 19(5) cage at R2C3 (step 16) = {12457/13456}
17a. 7 of {12457} only in R3C123, 4 of {13456} must be in R3C123 (R3C123 cannot be {356} which clashes with 9(3) cage at R3C4) -> R3C123 must contain at least one of 4,7
17b. R12C1 = {29/38/56} (cannot be {47} which clashes with R3C123), no 4,7 in R12C1
[nd’s solving path for #10 was a bit more direct than my one because he used the following step, so I’ll use it here.]
18. 45 rule on R3 3 innies R3C789 = 1 remaining outie R4C3 + 18
18a. R4C3 = {456} -> R3C789 = 22,23,24 and must contain 8,9 (step 16c) = {589/689/789}, no 4
19. 4 in R3 only in R3C123, locked for N1 and 19(5) cage at R2C3, no 4 in R4C3
19a. R4C3 = {56} -> R3C789 (step 18a) = 23,24 = {689/789}, no 5
19b. R4C3 + R3C789 = 5+{689}/6+{789}, CPE no 6 in R3C123
20. 45 rule on N12 2 innies R12C6 = 1 outie R4C3 + 7
20a. Min R4C3 = 5 -> min R12C6 = 12, no 2 in R12C6
21. 45 rule on N12 3(2+1) outies R23C7 + R4C3 = 17
21a. R4C3 = {56} -> R23C7 = 11,12 must contain one of 2,3,4,5 -> R2C7 = {2345}
22. 45 rule on R123 2 innies R23C9 = 1 outie R4C3 + 8
22a. R4C3 = {56} -> R23C9 = 13,14 -> R2C9 = {4567}
23. 45 rule on R123 3(2+1) outies R4C3 + R45C9 = 16
23a. Max R4C3 = 6 -> min R45C9 = 10, no 1
24. 45 rule on N4578 4(3+1) outies R489C7 + R4C8 = 1 innie R4C3 + 6
24a. Max R4C3 = 6, min R489C7 = 7 (R489C7 cannot total 6, IOU) -> max R4C8 = 5
24b. Max (R4C3 – R4C8) = 5 -> max R489C7 = 10 (R489C7 cannot total 11 which would clash with R789C7, CCC) -> no 8 in R4C7
25. 35(6) cage at R4C4 must contain 8, locked for N5
26. 33(5) cage at R1C2 = {36789/45789}, 19(5) cage at R2C3 (step 16) = {12457/13456}
26a. 45 rule on N1 3(2+1) outies R12C4 + R4C3 = 18
26b. Consider placements for R4C3
R4C3 = 5 => R3C123 = {247}, locked for R3 and N1, R12C4 = 13 must contain 7 for 33(5) cage = {67}
or R4C3 = 6 => R3C123 = {345}, 2 in N1 only in R12C1 = {29}, locked for N1, R12C4 = 12 must contain 9 for 33(5) cage = {39}
-> R12C4 = {39/67}, no 4,5,8
26c. R12C4 = {39/67}, 9(3) cage at R3C4 = {126/135}, killer pair 3,6 locked for N2; clean-up: no 8 in R12C5
26d. R12C4 = {39/67}, R12C5 = {29/47}, killer pair 7,9 locked for N2
[Also, in the same way as step 19b …]
26e. R12C4 + R4C3 = {67}+5/{39}+6, CPE no 6 in R4C4
27. 33(5) cage at R1C2 = {36789} (only remaining combination), no 5, 8 locked for N1, clean-up: no 3 in R12C1
27a. Extending step 26b slightly
R4C3 = 5 => R3C123 = {247} => R12C1 = {56}
or R4C3 = 6 => R3C123 = {345} => R12C1 = {29}
-> R12C1 + R4C3 = {56}+5/{29}+6, CPE no 6 in R1C3 + R45C1
28. 8 in N2 only in R12C6, locked for C6 and 24(4) cage at R1C6, no 8 in R3C7
28a. 24(4) cage at R1C6 contains 8 = {2589/3489/4578} (cannot be {3678} because 6,7 only in R3C7), no 6
29. 9 in N7 only in R7C12
29a. 45 rule on N7 4 innies R7C123 + R9C3 = 24 = {2589/2679/3489/3579/4569}
29b. 2 of {2589/2679} must be in R7C3 -> no 2 in R9C3
30. 45 rule on R89 4 innies R8C5678 = 24 = {1689/2589/2679/3579/3678/4569/4578} (cannot be {3489} because R8C5 only contains 5,6)
30a. When R8C7 = 3 or 4, then the same number must be in R8C345, therefore from 12(4) cage at R8C3 -> R8C57 = [54/63] when R8C7 = 3 or 4
30b. R8C5678 = {1689/2589/2679/3678/4569/4578} (cannot be {3579} because of step 30a)
30c. 5,6 of R8C5678 = {1689/2589/2679/3678/4578} must be in R8C5, 5 of {4569} must be in R8C5 (because of step 30a) -> no 5 in R8C6
31. 12(4) cage at R7C3 = {1236/1245}
31a. 17(3) cage at R7C8 (step 9c) = {359/368/458/467}
31b. 7 of {467} must be in R8C8 (cannot be {47}6 which clashes with 12(4) cage), no 7 in R7C89
32. 33(7) cage at R5C2 = {1234689/1235679} contains 6
32a. Consider combinations for 35(6) cage at R4C6 = {146789/236789/245789/345689}
35(6) = {146789/236789/345689} contain 6 -> either 6 in N5 so 33(7) and 35(6) cages contain both 6s for N45 or 6 in R4C7 => R4C3 = 5, R45C9 = 11 (step 23), no 2 in R45C9
or 35(6) cage = {245789}, 2 locked for N6
-> no 2 in R45C9
33. Hidden killer pair 1,2 in 35(6) cage at R4C6 and 34(6) cage at R5C7 for N6, each cage can only contain one of 1,2 -> 35(6) cage = {146789/236789/245789}, 34(6) cage = {136789/145789/235789/245689}
34. 35(6) cage at R4C6 (step 33) = {146789/236789/245789}, 33(7) cage at R5C2 (step 32) = {1234689/1235679} contains 6
34a. Consider combinations for 9(3) cage at R4C1
9(3) cage = {126}, locked for N4 => 6 in 33(7) cage must be in N5 => 35(6) cage cannot be {236789} because 2,3,6 only in R4C78
or 9(3) cage = {135}, locked for N4 -> R4C3 = 6 => 6 in 33(7) cage must be in N5 => 35(6) cage cannot contain 6
or 9(3) cage = {234} => 35(6) cage cannot be {236789} because R4C78 = {23} clashes with 9(3) cage, ALS block
-> 35(6) cage at R4C6 = {146789/245789}, no 3
35. 8 in N8 only in R8C4 + R9C45, CPE no 8 in R9C3
35a. 8 in N8 must be in either 14(3) cage at R8C4 = {248} or in 14(3) cage at R9C5 = {158/248}
[I can see a contradiction move here but, since these days I prefer forcing chains to contradiction moves I’ll try one …]
35b. 14(3) cage at R8C4 = {239/248/257/347/356}
35c. 45 rule on N7 2 outies R6C12 = 2 innies R79C3 + 5
35d. Consider placement of 2 in R7
R7C3 = 2, min R6C12 = {57} = 12 => min R79C3 = 7 => min R9C3 = 5 => 14(3) cage at R8C4 = {257/347/356}
or 2 in R7C45 => 14(3) cage at R8C4 = {347/356}
-> 14(3) cage at R8C4 = {257/347/356}, no 8,9
36. R9C5 = 8 (hidden single in N8), R9C67 = 6 = [24/42/51]
37. R8C6 = 9 (hidden single in N8)
37a. 9 in N9 only in 17(3) cage at R8C9 = {179/269/359}, no 4,8
37b. 17(3) cage at R7C8 (step 9c) = {368/458/467}
37c. R789C7 (step 4) = {128/137/236/245} (cannot be {146} which clashes with 17(3) cage at R7C8)
38. 7 in R7 only in R7C1267
38a. R8C6 = 9 -> R67C6 + R8C7 = 10
38b. R6C6 + R7C127 (step 6c) = 1+{589}/1+{679}/2+{689}/3+{789} -> R6C6 + R7C6 + R8C7 = [172/154/271/352/361] -> R8C7 = {124}
39. R789C7 (step 37c) = {128/245}, no 6,7, 2 locked for C7 and N9, clean-up: no 6 in 17(3) cage at R8C9 (step 37a)
39a. Killer pair 1,5 in R789C7 and 17(3) cage at R8C9, locked for N9
40. Hidden killer pair 5,7 in R7C127 and R7C6 for R7, neither can contain both of 5,7 -> R7C127 (step 6c) = {589/679/789} = 22,24, R7C6 = {57}, R6C6 (step 6) = {13}
40a. R678C6 are all odd, cage sum odd -> R8C7 must be even = {24}
41. 2 in N5 only in R5C56 + R6C45, locked for 33(7) cage at R5C2, no 2 in R5C23
42. 2 in N4 only in 9(3) cage at R4C1 = {126/234}, no 5
42a. 6 of {126} must be in R4C2 -> no 1 in R4C2
43. Consider combinations for R789C7 (step 39) = {128/245}
43a. R789C7 = {128} = [821] => R9C6 = 5
or R789C7 = {245} => R7C7 = 5
-> 5 in R7C7 or R9C6, CPE no 5 in R7C6
[I think the puzzle is now almost cracked.]
44. R7C6 = 7, R8C6 = 9 -> R6C6 + R8C7 = 3 -> R6C6 = 1, R8C7 = 2, clean-up: no 4 in R9C6 (step 36)
45. 33(7) cage at R5C2 (step 32) = {1234689/1235679} -> R5C2 = 1
46. 9(3) cage at R4C1 (step 42) = {234} (only remaining combination), locked for N4
47. 29(4) cage at R6C1 = {5789}, 7 locked for R6 and N4
48. Naked triple {589} in R7C127, locked for R7
49. 6 in R7 only in R7C89, locked for N9
50. 14(3) cage at R8C4 (step 35d) = {347/356} (cannot be {257} which clashes with R9C6), no 2
50a. 7 of {347} must be in R9C3, 5 or 6 of {356} must be in R9C3 (R89C4 cannot be {56} which clashes with R8C5) -> R9C3 = {567}
50b. 14(3) cage = {347/356}, 3 locked for C4 and N8, clean-up: no 9 in R12C4 (step 26b)
51. Naked pair {67} in R12C4, locked for C4, N2 and 33(5) cage at R1C2, no 6,7 in R1C23 + R2C2, clean-up: no 4 in R12C5
51a. Naked pair {29} in R12C5, locked for C5 and N2
51b. 9(3) cage at R3C4 = {135} (only remaining combination), locked for R3 and N2
51c. Naked pair {48} in R12C6, locked for C6 and 24(4) cage at R1C6, no 4 in R2C7
52. Naked triple {247} in R3C123, locked for R3 and N1 -> R3C7 = 9, R2C7 = 3 (cage sum), clean-up: no 9 in R12C1
52a. R2C3 = 1, R3C123 = {247} = 13 -> R4C3 = 5 (cage sum), R4C6 = 6
53. Naked pair {78} in R6C12, locked for R6, N4 and 29(4) cage at R6C1, no 8 in R7C12
53a. Naked pair {69} in R56C3, locked for C3
53b. Naked pair {59} in R6C12, locked for R6 and N7 -> R7C7 = 8, R9C7 = 1 (step 39), R9C6 = 5 (step 36)
54. R8C8 = 7 -> R7C89 = 10 = {46}, locked for R7 and N9, R7C345 = [321], R8C5 = 6
55. 33(7) cage at R5C2 (step 32) = {1235679} (only remaining combination) -> R5C5 = 7, R4C5 = 4, R4C7 = 7
56. Naked pair {23} in R4C12, locked for R4 and N4 -> R5C1 = 4, R4C89 = [18]
and the rest is naked singles.
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