Prelims
a) R1C12 = {15/24}
b) R12C9 = {16/25/34}, no 7,8,9
c) R4C56 = {59/68}
d) R5C89 = {18/27/36/45}, no 9
e) R6C56 = {15/24}
f) R6C89 = {79}
g) R78C2 = {19/28/37/46}, no 5
h) R89C8 = {49/58/67}, no 1,2,3
i) 11(3) cage in N3 = {128/137/146/236/245}, no 9
j) 27(4) cage at R1C6 = {3789/4689/5679}, no 1,2
1. Naked pair {79} in R6C89, locked for R6 and N6, clean-up: no 2 in R5C89
2. 45 rule on R6789 2 innies R6C23 = 8 = {26/35}
2a. Killer pair 2,5 in R6C23 and R6C56, locked for R6
3. 45 rule on N3 1 outie R1C6 = 1 innie R3C9, no 1,2 in R3C9
3a. 45 rule on N3 4 innies R1C78 + R2C8 + R3C9 = 27 = {3789/4689/5679}
4. 28(5) cage at R4C3 = {34579/34678} (cannot be {13789/14689/15679/23689/24589/24679/25678} which clash with R4C56), no 1,2
5. 2 in N5 only in R6C56 = {24}, locked for R6 and N5, clean-up: no 6 in R6C23 (step 2)
5a. Naked pair {35} in R6C23, locked for R6 and N4
5b. R6C4 = 1 (hidden single in N5)
6. 28(5) cage at R4C3 (step 4) = {34579/34678} -> R4C3 = 4
7. 45 rule on R6789 3 outies R5C123 = 18 = {189/279}, no 6, 9 locked for R5 and N4
7a. 6 in N4 only in R4C12 + R6C1, CPE no 6 in R23C1
8. R6C4 = 1 -> 28(5) cage at R6C4 = {13789/14689/15679}, no 2
9. 45 rule on N6 1 outie R3C9 = 1 innie R6C7 -> R3C9 = {68}, R1C6 = {68} (step 3)
9a. Naked pair {68} in R6C17, R3C9 = R6C7 -> naked pair {68} in R3C9 + R6C1, CPE no 8 in R3C1
10. 45 rule on N7 2 innies R78C3 = 1 outie R6C1 + 9
10a. R6C1 = {68} -> R78C3 = 15,17 = {69/78/89}, no 3,5
11. R5C123 = 18, R4C4 + R5C456 = 24
11a. 45 rule on R5 1 outie R4C4 = 1 innie R5C7 + 6, R4C4 = {789}, R5C7 = {123}
12. 3 in N5 only in R5C456, locked for R5, clean-up: no 9 in R4C4, no 6 in R5C89
13. 9 in R4 only in R4C56 = {59}, locked for R4 and N5
14. R5C89 = {45} (hidden pair in R5)
15. 11(3) cage in N3 = {128/137/146/245} (cannot be {236} which clashes with R12C9)
15a. 8 of {128} must be in R23C7 (R23C7 cannot be {12} which clashes with R5C7) -> no 8 in R2C8
16. 45 rule on C1234 2 outies R5C56 = 1 innie R9C4 + 8
16a. Hidden killer pair 3,6 in R5C4 and R5C56 for R5, R5C56 must contain one of 3,6 (cannot be {36} = 9 because no 1 in R9C4) -> R5C4 = {36}
16b. R5C56 = {37/38/67/68} = 10,11,13,14 -> R9C4 = {2356}
17. 5 in N7 only in 26(6) cage at R6C1 = {123569/123578/124568/134567}, 1 locked for N7, clean-up: no 9 in R78C2
18. 45 rule on C789 4 outies R1789C6 = 25 = {1789/2689/3679/4678} (cannot be {3589/4579} which clash with R4C6), no 5
18a. R1789C6 = {1789/2689/4678} contain 8, R1789C6 = {3679} => R5C6 = 8 -> R15789C6 must contain 8, locked for C6
19. 45 rule on N14 2(1+1) remaining innies R3C3 + R6C1 = 1 outie R1C4 + 2
19a. Min R3C3 + R6C1 = 7 -> min R1C4 = 5
19b. Max R1C4 = 9 -> max R3C3 + R6C1 = 11, min R6C1 = 6 -> max R3C3 = 5
20. 17(3) cage at R6C7 = {269/278/368/458/467} (cannot be {179/359} because R6C7 only contains 6,8), no 1
21. Hidden killer pair 7,9 in R6C9 and R789C9 for C9, R6C9 = {79} -> R789C9 must contain one of 7,9
21a. 17(4) cage in N9 = {1259/1349/1367/2357} (other combinations don’t contain 7 or 9, cannot be {1457} which clashes with R89C8), no 8
21b. 17(4) cage only contains one of 7,9 -> no 7,9 in R7C8
22. 8 in C9 only in R34C9, locked for 20(5) cage at R2C9, no 8 in R4C78
23. 45 rule on N2 4 innies R123C4 + R1C6 = 24 = {2589/2679/3489/4569/4578} (cannot be {3579} because R1C6 only contains 6,8, cannot be {3678} because {367}8 clashes with R4C5 and {378}6 clashes with R4C4)
23a. R1C6 = {68} -> no 6,8 in R123C4
24. 12(3) cage at R2C4 = {129/147/237/345}
24a. 1,3 of {129/237} must be in R3C3 (R23C4 cannot be {37} because no combination of R123C4 + R1C6 (step 23) contains both of 3,7) -> no 2 in R3C3
25. 45 rule on N9 3 innies R789C7 = 15 = {249/258/267/348} (cannot be {168} which clashes with R6C7, cannot be {456} which clashes with R89C8, cannot be {159/357} which clash with 17(4) cage), no 1
26. 25(4) cage at R1C3 = {1789/2689/3589/3679} (cannot be {4579} which clashes with R1C12, cannot be {4678} because R12C3 = {68} clashes with R78C3), no 4
[The {4579} clash was easy but I’ve only just spotted the {4678} clash.]
26a. 2 of {2689} must be in R12C3 (R12C3 cannot be {68} which clashes with R78C3) -> no 2 in R2C2
[The next step was even harder to spot. I’d realised for a while that 7 in N1 must be in either 25(4) cage at R1C3 or in 27(5) cage at R2C1 but it still took time to spot ...]
27. Consider the placement for 7 in N1
7 in 25(4) cage at R1C3 in N1 => no 7 in R1C4
or 7 in 27(5) cage at R2C1 in N1 => no 7 in R4C12 => R4C4 = 7 (hidden single in R4) => no 7 in R1C4
27b. -> no 7 in R1C4
28. 25(4) cage at R1C3 (step 26) = {1789/2689/3589/3679}, R1C12 = {15/24}
28a. Consider combinations for R1C12
28b. R1C12 = {15} => R3C3 = 3 => killer triple 1,2,3 in R1C12 + R3C3 + 25(4) cage at R1C3, locked for N1
or R1C12 = {24}, locked for N1
28c. -> no 2 in R2C1 + R3C12
[Getting a bit heavier, since I can’t see anything easier.]
29. R789C7 (step 25) = {249/258/267/348}
29a. Consider combinations for R789C6
R789C6 = {249}, locked for N9, 8 in N9 only in R89C8 = {58}, locked for C8 => R5C89 = [45], R12C9 = {16/34}, R4C9 = 2 (hidden single in C9) => R6C7 = 8 (hidden single in N6)
or R789C7 {258/267/348}, killer pair 6,8 in R6C7 and R789C7, locked for C7
29b. -> 8 must be in R6789C7, locked for C7
30. 11(3) cage in N3 (step 15) = {137/146/245}
Hidden killer pair 1,2 in R12C9 and 11(3) for N3, 11(3) cage contains one of 1,2 -> R12C9 must contain one of 1,2 -> R12C9 = {16/25}, no 3,4
31. R1C78 + R2C8 + R3C9 (step 3a) = {3789/4689} (cannot be {5679} which clashes with R12C9), no 5
[Alternatively 8 in N3 only in R1C78 + R2C8 + R3C9 ...]
31a. 3 of {3789} must be in R12C8 (R12C8 cannot be {79} which clashes with R6C8 -> no 3 in R1C7
32. 1 in N9 only in 17(4) cage (step 21a) = {1259/1349/1367}
32a. 1 of {1259} must be in R7C8 (R789C9 cannot be {129/159} which clash with R12C9) -> no 2,5 in R7C8
[I’m probably missing something better but the only way forward I can see is a contradiction move; while contradiction moves can often be replaced by forcing chains, I couldn’t see how to do that this time.]
33. 45 rule on N2 2 innies R1C46 = 1 outie R3C3 + 12
33a. R1C12 cannot be {15}, here’s how
R1C12 = {15}, locked for R1 and N1 => R3C3 = 3, R1C46 = 15 = [96] => 27(4) cage at R1C6 = {4689} = [6489], 25(4) cage at R1C3 (step 26) = {2689} => R1C3 = 2 => no remaining candidate for R1C9
33b. -> R1C12 = {24}, locked for R1 and N1, clean-up: no 5 in R2C9
[Followed by something similar. This was intended to be a forcing chain but happens to contain a contradiction; still it’s better than my original contradiction move which gave the same result.]
34. R1C46 = R3C3 + 12 (step 33), R3C3 = {135} -> R1C46 = 13,15,17 = [58/96/98]
34a. Consider the permutations for R1C46
R1C46 = [58/98] = 27(4) cage at R1C6 = {3789}
R1C46 = [96] => 9 in 27(4) cage at R1C6 must be in R2C8 => 27(4) cage cannot be {4689}
34b. -> 27(4) cage at R1C6 = {3789} (only remaining combination), no 4,6 -> R1C6 = 8, R1C78 + R2C8 = {379}, locked for N3, 3 also locked for C8, R3C9 = 8 (step 3), R6C7 = 8 (step 9), R6C1 = 6
35. 8 in C8 only in R89C8 = {58}, locked for C8 and N9 -> R5C89 = [45], clean-up: no 2 in R2C9
35a. Naked pair {16} in R12C9, locked for C9 and N3 -> R3C8 = 2, R23C7 = {45}, locked for C7
36. 1,4 in N9 only in 17(4) cage (step 21a) = {1349} (only remaining combination) -> R7C8 = 1, R789C9 = {349}, locked for C9 and N9 -> R4C9 = 2, R5C7 = 1, R4C78 = [36], R6C89 = [97], R4C4 = 7 (step 11a)
36a. R5C7 = 8 (hidden single in N5)
37. 45 rule on C1234 2 remaining innies R59C4 = 9 = {36} (locked for C4)
38. 45 rule on N7 2 remaining outies R78C4 = 12 = {48}, locked for C4, N8 and 28(4) cage at R6C4, no 8 in R78C3
39. R2C4 = 2 (hidden single in C4) -> 12(3) cage at R2C4 (step 24) = {129} (only remaining combination) -> R3C34 = [19], R1C4 = 5
40. R1C7 = 9 (hidden single in N3)
40a. R1C4 = 5 -> 25(4) cage at R1C3 (step 26) = {3589} (only remaining combination) -> R1C3 = 3, R2C23 = {89}, locked for N1, R6C23 = [35], clean-up: no 7 in R78C2
41. Naked pair {57} in R23C1, locked for C1 and N1 -> R3C2 = 6, clean-up: no 4 in R78C2
41a. Naked pair {28} in R78C2, locked for C2 and N7 -> R1C12 = [24], R2C23 = [98], R4C12 = [81], R5C123 = [972], R9C2 = 5, R89C8 = [58]
42. R78C3 = R6C1 + 9 (step 10)
42a. R6C1 = 6 -> R78C3 = 15 = {69}, locked for C3 -> R9C3 = 7
43. R7C5 = 5 (hidden single in N8), R4C56 = [95]
43a. R7C5 = 5 -> 16(4) cage in N8 = {1357/2356}, 3 locked for N8
43b. 7 of {1357} must be in R8C5 -> no 1 in R8C5
44. R6C7 = 8 -> 17(3) cage at R6C7 = {278} (only remaining combination) -> R7C67 = {27}, locked for R7 -> R78C2 = [82], R78C4 = [48], R7C1 = 3, R7C9 = 9, R78C3 = [69]
45. 6 in C7 only in R89C7 -> 23(4) cage at R8C6 = {1679} (only remaining combination) -> R9C6 = 9, R8C6 = 1, R9C7 = 6, R8C7 = 7
and the rest is naked singles.