Prelims
a) R1C89 = {49/58/67}, no 1,2,3
b) R23C6 = {49/58/67}, no 1,2,3
c) R3C78 = {14/23}
d) R6C23 = {49/58/67}, no 1,2,3
e) R6C45 = {14/23}
f) R78C3 = {14/23}
g) R89C1 = {69/78}
h) 10(3) cage at R6C6 = {127/136/145/235}, no 8,9
i) 13(4) cage at R2C9 = {1237/1246/1345}, no 8,9
1a. 45 rule on N1 1 innie R3C3 = 5, clean-up: no 8 in R2C6, no 8 in R6C2
1b. 45 rule on N1245 1 innie R6C6 = 1, clean-up: no 4 in R6C45
1c. Naked pair {23} in R6C45, locked for R6 and N5
1d. R6C7 + R7C6 = 9 = {45}/[63/72], no 6,7 in R7C6
1e. 45 rule on N3 2 outies R45C9 = 3 = {12}, locked for C9 and N6
1f. R45C9 = 3 -> R23C9 = 10 = {37/46}
1g. Killer pair 3,4 in R23C9 and R3C78, locked for N3, clean-up: no 9 in R1C89
1h. R1C89 = {58} (cannot be {67} which clashes with R23C9), locked for R1 and N3
1i. 37(7) cage at R5C8 must contain at least one of 1,2 -> R9C8 = {12}
1j. 45 rule on N78 1 innie R7C6 = 1 outies R9C7 = {2345}
1k. 45 rule on N7 2 outies R9C45 = 12 = {39/48/57}, no 1,2,6
1l. 45 rule on N7 2 innies R9C23 = 9 = {18/27/36}/[54], no 9, no 4 in R9C2
1m. 45 rule on N2 3 innies R2C5 + R3C45 = 10 = {127/136/145/235}, no 8,9
1n. 5 of {145} must be in R2C5 -> no 4 in R2C5
2a. 9 in C9 only in R6789C9, locked for 37(7) cage, no 9 in R56C8
2b. 37(7) cage = {1345789/2345689} (cannot be {1246789} because 1,2 only in R9C8) must contain 3,4,5,8,9
2c. Hidden killer pair 6,7 in R23C9 and R6789C9 for C9, R23C9 contains one of 6,7 -> R6789C9 must contain one of 6,7, 37(7) cage only contains one of 6,7 -> no 6,7 in R56C8
2d. Caged X-Wing for 5,8 in R1C89 and 37(7) cage, no other 5,8 in C8
2e. Hidden killer pair 5,8 in R1C8 and R56C8 for C8, R1C8 = {58} -> R56C8 must contain one of 5,8 -> must also contain one of 3,4
2f. 45 rule on N36 4 innies R5C8 + R6C789 = 24 = {3489/3678/4569/4578} (cannot be {3579} which clashes with R6C23)
2g. 45 rule on R6789 1 innie R6C1 = 1 outie R5C8 + 2 -> R5C8 = {345}, R6C1 = {567}
2h. 45 rule on N78 2 remaining outies R69C7 = 9 = [54/63/72] (cannot be [45] because puts 5 in R5C8 + R6C789 when [5487] clashes with R1C8 and cannot be {4569} because 6,9 only then in R6C9) -> R6C7 = {567}, R9C7 = {234}, clean-up: no 5 in R7C6 (step 1d)
2i. R6C23 = {49}/[58] (cannot be {67} which clashes with R6C17, ALS block), no 6,7
2j.Consider permutations for R69C7 = [54/63/72]
R69C7 = [54] => R5C8 + R6C789 = [4587] (cannot be {4569} because 6,9 only then in R6C9)
or R69C7 = [63] => R5C8 = 3, R6C789 = [687]
or R69C7 = [72] => R5C8 + R6C789 = [3768]/[47]{58} (cannot be [57]{48} which clashes with R6C1, step 2g)
-> R5C8 + R6C789 = 3{678}/4{578}, no 5 in R5C8, 7,8 locked for R6 and N6, clean-up: no 5,6 in R6C2
2k. Naked pair {49} in R6C23, locked for R4 and N4
2l. 9 in C9 only in R789C9, locked for N9
2m. 15(4) cage at R4C1 = {1257/1356}, no 8, 1 locked for N4
2n. Consider placement for 7 in N6
R6C7 = 7 => R9C7 = 2 -> 37(7) cage = {1345789}, 7 locked for C9
or R6C9 = 7 -> 37(7) cage = {1345789}
-> R9C8 = 1, 37(7) cage = {1345789}, 7 locked for C9, clean-up: no 3 in R23C9, no 8 in R9C23 (step 1l)
[Cracked, fairly straightforward from here.]
2o. Naked pair {46} in R23C9, locked for C9 and N3, clean-up: no 1 in R3C7
2p. Naked pair {23} in R3C78, locked for R3, 2 locked for N3
2q. 37(7) cage = {1345789} -> R5C8 = 4, R6C789 = {578}, 5 locked for R6 and N6, 3 locked for N9, clean-up: no 3 in R7C6 (step 1d)
2r. R6C1 = 6 -> R4C12 + R5C1 = {135}, 3,5 locked for N4, clean-up: no 9 in R89C1
2s. Naked pair {78} in R89C1, locked for C1 and N7, clean-up: no 2 in R9C23 (step 1l)
2t. Killer pair 3,4 in R78C3 and R9C23, locked for N7
2u. 21(4) cage at R9C2 must contain 3, locked for R9
2v. 4 in C1 only in R123C1, locked for N1
2w. 8 in N1 only in 15(3) cage at R2C2 = {168}, 1,6 locked for N1
2x. R2C5 + R3C45 (step 1m) = {127/136/145} (cannot be {235} because 2,3,5 only in R2C5)
2y. 2,3,5 only in R2C5 -> R2C5 = {235}, R3C45 = {14/16/17}, 1 locked for R3 and N2
2z. 15(3) cage = {168}, 1 locked for R2 -> R2C78 = {79}, locked for R2 and N3, clean-up: no 4,6 in R3C6
[Some clean-ups omitted from here.]
3a. 6 in R1 only in R1C456, locked for N2, clean-up: no 7 in R3C6
3b. R3C45 = {17} (hidden pair in R3), 7 locked for N2 -> R2C5 = 2 (step 2x), R6C45 = [23]
3c. Naked pair {78} in R5C23, locked for R5, N4 and 31(6) cage at R2C5 -> R3C5 = 1
3d. R3C34 = [57], R4C3 = 2 -> R4C4 = 8 (cage sum), clean-up: no 3 in R78C3
3e. R23C5 = [21], R5C23 = {78} -> R4C5 + R5C4 = 13 = [49]
3f. R3C6 = 8 (hidden single in N2) -> R2C6 = 5, R3C2 = 6, R23C9 = [64], R3C1 = 9
3g. R5C56 = [56], R4C6 = 7, R5C17 = [13], R3C78 = [23]
3h. R9C7 = 4, R9C23 (step 1l) = [36], R9C4 = 5 -> R9C5 = 7 (step 1k), R4C12 = [35], R12C1 = [24], R1C23 = [73], R2C4 = 3
3i. R8C6 = 3 (hidden single in N8), R9C7 = 4 -> R8C5 + R9C6 = 10 = [82], R67C6 = [14] -> R6C7 = 5 (cage sum)
and the rest is naked singles.
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