Prelims
a) R23C1 = {39/48/57}, no 1,2,6
b) R4C89 = {79}
c) 5(2) cage at R6C8 = {14/23}
d) R89C5 = {15/24}
e) 10(3) cage at R2C2 = {127/136/145/235}, no 8,9
f) 20(3) cage at R2C9 = {389/479/569/578}, no 1,2
g) 19(3) cage at R6C5 = {289/379/469/478/568}, no 1
h) 9(3) cage at R8C1 = {126/135/234}, no 7,8,9
i) 12(4) cage at R1C8 = {1236/1245}, no 7,8,9
j) 30(4) cage at R6C3 = {6789}
1a. Naked pair {79} in R4C89, locked for R4 and N6
1b. 45 rule on N3 2 innies R12C7 = 13 = {49/58/67}, no 1,2,3
1c. 45 rule on N3 2 outies R23C6 = 11 = {29/38/47/56}, no 1
1d. 45 rule on C1234 2 outies R1C56 = 13 = {49/58/67}, no 1,2,3
1e. 45 rule on C1234 2 innies R1C34 = 7 = {16/25/34}, no 7,8,9
1f. 45 rule on N6 2(1+1) outies R5C6 + R7C9 = 5 = {14/23}
1g. 45 rule on C1 1 innie R1C1 = 1 outie R9C2 + 5 -> R1C1 = {6789}, R9C2 = {1234}
1h. 45 rule on C89 2 outies R38C7 = 6 = {15/24}
1i. 45 rule on C6789 2 innies R17C6 = 1 outie R5C5 + 10
1j. Max R17C6 = 17 -> max R5C5 = 7
1k. 45 rule on R89 2 innies R8C23 = 1 outie R7C8 + 12
1l. Max R8C23 = 17 -> max R7C8 = 5
1m. Min R8C23 = 13, no 1,2,3 in R8C23
1n. Max R7C8 + R8C7 = 9 -> min R8C8 = 3
1o. 45 rule on N69 2 innies R79C7 = 1 outie R5C6 + 10
2a. 45 rule on N9 3 innies R7C79 + R9C7 = 15
2b. Max R7C9 = 4 -> min R79C7 = 11, no 1 in R79C7
2c. Hidden killer pair 7,9 in R12C7 and R79C7 for C7, neither can contain both of 7,9 -> each must contain one of 7,9 -> R12C7 (step 1b) = {49/67}, no 5,8
2d. R23C6 (step 1c) = {29/38/56} (cannot be {47} which clashes with R12C7), no 4,7
2e. 8 in N3 only in 20(3) cage at R2C9 = {389/578}, no 4,6
2f. R79C7 contains one of 7,9 so cannot also contain 8 -> R456C7 must contain 8, locked for C7 and N6
2g. 16(3) cage at R4C7 = {1258/1348}, no 6
2h. 6 in N6 only in 13(3) cage at R5C8 = {256/346}, no 1
2i. 3 in C7 must be in R456C7 (cannot be in R79C7 because R79C7 + R7C9 cannot be {39}3 and max R7C9 = 4), locked for C7, N6 and 16(3) cage at R4C7, no 3 in R5C6
2j. 16(3) cage = {1348}, no 2,5
2k. 5 in N6 only in 13(3) cage = {256}
2l. R5C6 + R7C9 (step 1f) = {14} -> 5(2) cage at R6C8 = {14}
2m. Naked pair {14} in R5C6 + R7C9, CPE no 4 in R7C6
2n. Naked pair {14} in 5(2) cage, CPE no 1,4 in R7C8
2o. R79C7 + R7C9 = {29}4/{56}4/{59}1, no 7, no 4 in R79C7
2p. 7 in C7 only in R12C7 = {67}, locked for N3, 6 locked for C7 and 24(4) cage at R1C7, clean-up: no 5 in R23C6
2q. 12(4) cage at R1C8 = {1245} (only remaining combination), no 3
2r. R1C56 (step 1d) = {49/58} (cannot be {67} which clashes with R1C7), no 6,7
2s. Killer pair 8,9 in R1C56 and R23C6, locked for N2
2t. Max R17C6 = 16 (cannot be {89} which clashes with R23C6) -> max R5C5 = 6 (step 1i)
2u. 9 in C7 only in R79C7, locked for N9
2v. 18(3) cage at R8C9 = {378/468/567}, no 1,2
2w. Hidden killer triple 6,7,8 in R8C8 and 18(3) cage for N9, 18(3) cage contains two of 6,7,8 -> R8C8 = {678}
2x. 45 rule on D\ 3 innies R1C1 + R8C8 + R9C9 = 1 outie R4C6 + 17
2y. Max R1C1 + R8C8 + R9C9 = 24 -> no 8 in R4C6
2z. R6C4 + R7C3 + R8C2 = {789} (hidden triple on D/)
3a. Hidden killer pair 8,9 in R1C12 and R1C56 for R1, R1C56 (step 2r) contains one of 8,9 -> R1C12 must contain one of 8,9
3b. Hidden killer triple 3,6,7 in R1C12, R1C34 and R1C7 for R1, R1C12 can only contain one of 3,6,7, R1C34 (step 1e) can only contain one of 3,6, R1C7 = {67} -> R1C2 = {36789} , R1C34 = {16/34}, no 2,5
3c. 2 in R1 only in R1C89, locked for N3, clean-up: no 4 in R8C7 (step 1h)
3d. 2 in C7 only in R789C7, locked for N9
3e. 12(3) cage at R7C8 = {138/156/237} = [318/327/516], clean-up: no 1 in R3C7 (step 1h)
3f. R7C8 = {35} -> R8C23 = 15,17 (step 1k) = [96]/{89} (cannot be {78} which clashes with [318/327] in 12(3) cage), no 4,5,7, 9 locked for R8, N7 and 30(5) cage at R5C2, no 9 in R56C2
3g. 7 on D/ only in R6C4 + R7C3, locked for 30(6) cage at R6C3
3h. Combined cage 12(3) cage + R8C23 = [318][96]/[327][96]/[516]{89}, 6 locked for R8
4a. R1C34 (step 3b) = {16/34}, R1C56 (step 2r) = {49/58}, R23C6 (step 2p) = {29/38}, R1C1 = R9C2 + 5 (step 1g)
4b. Consider placement for 3 in R1
R1C2 = 3, no 3 in R9C2 => no 8 in R1C1 => 8 in R1 only in R1C56 = {58}
or R1C34 = {34}, 4 locked for R1, no 9 in R1C56
-> R1C56 = {58}
[This is neater than consider combinations for R1C34, which was the first way I found to do this breakthrough]
4c. 9 in R1 only in R1C12, locked for N1, clean-up: no 3 in R23C1
4d. Naked pair {58} in R1C56, locked for R1 and N2, clean-up: no 3 in R23C6, no 3 in R9C2
4e. Naked pair {29} in R23C6, locked for C6, 2 locked for N2
4f. 5 in N3 only in R2C8 + R3C7, locked for D/
4g. 17(3) cage at R2C5 = {368/467} (cannot be {278/458} because 2,5,8 only in R4C5), no 1,2,5, 6 locked for C5
4h. 8 of {368} only in R4C5 -> no 3 in R4C5
4i. 1 in N2 only in R123C4, locked for C4
4j. Combined half-cage R1C5 + 17(3) cage = 5{368}/8{467} (cannot be 5{467} which clashes with R89C5), 8 locked for C5
4k. Killer pair 4,5 in R1C5 + 17(3) cage + R89C5, locked for C5
4l. 9 in C5 only in R67C5 -> 19(3) cage at R6C5 = {289/379}, no 5,6
4m. Consider combinations for 19(3) cage
19(3) cage = {289} => R7C6 = 8, R7C3 = 7
or 19(3) cage = {379}, Caged X-Wing for 7 in 30(4) cage and 19(3) cage, no other 7 in R67
-> no 7 in R7C12
4n. 7 in N7 only in R79C3, locked for C3
4o. Min R1C12 = 12 -> max R2C3 + R23C4 = 9, no 7,8 in R2C3 + R23C4
4p. 7 in N2 only in 17(3) cage at R2C5 = {467}, 4,7 locked for C5, clean-up: no 2 in R89C5
4q. Naked pair {15} in R89C5, locked for C5 and N8 -> R1C56 = [85]
4r. 7,8 of 19(3) cage only in R7C6 -> R7C6 = {78}
4s. Naked pair {78} in R7C36, 8 locked for R7
4t. Min R2C3 + R23C4 = 6 -> max R1C12 = 15 -> R1C12 = {69}/[93], no 7, clean-up: no 2 in R9C2
4u. R1C7 = 7 (hidden single in R1) -> R2C7 = 6
4v. 9(3) cage at R8C1 = {135/234} (cannot be {126} = [261] which clashes with R1C1 + R9C2 = [61]), no 6, 3 locked for C1 and N7
5a. R4C4 = 6 (hidden single on D/) -> R4C5 = 4, R5C6 = 1, R6C8 = 1 (hidden single in N6), R7C9 = 4
5b. R8C7 = 1 (hidden single in C7) -> R3C7 = 5 (step 1h), R2C8 = 4, placed for D/, R1C8 = 2, R1C9 = 1, placed for D/, clean-up: no 6 in R1C34, no 7 in R2C1, no 8 in R3C1
5c. R1C12 = {69} (hidden pair in R1), 6 locked for N1
5d. R1C12 = {69} = 15 -> R2C3 + R23C4 = 6 -> R2C3 = 2, R23C4 = {13}, 3 locked for C4 -> R1C34 = [34]
5e. R4C6 = 6 -> R5C5 + R6C6 + R7C7 = 12 = {237} (only remaining combination) = [372], placed for D\, 3 placed for D/, R4C4 = 5, placed for D\
5f. Naked pair {68} in R8C8 + R9C9, locked for N9, 6 locked for D\
5g. R9C1 = 2 -> R8C1 + R9C2 = 7 = [34]
and the rest is naked singles.
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