Prelims
a) R3C12 = {59/68}
b) R5C78 = {39/48/57}, no 1,2,6
c) R67C2 = {17/26/35}, no 4,8,9
d) R6C34 = {39/48/57}, no 1,2,6
e) R78C4 = {15/24}
f) R7C56 = {15/24}
g) 10(3) cage at R5C9 = {127/136/145/235}, no 8,9
h) 20(3) cage at R6C8 = {389/479/569/578}, no 1,2
i) 23(3) cage at R8C5 = {689}
j) 20(3) cage at R9C2 = {389/479/569/578}, no 1,2
1a. 45 rule on R6789 1 outie R5C9 = 1
1b. 19(5) cage at R1C6 = {12349/12358/12367/12457/13456}, CPE no 1 in R1C45
2a. R9C46 = {37} (hidden pair in N8), locked for R9
2b. 20(3) cage at R9C2 = {389/479/578} (cannot be {569} which doesn’t contain 3 or 7), no 6
2c. 45 rule on N69 1 innie R6C7 = 1 outie R9C6 -> R6C7 = {37}
2d. 45 rule on N689 2(1+1) innies R6C7 + R9C4 = 10 = {37}
2e. Naked pair {37} in R6C7 + R9C4, CPE no 3,7 in R6C4, clean-up: no 5,9 in R6C3
2f. 45 rule on N6 3 remaining innies R6C456 = 16 = {349/367/457} (cannot be {259/268} because R6C7 only contains 3,7, cannot be {358} which clashes with R5C78), no 2,8
2g. R5C9 = 1 -> R67C9 = 9 = {36/45}/[72], no 7 in R7C9
2h. 2 in N6 only in 16(3) cage at R4C7 = {259/268}, 2 locked for N6
2i. R6C34 = [39/75]{48}, R6C789 = {349/367/457} -> combined cage R6C34789 = [39]{457}/{48}{367}/[75]{349}, 3,4,7 locked for R6, clean-up: no 1,5 in R7C2
[First time through, while manually deleting 3,4,7 in R6, I carelessly also deleted them from R6C9.]
2j. 14(3) cage at R6C5 = {167/239/257/356} (cannot be {158} because R6C7 only contains 3,7), no 8
2k. Overlapping combined cage 14(3) cage + R6C789 = {16}7{45}/{29}3{67}/{25}7{36} (cannot be {56}3[94] which clashes with R6C34)
-> 14(3) cage = {167/239/257}, R6C789 = {367/457}, no 9, 7 locked for R6 and N6, clean-up: no 5 in R5C78, no 5 in R6C4
2l. 20(3) cage at R6C8 = {389/479/569/578}
2m. 3 of {389} must be in R6C8 -> no 3 in R7C78
3a. 45 rule on C56789 2 outies R23C4 = 12 = {39/48/57}, no 1,2,6
3b. 45 rule on C56789 2 innies R12C5 = 12 = {39/48/57}, no 1,2,6
3c. 45 rule on R12 2 outies R3C49 = 7 = [34/43/52] -> R2C4 = {789}
3d. 45 rule on R1234 2 innies R4C16 = 11 = {38/47} (cannot be {56} which clashes with 16(3) cage at R4C7)
3e. 45 rule on R123 1 outie R4C5 = 1 innie R3C3 + 1, no 1,9 in R3C3, no 1 in R4C5
3f. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 + 4 -> R1C4 = {6789}, R3C3 = {2345}, R4C5 = {3456}
4. 45 rule on N7 3(2+1) outies R6C12 + R9C4 = 13 = [157/517/823/913], no 2,6 in R6C1, no 6 in R6C2, clean-up: no 2 in R7C2
5a. Hidden killer pair 1,2 in R45C4 and R78C4 for C4, R78C4 contains one of 1,2 -> R45C4 must contain one of 1,2
5b. Killer pair 1,2 in R45C4 and 14(3) cage at R6C5 in R6C56, locked for N5
5c. 17(3) cage at R4C6 = {359/368/458/467}
[My original breakthrough, which had followed from my careless deletions in R6C9, no longer works at this stage. I’ve therefore reworked from here.]
5d. R23C4 = [75/84/93] (step 3c), 14(3) cage at R6C5 = {167/239/257}, R6C789 = {367/457} (both step 2k)
5e. Consider placement for 7 in N5
7 in R45C4 => R9C4 = 3, R23C4 = [84], R6C4 = 9 => R6C3 = 3, R6C7 = 7, R6C89 = {45}, 5 locked for R6 => 14(3) cage at R6C5 = {16}7, 6 locked for N5 -> 17(3) cage = {458}
or 7 in 17(3) cage = {467}
-> 17(3) cage = {458/467}, no 3,9, 4 locked for N5, clean-up: no 8 in R6C3
5f. Also from step 5e, 6 in 17(3) cage or R6C56 = {16}, 6 locked for N5
6a. R1C4 = 6 (hidden single in C4) -> R3C3 = 2 (step 3f) -> R4C5 = 3 (step 3e), clean-up: no 9 in R12C5 (step 3b), no 5 in R3C4 (step 3c), no 8 in R4C16 (step 3d), no 7 in R2C4 (step 3a)
6b. R1C4 = 6 -> R12C3 = 12 = {39/48/57}, no 1
6c. Naked pair {34} in R3C49, 4 locked for R3
6d. Naked pair {89} in R26C4, locked for C4
6e. Naked pair {47} in R4C16, locked for R4
7a. R78C4 = {24} (cannot be {15} which clashes with R4C4), locked for C4 and N8 -> R3C4 = 3, R2C4 = 9 (step 3a), R6C4 = 8 -> R6C3 = 4, R9C46 = [73], R45C4 = [15], R4C16 = [74], R3C9 = 4
7b. Naked pair {67} in R5C56, 6 locked for R5 and N5
7c. R6C5 = 2 (hidden single in C5), R6C6 = 9 -> R6C7 = 3 (cage sum), clean-up: no 9 in R5C78, no 6 in R7C2
7d. Naked pair {48} in R5C78, 8 locked for R5 and N6
7e. R6C89 = {67} (hidden pair in R6), 6 locked for N6
7f. Naked pair {15} in R7C56, locked for R7
7g. R4C23 = {68} (hidden pair in N4)
7h. 20(3) cage at R6C8 = {479} (only remaining combination) -> R6C8 = 7, R7C67 = {49}, locked for R7 and N9 -> R78C4 = [24]
7i. 10(3) cage at R5C9 = [163], R7C2 = 7 -> R6C2 = 1
7j. R12C3 (step 6b) = {57} (cannot be [93] which clashes with R5C3), 5 locked for C3 and N1, clean-up: no 9 in R3C12
7k. Naked pair {68} in R3C12, locked for R3 and N1
7l. Naked pair {68} in R7C13, locked for N7 -> R9C34 = [97], R9C2 = 4 (cage sum), R12C2 = [93]
7m. R5C23 = [23], R8C23 = [51] -> R7C3 = 8 (cage sum)
7n. 4 in N2 only in R12C5 (step 3a) = {48}, 8 locked for C5 and N2 -> 23(3) cage at R8C5 = [986], R5C56 = [76]
8a. 6 in N9 only in R8C78, R9C6 = 3 -> 17(4) cage at R8C7 = [7631]
8b. R2C6 = 7 (hidden single in R2)
9a. 2 in R1 and C6 only in 19(5) cage at R1C6 -> R1C6 = 2
9b. R1C6 = 2, R1C8 = 3 (hidden single in R1) -> 19(5) cage at R1C6 = {12358} (only possible combination) -> R2C6 = 1, R1C79 = {58}, locked for R1 and N3
and the rest is naked singles.