Prelims
a) R23C4 = {17/26/35}, no 4,8,9
b) R3C56 = {39/48/57}, no 1,2,6
c) R7C45 = {49/58/67}, no 1,2,3
d) R78C6 = {15/24}
e) 20(3) cage at R1C6 = {389/479/569/578}, no 1,2
f) 21(3) cage at R2C1 = {489/579/678}, no 1,2,3
g) 19(3) cage at R9C2 = {289/379/469/478/568}, no 1
h) 12(4) cage at R3C1 = {1236/1245}, no 7,8,9
1a. 12(4) cage at R3C1 = {1236/1245}, CPE no 1,2 in R1C2
1b. 45 rule on N1 2(1+1) outies R1C4 + R4C2 = 6 = {15/24}/[33], no 6,7,8,9
[I missed 3 in N1 in 18(4) cage at R1C1 or in 12(4) cage at R3C1 blocks [33], an implied forcing chain.]
1c. 45 rule on C123 3 outies R189C4 = 11 = {128/137/146/236/245}, no 9
1d. 45 rule on N9 2(1+1) outies R6C8 + R9C4 = 7 = {16/25/34}, no 7,8,9
1e. 45 rule on C789 3 outies R129C6 = 19 = {289/379/469/478/568}, no 1, clean-up: no 6 in R6C8
1f. Max R9C6 = 6 -> min R12C6 = 13, no 2,3
1g. 45 rule on R1 2 innies R1C59 = 7 = {16/25/34}, no 7,8,9
1h. 45 rule on R9 2 innies R9C15 = 6 = {15/24}
1i. 9 in C4 only in R4567C4
1j. 45 rule on C1234 4 innies R4567C4 = 26 = {2789/3689/4589/4679}, no 1
2a. Hidden killer quad 1,2,4,5 for N8, R78C6 = {15/24}, R9C5 = {1245} -> the remaining cells in N8 can only contain one of 1,2,4,5
2b. R189C4 (step 1c) = {128/137/146/236} (cannot be {245} which requires two of 2,4,5 in R89C4), no 5, clean-up: no 1 in R4C2 (step 1b)
2c. 12(4) cage at R3C1 = {1236/1245}, 1 locked for R3 and N1, clean-up: no 7 in R2C4
3a. R78C6 = {15/24} and R9C15 (step 1h) = {15/24} interact to form combined cage R78C4 + R9C15 = {15/24}, CPE no 2,4,5 in R9C6, clean-up: no 2,3,5 in R6C8 (step 1d)
[Ed pointed out that the CPE also gives no 2,4 in R9C4; fortunately this didn’t significantly affect my solving path.]
3b. R129C6 (step 1e) = {379/469/568} (cannot be {478} because R9C6 only contains 3,6)
3c. R9C6 = {36} -> no 6 in R12C6
3d. R189C4 (step 2b) = {128/137/146/236}
3e. Consider combinations for R129C6
R129C6 = {379}, 7 locked for N2
or R129C6 = {469/568}, R9C6 = 6 => R189C4 = {128/137}, 1 locked for C4
-> R23C4 = {26/35}, no 1,7
3f. R189C4 = {128/137/146} (cannot be {236} which clashes with R23C4)
3g. R4567C4 (step 1j) = {2789/4589/4679} (cannot be {3689} which clashes with R23C4), no 3
[Continuing in that area.]
4a. R189C4 (step 3f) = {128/137/146}
4b. Consider combinations for R7C45 = {49/58/67}
R7C45 = {49/58} => killer quad 1,2,4,5 in R7C45, R78C6 and R9C5, locked for N8 => R189C4 = 1{37}, R8C5 = {89}
or R7C45 = {67} => R189C4 = {128}, R8C5 = 9 (hidden single in N8)
-> R189C4 = {128}/1{37}, no 4,6, no 3 in R1C4, clean-up: no 2,3 in R4C2 (step 1b) and
R8C5 = {89}
4c. Killer pair 2,3 in R189C4 and R23C4, locked for C4
4d. 12(4) cage at R3C1 = {1245} (only remaining combination), 2 locked for R3 and N1, clean-up: no 6 in R2C4
4e. 12(4) cage at R3C1 = {1245}, CPE no 4,5 in R12C2
4f. 3 in N1 only in R1C123, locked for R1, clean-up: no 4 in R1C59 (step 1g)
4g. 20(3) cage at R1C6 = {479/578} (cannot be {569} which clashes with R1C59), no 6, 7 locked for R1
4g. 7 in N1 only in 21(3) cage at R2C1 = {579/678}, no 4, 7 locked for R2
4h. 3 in R1C123, R1C4 = {12} -> 18(4) cage at R1C1 = {349}2/{368}1 (cannot be {358}2 which clashes with 21(3) cage, cannot be {359}1 which clashes with 20(3) cage), no 5
[Continuing those combinations …]
4i. 18(4) cage = {349}2, 4 locked for N1 => R3C123 = {125}, 5 locked for R3 => R23C4 = [53]
or 18(4) cage = {368}1 => 21(3) cage = {579}
-> 5 in R2C134, locked for R2
4j. 21(4) disjoint cage at R1C5 cannot be {1389} because 3,8,9 only in R28C5, R8C5 = {89} -> no 8,9 in R2C5
4k. Max R12C5 = [64] = 10 (cannot be [56] which clashes with R23C4) -> min R89C5 = 11, no 1 in R9C5, clean-up: no 5 in R9C1 (step 1h)
5a. 18(4) cage at R1C1 (step 4h) = {349}2/{368}1
5b. Consider combinations for R189C4 (step 4b) = {128}/1{37}
R189C4 = {128}, 2 locked for C4 => R23C4 = {35}, 5 locked for N2
or R189C4 = 1{37} => 18(4) cage = {368}1, 8 locked for R1 => 20(3) cage = {479}
-> no 5 in R1C6
5c. R129C6 (step 3b) = {379/469}, no 8
5d. 8 in N2 only in R3C56 = {48}, locked for R3, 4 locked for N2
5e. R12C6 = [79] -> R9C6 = 3 (hidden cage sum, step 1e) -> R6C8 = 4 (step 1d)
[The rest is fairly straightforward, except for the final step.]
5f. 21(3) cage at R2C1 = {678}, 6,8 locked for R2 and N1
5g. Naked triple {349} in R1C123, 4,9 locked for R1
5h. Naked pair {58} in R1C78, 5 locked for R1 and N3, clean-up: no 2 in R1C59 (step 1g)
5i. R1C4 = 2 (hidden single in R1) -> R89C4 (step 1c) = 9 = [18], R8C8 = 9, R4C2 = 4 (step 1b)
5j. R7C45 = {67} (hidden pair in N8), locked for R7
5k. R78C6 = {24}, locked for C6 and N8 -> R3C56 = [48], R9C5 = 5 -> R9C1 = 1 (step 1h)
5l. R89C5 = [95] = 14 -> R12C5 = 7 = [61], R1C9 = 1, R7C56 = [67]
5m. R2C4 = 5 (hidden single in R2), R3C4 = 3
5n. R9C1 = 1 -> 22(4) cage at R7C2 = {1489/1579/1678}, no 2,3
5o. R9C4 = 8 -> R9C23 = 11 = {29}/[74]
5p. Killer pair 7,9 in 22(4) cage and R9C23, locked for N7
6a. 45 rule on N6 1 remaining outie R3C9 = 1 innie R4C7 = {67}
6b. R3C8 = 9 (hidden single in N3), R1C9 = 1 -> R2C89 = 7 = [34]
6c. R2C7 = 2
6d. Naked pair {67} in R34C7, locked for C7
6e. 6 in R9 only in R9C89, locked for N9
6f. 20(4) cage at R9C6 contains 3,6 = {2369/3467}
6g. R9C7 = {49} -> no 9 in R9C9
7a. 1 in N9 only in 19(4) cage at R6C8 = 4{159}, 5,9 locked for R7 and N9
7b. R9C67 = [34] -> R9C89 = 13 = {67}, 7 locked for R9 and N9
7c. Naked pair {67} in R39C9, locked for C9
7d. R7C2 = 8, R9C1 = 1 -> R8C12 = 13 = {67}, 6 locked for R8
7e. R8C36 = [54] (hidden pair in R8) -> R7C6 = 2
7f. R8C34 = [51] = 6 -> R67C3 = 9 = [63]
7g. R7C1 = 4 -> R456C1 = 12 = {237} (only remaining combination), locked for C1 and N4
8a. 16(3) cage at R4C4 = {178/259} (cannot be {268/358} because R4C4 only contains 7,9, cannot be {367} which clashes with R4C7, cannot be {169} because 1,6 only in R4C6), no 3,6
8b. R5C46 = [46] (hidden pair in N5) -> R5C5 = 3 (cage sum)
8c. 45 rule on C9 3 remaining innies R789C9 = 19 = [586/937] (cannot be {289} because R9C9 only contains 6,7), no 2
8d. R8C8 = 2 (hidden single in N9)
[Then a sting in the tail, now to fix the final combinations in N6.]
9a. 20(4) cage at R4C8 = {1379/1568}
9b. 6 of {1568} must be in R4C8 -> no 5,8 in R4C8
9c. 16(3) cage at R4C4 (step 8a) = {259} (only remaining combination, cannot be {178} which clashes with R4C78, ALS block) = [925] -> 16(3) cage at R6C4 = [781]
9d. 20(4) cage = {1379} (cannot be {1568} = 6{18}5 because R5C78 = {18} clashes with 23(4) cage at R4C3 ALS block since no 1,8 in R6C2) -> R56C7 = [93], R45C8 = {17}, locked for C8
and the rest is naked singles.