Prelims
a) R34C6 = {39/48/57}, no 1,2,6
b) R67C4 = {19/28/37/46}, no 5
c) 9(3) cage at R1C7 = {126/135/235}, no 7,8,9
d) 22(3) cage at R3C9 = {589/679}
e) 11(3) cage at R6C5 = {128/137/146/2236/245}, no 9
f) 21(3) cage at R8C6 = {489/579/678}, no 1,2,3
g) 12(4) cage at R2C2 = {1236/1245}, no 7,8,9
1a. 45 rule on N69 2(1+1) outies R3C9 + R8C6 = 18 -> R3C9 = 9, R8C6 = 9, clean-up: no 3 in R34C6, no 1 in R6C4
1b. R8C6 = 9 -> R89C7 = 12 = {48/57}, no 6
1c. 45 rule on C789 2 outies R28C6 = 11 = [29]
1d. R2C6 = 2 -> R12C7 = 7 = {16/34}, no 5
1e. 45 rule on C123 2 outies R28C4 = 8 = {17/35}/[62], no 4,8,9, no 6 in R8C6
1f. 45 rule on N7 1 outie R8C4 = 1 innie R7C1 + 2 -> R8C4 = {357}, R7C1 = {135}, clean-up: no 6,7 in R2C5
1g. Max R2C4 = 5 -> min R12C3 = 12, no 1,2 in R12C3
1h. Max R7C1 = 5 -> min R6C12 = 12, no 1,2 in R6C12
1i. 40(7) cage at R3C4 must contain 9, locked for N5, clean-up: no 1 in R7C4
1j. R67C4 = {28/46} (cannot be {37} which clashes with R28C4), no 3,7
1k. 45 rule on N6 1 outie R3C9 = 3 innies R5C9 + R6C89 = 9 = {126/135/234}, no 7,8,9
1l. 45 rule on N9 1 innie R7C9 = 1 remaining outie R6C8 + 5 -> R7C9 = {678}, R6C8 = {123}
1m. 45 rule on C5 3 innies R159C5 = 19 = {289/379/469/478/568}, no 1
1n. 45 rule in R12 3 innies R2C258 = 17 = {179/359/368/458/467}
1o. 1 of {179} must be in R2C2 -> no 1 in R2C58
1p. 45 rule on C1 3 outies R169C2 = 19 = {289/379/469/478/568}, no 1
2a. 16(4) cage at R6C8 = {1249/1267/1348/1357/2356} (cannot be {1258/1456/2347} which clash with R89C7)
2b. Consider placement of 9 in N9
16(4) cage at R6C8 = {1249}, no 3
or 12(3) cage at R8C9 = {129} => 16(4) cage must contain at one of 1,2 in R6C8
-> R6C8 = {12}, clean-up: no 8 in R7C9 (step 1l)
2c. 16(4) cage at R6C8 = {1249/1348/1357/2356} (cannot be {1267} which clashes with R7C9)
2d. Killer pair 4,5 in 16(4) cage and R89C7, locked for N9
2e. Killer triple 4,6,7 in 16(4) cage, R7C9 and R89C7, locked for N9
[I saw steps 2d and 2e in that order; together they form
Killer quad 4,5,6,7 in 16(4) cage, R7C9 and R89C7, locked for N9]
2f. 12(3) cage at R8C9 = {129/138}, 1 locked for N9
3a. R5C9 + R6C89 = {126/135/234} (step 1k), R7C9 = R6C8 + 5 (step 1l)
3b. Consider placements for R6C8 = {12}
R6C8 = 1 => R7C9 = 6, R56C9 = 8 = {35}, 5 locked for N6 => R4C89 = [67] => 1,6 in C7 only in R12C7 = {16} (cannot both be in R3C7)
or R6C8 = 2, R7C9 = 7 => R89C7 = {48}, 4 locked for C7
-> R12C7 = {16}, locked for C7 and N3
and 7 in R47C9, locked for C9
[Looking at those placements slightly differently.]
3c. R6C8 = 1 => R7C9 = 6, R56C9 = {35}
or R6C8 = 2 => 16(4) cage at R6C8 (step 2c) = {2356} => R9C8 = 9 (hidden single in N9), R89C9 = 3 = {12}, locked for C9 => R56C9 = 7 = {34}
-> R56C9 = {34/35}, no 1,2,6, 3 locked for C9 and N6
3d. 12(3) cage at R8C9 (step 2f) = {129/138} -> R9C8 = {39}
3e. 14(3) cage at R1C8 = {248/257} (cannot be {347} because 3,7 only in R1C8), no 3, 2 locked for R1 and N3
3f. 23(4) cage at R4C7 = {2489} (only possible combination, cannot be {1679} because 1,6 only in R5C8, cannot be {1589} = {589}1 which clashes with R89C7, cannot be {2579/2678/4568} which clash with R4C89), locked for N6
3g. R6C8 = 1 -> R7C9 = 6
3h. Naked pair {35} in R56C9, 5 locked for C9 and N6 -> R4C89 = [67], clean-up: no 5 in R3C6, no 4 in R6C4
3i. 14(3) cage at R1C8 = {248} (cannot be {257} because 5,7 only in R1C8), 4,8 locked for N3
3j. 4 in C9 only in R12C9, locked for N3
3k. 40(7) cage at R3C4 = {1456789/2356789}, CPE no 6 in R6C4, clean-up: no 4 in R7C4
3l. Naked pair {28} in R67C4, locked for C4
3m. 40(7) cage at R3C4 = {1456789/2356789}, CPE no 8 in R4C6, clean-up: no 4 in R3C6
3n. 16(3) cage at R1C4 = {349/358/367/457} (cannot be {169} which clashes with R1C7, cannot be {178} which clashes with R3C6), no 1
4a. 40(7) cage at R3C4 = {1456789/2356789}
4b. Consider placement for 5
5 in R3C4 => R3C78 = {37}, 7 locked for R3 => R3C6 = 8
or 5 in R4C4 + R5C456 + R67C6, CPE no 5 in R4C6
-> R34C6 = [84]
4c. 4 of {1456789} only in R3C4 -> no 1 in R3C4
[Here I spotted 4 in R3C4 or 2 in R5C5 which looked useful, but not needed because …]
4d. R5C5 = 8 (only remaining position in 40(7) cage) -> R67C4 = [28]
4e. 40(7) cage at R3C4 = {1456789}, no 3 -> R3C4 = 4
4f. 16(3) cage at R1C4 (step 3n) = {367} (only remaining combination), locked for R1 and N2 -> R12C7 = [16], clean-up: no 5 in R8C4 (step 1e), no 3 in R7C1 (step 1f)
[Note that 45 rule on N14 2(1+1) outies R2C4 + R7C1 = 6 now form naked pair {15} but I didn’t need to use this.]
4g. R2C5 = 9 (hidden single in N2) -> R34C5 = 6 = {15}, locked for C5
4h. R6C5 = 3 (hidden single in N5) -> R78C5 = 8 = [26], R56C9 = [35]
4i. 17(3) cage at R1C1 = {359/458} (cannot be {179} because 1,7 only in R2C1), no 1,7, 5 locked for N1
4j. 5 in R1 only in R1C12, locked for N1
5a. 8 in N6 only in R46C7, locked for C7, clean-up: no 4 in R89C7
[With hindsight I could have got this after step 3f, at least one of 4,8 in R456C7, but I was focussing on other steps.]
5b. Naked pair {57} in R89C7, locked for C7 and N9 -> R3C7 = 3
5c. R1C5 = 7, R9C5 = 4 -> R9C46 = 8 = {17/35}
5d. Killer pair 5,7 in 12(3) cage and R9C7, locked for R9
6a. 17(3) cage at R6C1 = {48}5/{79}1, no 6
6b. 15(3) cage at R3C1 = {258/267/357} (cannot be {159} which clashes with R7C1, cannot be {168/249} which clash with 17(3) cage, cannot be {348} because no 3,4,8 in R3C1, cannot be {456} = [654] which clashes with 12(4) cage at R2C2), no 1,4,9
6c. 2 of {258} must be in R3C1, 2 of {267} must be in R4C1 -> no 2 in R5C1
6d. 8 of {258} must be in R4C1, 3 of {357} must be in R4C1 -> no 5 in R4C1
6e. 1 in C1 only in R789C1, locked for N7
6f. 1 in N1 only in R2C2 + R3C23, locked for 12(4) cage at R2C2, no 1 in R4C2
7a. R8C4 = R7C1 + 2 (step 1f) -> R7C1 + R8C4 = [13/57]
7b. 14(3) cage at R8C3 = {239/347/356} (cannot be {248} because R8C4 only contains 3,7, cannot be {257} which clashes with R7C1 + R8C4), no 8
7c. 9 of {239} must be in R9C3 -> no 2 in R9C3
7d. R8C3 = {245} (only remaining cell for 2,4,5)
7e. 18(3) cage at R7C2 = {279/378} (cannot be {459} which clashes with 14(3) cage), no 4,5
7f. 2,8 only in R8C2 -> R8C2 = {28}, R7C23 = {37/79}, 7 locked for R7 and N7
7g. 16(4) cage at R6C8 (step 2c) = {1249/1348} -> R8C8 = {28}
7g. Naked pair {28} in R8C28, locked for R8 -> R8C9 = 1
7h. 14(3) cage = {347/356}, no 9
7i. 14(3) cage = {347/356}, CPE no 3 in R8C1 + R9C46
8a. Naked pair {45} in R8C13, 5 locked for R8 and N7 -> R89C7 = [75], R8C4 = 3, R9C3 = 6 -> R8C3 = 5 (cage sum)
8b. R7C1 = 1 -> R6C12 = 16 = {79}, locked for R6 and N4
8c. R8C4 = 3 -> R2C4 = 5 (step 1e), R12C3 = 12 = [93] (cannot be {48} which clashes with R6C3)
8d. R128C1 = [584], R1C2 = 4, R2C2 = 1, R3C23 = [62] -> R4C2 = 3 (cage sum)
8e. R7C23 = [97] -> R8C2 = 2 (cage sum)
and the rest is naked singles.