Prelims
a) R12C5 = {29/38/47/56}, no 1
b) R3C45 = {15/24}
c) R45C3 = {18/27/36/45}, no 9
d) R5C12 = {49/58/67}, no 1,2,3
e) R56C7 = {17/26/35}, no 4,8,9
f) R5C89 = {14/23}
g) R7C56 = {59/68}
h) R89C5 = {39/48/57}, no 1,2,6
i) 11(3) cage at R1C6 = {128/137/146/236/245}, no 9
j) 20(3) cage at R1C8 = {389/479/569/578}, no 1,2
k) 19(3) cage at R3C6 = {289/379/469/478/568}, no 1
l) 21(3) cage at R3C9 = {489/579/678}, no 1,2,3
m) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
n) 9(3) cage at R8C1 = {126/135/234}, no 7,8,9
o) 27(4) cage at R7C2 = {3789/4689/5679}, no 1,2
1. 27(4) cage at R7C2 = {3789/4689/5679}, 9 locked for N7
1a. 45 rule on N1 2 innies R1C3 + R3C1 = 9 = {18/27/36/45}, no 9
1b. 45 rule on N3 2 innies R1C7 + R3C9 = 10 = [19/28/37/46/64]
1c. 45 rule on N7 2 innies R7C1 + R9C3 = 9 {18/27/45} (cannot be {36} which clashes with 27(4) cage), no 3,6
1d. 45 rule on N9 2 innies R7C9 + R9C7 = 9 = {18/27/36/45}, no 9
2. 45 rule on N5 2 innies R4C6 + R6C4 = 12 = {39/48/57}, no 1,2,6
3a. 45 rule on R123 3 innies R3C169 = 20 = {389/479/569/578}, no 1,2, clean-up: no 7,8 in R1C3 (step 1a)
3b. 45 rule on R1234 3 innies R4C345 = 8 = {125/134}, 1 locked for R4, clean-up: no 1,2,3 in R5C3
3c. 45 rule on R789 3 innies R7C149 = 8 = {125/134}, 1 locked for R7, clean-up: no 1,2 in R9C3 (step 1c), no 1,2,3 in R9C7 (step 1d)
3d. Min R9C7 = 4 -> max R89C6 = 9, no 9 in R89C6
4a. 45 rule on C12 3 innies R378C2 = 22 = {589/679}, 9 locked for C2, clean-up: no 4 in R5C1
4b. 45 rule on R89 3 innies R8C237 = 22 = {589/679}, 9 locked for R8, clean-up: no 3 in R9C5
4c. Consider combinations for 27(4) cage at R7C2 = {3789/4689/5679}
27(4) cage = {3789} = [7398/8397] (R78C2 and R8C23 cannot be {78} because 22(3) hidden cages only contain one of 7,8)
or 27(4) cage = {4689} = [6498/8496] (R78C2 and R8C23 cannot be {68} because 22(3) hidden cages only contain one of 6,8)
or 9 of 27(4) cage = {5679} must be in R7C2 + R8C23 (R7C2 + R8C23 cannot be {567} because 22(3) hidden cages cannot contain 5 and one of 6,7)
-> no 8,9 in R7C3, no 8 in R8C2
4d. Consider combinations for R7C149 (step 3c) = {125/134}
R7C149 = {125}, 5 locked for R7 => R7C56 = {68}, locked for R7 => 27(4) cage = [7398]/{5679}
or R7C149 = {134}, 3,4 locked for R7 => 27(4) cage = {5679}
-> 27(4) cage = {3789/5679}, no 4, no 8 in R7C2, 7 locked for N7, clean-up: no 2 in R7C1 (step 1c)
4e. 9(3) cage at R8C1 = {126/234} (cannot be {135} which clashes with 27(4) cage), no 5
4f. 8 in N7 only in R89C3, locked for C3, clean-up: no 1 in R4C3
4g. 8 in N7 only in R89C3, CPE no 8 in R8C4
4h. 1 in R4 only in R4C45, locked for N5
4i. 1 in R5 only in R5C789, locked for N6, clean-up: no 7 in R5C7
4j. 8 of R378C2 = {589} must be in R3C2 -> no 5 in R3C2
5. R7C149 (step 3c) = {125/134}
5a. Hidden killer pair 2,4 in R7C149 and 18(3) cage at R7C7 for R7, R7C149 contains one of 2,4 -> 18(3) cage must contain one of 2,4 = {279/459/468}, no 3
5b. 18(3) cage = {279/459} (cannot be {468} which clashes with R7C149 + R7C56, killer ALS block), no 6,8, 9 locked for N9
5c. 5 of {459} must be in R8C7 (R7C78 cannot be {45} which clashes with R7C149) -> no 5 in R7C78
5d. 8 in R7 only in R7C56 = {68}, locked for R7 and N8, clean-up: no 4 in R89C5
6. 45 rule on C123 3 innies R169C3 = 13 = {148/157/238/247/256/346} (cannot be {139} because R9C3 only contains 4,5,8), no 9
6a. 5 of {157/256} must be in R9C3 -> no 5 in R16C3, clean-up: no 4 in R3C1 (step 1a)
6b. 9 in N4 only in R45C1, locked for C1
7. 27(4) cage at R7C2 (step 4d) = {3789/5679}
7a. Consider combinations for R45C3
R45C3 = [27]/{45} => 2 or 4 locked for N4 => 11(3) cage at R6C1 cannot be {24}5
or R45C3 = [36] => 27(4) cage = {5679}, 5 locked for N7
-> no 5 in R7C1, clean-up: no 4 in R9C3 (step 1c)
8. R7C149 (step 3c) = {125/134}, R7C1 + R9C3 (step 1c) = [18/45], 27(4) cage at R7C2 (step 4d) = {3789/5679}
8a. Consider combinations for 18(3) cage at R7C7 = {279/459}
18(3) cage = {279}, 2 locked for R7 => R7C149 = {134}, locked for R7 => 27(4) cage = {5679}
or 18(3) cage = {459}, 4 locked for R7 => R7C1 = 1, R9C3 = 8 => 27(4) cage = {5679}
-> 27(4) cage = {5679}, locked for N7 -> R9C3 = 8, R7C1 = 1
8b. R6C3 = 1 (hidden single in R6), R9C3 = 8 -> R1C3 = 4 (step 6), R3C1 = 5 (step 1a)
8c. 6 in N7 only in R8C23, locked for R8 -> R8C237 (step 4b) = {679}, 7 locked for R8
8d. 27(4) cage = {5679}, 5 locked for R7 => R7C149 = 1{34}, 4 locked for R7
8e. Naked triple {279} in 18(3) cage at R7C7, locked for N9
8f. R9C3 = 8 -> R89C4 = 8 = [17] (cannot be {35} which clashes with R89C5)
8g. R9C5 = 9 (hidden single in R9) -> R8C5 = 3, R7C4 = 4, R6C4 = 8 (cage sum), R4C6 = 4 (step 2), both placed for D/, R7C9 = 3 -> R9C7 = 6 (step 1d), R3C4 = 2 -> R3C5 = 4
8h. R12C5 = {56} (only remaining combination), locked for C5 and N2 -> R7C56 = [86]
8i. Naked pair {39} in R12C4, locked for C4 and N2 -> R4C4 = 5, placed for D\, R5C4 = 6
8j. Naked pair {27} in R56C5, locked for N5
8k. 1 in N2 only in R12C6, locked for 11(3) cage at R1C6 -> R1C7 = {23}
8l. Clean-ups: no 6,9 in R3C9 (step 1b), no 3 in R4C3, no 7 in R5C1, no 7,8 in R5C2, no 5 in R5C3, no 2 in R56C7, no 2 in R5C8
9. R3C6 = {78}, R4C6 = 4 -> R4C7 = {78} (cage sum)
9a. Naked pair {78} in R3C69, locked for R3
9b. R378C2 (step 4a) = {679} (only remaining combination), locked for C2
9c. R45C3 = [27], R7C3 = 5, placed for D/, R5C5 = 2, placed for both diagonals, R6C5 = 7, R9C1 = 3, placed for D/, clean-up: no 1 in R5C7, no 3 in R5C8
9d. Naked pair {14} in R5C89, locked for R5 and N6, R5C2 = 5 -> R5C1 = 8, R4C2 = 3, R6C12 = [64], R4C1 = 9, R56C7 = [35], R1C7 = 2 -> R3C9 = 8 (step 1b)
9e. Naked pair {79} in R78C7, locked for C7 and N9
9f. R23C7 = [41] = 5 -> R23C8 = 10 = [73], 7 placed for D/
9g. R1C1 = 7, placed for D\, R7C7 = 9, placed for D\
9h. R8C8 = 8 (hidden single in C8), placed for D\ -> R2C2 = 1, placed for D\
and the rest is naked singles, without using the diagonals.