Prelims
a) R12C9 = {18/27/36/45}, no 9
b) R3C89 = {18/27/36/45}, no 9
c) R5C45 = {29/38/47/56}, no 1
d) R56C6 = {59/68}
e) R6C45 = {17/26/35}, no 4,8,9
f) R78C5 = {39/48/57}, no 1,2,6
g) R78C6 = {13}
h) R9C23 = {89}
i) 11(3) cage at R5C3 = {128/137/146/236/245}, no 9
j) 21(3) cage at R7C4 = {489/578/678}, no 1,2,3
k) 20(3) cage at R8C8 = {389/479/569/578}, no 1,2
l) 14(4) cage at R3C3 = {1238/1247/1256/1346/2345}, no 9
m) Disjoint 34(5) cage at R1C3 = {46789}
1a. Naked pair {13} in R78C6, locked for C6 and N8, clean-up: no 9 in R78C5
1b. Naked pair {89} in R9C23, locked for R9 and N7
1c. R5C45 = {29/38/47} (cannot be {56} which clashes with R56C6), no 5,6
1d. 21(3) cage at R7C4 = {489/579} (cannot be {678} which clashes with R78C5), no 6, 9 locked for C4, clean-up: no 2 in R5C5
1e. R9C56 = {26} (hidden pair in N8), locked for R9 and 20(5) cage at R8C7
1f. 45 rule on N3 1 innie R1C7 = 9
1g. R1C356 + R2C3 = {4678}, CPE no 4,6,7,8 in R1C12
1h. 4,7 in C6 only in R1234C6, CPE no 4,7 in R1C4 + R2C45 + R3C5
1i. 9 in N2 only in R23C56, locked for 36(7) cage at R1C4, no 9 in R4C6
1j. R9C56 = {26} = 8 -> R8C7 + R9C78 = 12 = {138/147/345}
1k. 2 in N9 only in R7C789, locked for R7
1l. 45 rule on N36 3 outies R7C789 = 13 contains 2 = {238/247/256}, no 1,9
1m. R7C4 = 9 (hidden single in R7) -> R89C4 = 12 = {57}/[84], no 4 in R8C4
1n. 9 in N9 only in 20(3) cage at R8C8 = {389/479/569}
1o. 3 of {389} must be in R9C9 -> no 3 in R8C89
1p. 1 in N9 only in R8C7 + R9C78 = {138/147}, no 5
1q. 8 of {138} must be in R8C7 -> no 3 in R8C7
1r. 45 rule on N1 3 innies R123C3 = 18
1s. Max R12C3 = 15 -> min R3C3 = 3
1t. 45 rule on R123 3 outies R4C456 = 12 = {138/147/246/345} (cannot be {156} which clashes with R56C6, cannot be {237} which clashes with both R5C45 and R6C45)
1u. 45 rule on R1234 3 innies R4C123 = 16
[Note that this must have a common value with R4C1 but a different combination.]
1v. 45 rule on R1234 1 outie R5C2 = 1 innie R4C1 + 2, no 8,9 in R4C1, no 1,2 in R5C2
1w. 45 rule on C1 1 innie R7C1 = R8C2 + 4 -> R7C1 = {567}, R8C2 = {123}
1x. 12(3) cage at R8C1 = {147/237/345} (cannot be {156} = [615] which clashes with R7C1 + R8C2 = [51], cannot be {246} = [624] which clashes with R7C1 + R8C2 = [62]), no 6
[My original step 1ab was incorrect so I’ve reworked from here, the next step being simpler than my original steps 1y and 1z.]
1y. 45 rule on C1 3 innies R789C1 = 16 = {367/457}, no 1,2, 7 locked for C1 and N7, clean-up: no 9 in R5C2
1z. 16(4) cage at R7C1 contains 6 for N7 = {1267/1456/2356}
1aa. R7C789 = {238/247} (cannot be {256} which clashes with 16(4) cage, ALS block), no 5,6
1ab. 20(3) cage at R8C8 = {569} (hidden triple in N9) -> R9C9 = 5, R8C89 = {69}, 6 locked for R7, clean-up: no 4 in R12C9, no 4 in R3C8, no 7 in R8C4
1ac. Variable hidden killer triple 1,3,7 in R7C123, R7C6 and R7C789 for R7, R7C6 = {13}, R7C789 contains one of 3,7 -> R7C123 cannot contain more than one of 1,3,7
1ad. 16(4) cage = {1456/2356} (cannot be {1267} = 7{16}2 which contains both of 1,7 in R7), no 7, clean-up: no 3 in R8C2
1ae. Hidden killer pair 7,8 in R7C5 and R7C789, R7C789 contains one of 7,8 -> R7C5 = {78}, R8C5 = {45}
1af. 5 in N8 only in R8C45, locked for R8
2a. R7C1 + R8C2 (step 1w) = [51/62], 16(3) cage at R4C1 = {169/259/268/358} (cannot be {349} which clashes with R89C1)
2b. Consider placement for R7C1 = {56}
R7C1 = 5 => R8C2 = 1 => 1 in N1 only in 13(3) cage at R1C1, locked for C1
or R7C1 = 6 => 16(3) cage = {259/358}, no 1
-> 1 in C1 only in 13(3) cage = {139/148}, no 2,5,6, 1 locked for C1 and N1, clean-up: no 3 in R5C2 (step 1v)
2c. 2 in N1 only in 14(3) cage at R1C2 = {239/248/257}, no 6, 2 locked for C2
2d. R8C2 = 1 -> R7C1 = 5, R78C6 = [13], R8C3 = 2 (hidden single in N7)
2e. 2,6 in C1 only in 16(3) cage at R4C1 = {268}, 6,8 locked for N4, 8 locked for C1
2f. R4C1 + 2 = R5C2 (step 1v) -> R4C1 = 2, R5C2 = 4, clean-up: no 7 in R5C45
2g. 13(3) cage = {139}, 3,9 locked for N1, 3 locked for C1
2h. 14(3) cage = {257}, 5,7 locked for C2 and N1 -> R46C2 = [93], R4C3 = 5 (cage sum), R7C2 = 6, R7C3 = 3 (cage sum), clean-up: no 5 in R6C45
2i. R7C789 (step 1aa) = {247}, 4,7 locked for N9, 7 locked for R7 -> R7C5 = 8, R8C45 = [54], R9C4 = 7, R8C7 = 8, clean-up: no 3 in R5C4
2j. R6C45 = {26} (cannot be {17} which clashes with R6C3), locked for R6 and N5 -> R56C1 = [68], R5C4 = 8 -> R5C5 = 3
2k. Naked pair {59} in R56C6, locked for C6
2l. Naked triple {147} in R4C456, locked for R4, 1 locked for 14(4) cage at R3C3, no 1 in R3C4
2m. 45 rule on R123 2 innies R3C34 = 1 outie R4C6 + 2
2n. R4C6 = {47} -> R3C34 = 6,9 = [42/63]
2o. R4C4 = 4 (hidden single in C4) -> R3C34 = [63], R4C56 = [17]
2p. Naked pair {48} in R12C3, locked for disjoint 34(5) cage -> R1C56 = [76], R9C56 = [62], R6C45 = [62], clean-up: no 2,3 in R2C9
2q. Naked pair {48} in R2C36, locked for R2, clean-up: no 1 in R1C9
2r. 18(4) cage at R5C7 = {2457} (only remaining combination), 4 locked for C7, 5 locked for N6
3a. R1C7 = 9, R12C9 = [27/36/81], R3C89 = {18/27}/[54] -> 18(4) cage at R1C8 must contain two 9(2) pairs from {18/27/36/45}
3b. Hidden killer triple 1,3,6 in R23C7, R4C7 and R9C7 for C7, R4C7 = {36}, R9C7 = {13} -> R23C7 must contain one of 1,3,6 -> R12C8 must contain one of 3,6,8
3c. R23C7 must also contain one of 2,5,7 (other values in those cells or from hidden killer triple 2,5,7 in C7) -> R12C8 must contain one of 2,4,7 -> no 5 in R12C8
3d. R1C2 = 5 (hidden single in R1)
3e. R3C89 = {18/27}/[54] (cannot be {27} which clashes with R3C2), no 2,7
3f. R3C27 = {27} (hidden pair in R3)
3g. Naked quad {2457} in R3567C7, 2,5,7 locked for C7, clean-up: no 4 in R1C8
3h. R3C89 = [54] (hidden pair in N3) -> R23C5 = [59], R23C6 = [48], R123C1 = [391]
3i. R12C9 = [81] (cannot be [27] which clashes with R3C7)
and the rest is naked singles.