Prelims
a) R1C23 = {18/27/36/45}, no 9
b) R4C12 = {17/26/35}, no 4,8,9
c) R56C1 = {29/38/47/56}, no 1
d) R89C1 = {17/26/35}, no 4,8,9
e) R89C9 = {59/68}
f) 19(3) cage at R3C3 = {289/379/469/478/568}, no 1
g) 21(3) cage at R7C7 = {489/579/678}, no 1,2,3
h) 10(3) cage at R8C4 = {127/136/145/235}, no 8,9
i) 20(3) cage at R9C5 = {389/479/569/578}, no 1,2
j) 11(4) cage at R7C8 = {1235}
1a. Naked quad {1235} in 11(4) cage at R7C8, locked for N9, clean-up: no 9 in R89C9
1b. Naked pair {68} in R89C9, locked for C9 and N9
1c. Naked triple {479} in R789C7, locked for C7
1d. 18(3) cage at R3C9 = {279/459}, no 1,3, 9 locked for C9
1e. 21(3) cage at R7C7 = {489/579} (cannot be {678} because 6,8 only in R8C6) -> R8C6 = {58}
1f. 45 rule on N9 3 outies R8C6 + R9C56 = 21 = {489/579/678}, no 3
1g. R8C6 = {58} -> no 5,8 in R9C56
1h. 20(3) cage at R9C5 = {479} (only remaining combination), locked for R9, clean-up: no 1 in R8C1
1i. R8C6 + R9C56 = {489/579}, 9 locked for R9 and N8
1j. 10(3) cage at R8C4 = {127/136/235} (cannot be {145} which clashes with R8C6 + R9C56), no 4
1k. 45 rule on N7 1 innie R7C3 = 9
1l. 19(3) cage at R3C3 = {478/568}, no 2,3, 8 locked for C3, clean-up: no 1 in R1C2
1m. R8C7 = 9 (hidden single in R8)
1n. 45 rule on N1 2 innies R23C3 = 11 = [38] (cannot be {47/56} which clash with 19(3) cage, combo crossover clash), clean-up: no 1,6 in R1C23
1o. 45 rule on N4 1 remaining outie R6C4 = 7, clean-up: no 4 in R5C1
2a. 45 rule on N5 3(1+2) outies R3C5 + R5C78 = 20
2b. Max R5C78 = 17 -> min R3C5 = 3
2c. Min R5C78 = 11, no 1 in R5C7, no 1,2 in R5C8
2d. 33(7) cage at R4C6 must contain 1, locked for N5
2e. 33(7) cage must contain 2,6, CPE no 2,6 in R5C4
3a. Hidden killer pair 8,9 in R56C1 and 22(4) cage at R5C2 for N4, R56C1 and 22(4) cage with R6C4 =7 cannot contain both of 8,9 -> each must contain one of 8,9
3b. R56C1 = {29/38}, no 4,5,6,7
3c. 22(4) cage with 7 and one of 8,9 = {1579/1678/2479/3478} (cannot be {2578} which clashes with R56C1)
4a. Hidden killer pair 6,8 in R7C12 and 23(4) cage at R7C3 for R7, 14(3) cage at R7C1 and 23(4) cage with R7C3 = 9 cannot contain both of 6,8 -> R7C12 and 23(4) cage must each contain one of 6,8
4b. 14(3) cage at R7C1 = {158/167/248/356} (cannot be {257/347} which don’t contain one of 6,8)
4c. 6,8 of 14(3) cage must be in R7C12 -> no 6,8 in R8C2
4d. 23(4) cage at R7C3 = {1679/2489/3569} (cannot be {1589} which clashes with R8C6, cannot be {2579/3479} which don’t contain one of 6,8)
4e. Consider permutations for R7C7 + R8C6 = [48/75]
R7C7 + R8C6 = [48] => 8 in R7 only in 14(3) cage = {158/248}, no 7 => 7 in R7 only in 23(4) cage = {1679}
or R7C7 + R8C6 = [75] => 23(4) cage = {2489}
-> 23(4) cage = {1679/2489}, no 3,5
4f. Killer pair 4,7 in 23(4) cage and R7C7, locked for R7
4g. 10(3) cage at R8C4 (step 1j) = {136/235} (cannot be {127} which clashes with 23(4) cage), no 7
4h. 14(3) cage at R8C3 = {158/167/248/356} (cannot be {257} which clashes with R89C1, cannot be {347} because 4,7 only in R8C3)
4i. 7 in N7 only in 14(3) cage at R7C1 = {167} or 14(3) cage at R8C3 = {167} or R89C1 = [71] -> R89C1 = {35}/[71] (cannot be {26} which clashes with {167}, blocking cages), no 2,6
4j. 14(3) cage at R7C1 = {167/248/356} (cannot be {158} which clashes with R89C1)
4k. 4,7 of {167/248} must be in R8C2 -> no 1,2 in R8C2
4l. 14(3) cage at R8C3 = {167/248/356} (cannot be {158} which clashes with R89C1)
4m. 4,7 of {167/248} must be in R8C3 -> no 1,2 in R8C3
4n. 3,8 of {248/356} must be in R9C2 -> no 2,5 in R9C2
4o. 1 of {167} must be in R9C3 (R89C3 cannot be [76] which clashes with R45C3), no 1 in R9C2
4p. Hidden killer pair 1,2 in R8C45 and R8C8 for R8, 10(3) cage at R8C4 and R8C8 cannot contain both of 1,2 -> R8C45 and R8C8 must each contain one of 1,2
4q. R8C8 = {12}
4r. 1,2 of 10(3) cage must be in R8C45 -> no 1,2 in R9C4
5a. 4 in C1 only in R123C1, locked for N1, clean-up: no 5 in R1C23
5b. Naked pair {27} in R1C23, locked for R1 and N1
5c. 30(7) disjoint cage at R1C5 must contain 2, CPE no 2 in R2C45
5d. 45 rule on R123 4 innies R3C5789 = 23 = {1679/2579/3479/3569}, 9 locked for R3
6a. 22(4) cage at R5C2 (step 3c) = {1579/1678/2479/3478}, 14(3) cage at R8C3 (step 4l) = {167/248/356}
6b. Consider combinations for R45C3 = {47/56}
R45C3 = {47}, locked for C3 => 14(3) cage = {356}
or R45C3 = {56}, locked for N4 and 22(4) cage = {2479}, caged X-Wing for 2 in R1C23 and 22(4) cage, no other 2 in C23 => 14(3) cage = {167/356}
or R45C3 = {56}, locked for N4 and 22(4) cage = {3478} => R6C3 = 4 => 14(3) cage = {167/356}
-> 14(3) cage = {167/356}, no 2,4,8
6c. R7C12 = {28}, R8C2 = 4 (hidden triple in N7), 2,8 locked for R7
6d. Killer pair 2,8 in R56C1 and R7C1, locked for C1, clean-up: no 6 in R4C2
6e. 23(4) cage at R7C3 (step 4e) = {1679} (only remaining combination), locked for N8, 1,7 locked for R7, R7C7 = 4, R8C7 = 9 -> R8C6 = 8 (cage sum), R89C9 = [68]
6f. R9C8 = 2 (hidden single in R9) -> R8C8 = 1
6g. Killer pair 5,7 in R45C3 and R8C3, locked for C3 -> R1C23 = [72], clean-up: no 1 in R4C1
6h. 2 in R2 only in R2C679, locked for 30(7) cage at R1C5, no 2 in R3C6
6i. 4,9 in C4 only in R12345C4, CPE no 4,9 in R3C5
6j. 9 in R3 only in R3C89, locked for N3
6k. R3C5 + R5C78 = 20 (step 2a)
6l. Max R3C5 = 7 -> min R5C78 = 13, no 2,3,4
6m. 33(7) cage at R4C6 contains 2, locked for N5
7a. 17(3) cage at R1C7 = {368/458/467}, no 1
7b. 1 in N3 only in R1C9 + R2C79 + R3C7, CPE no 1 in R3C6
7c. R3C5789 (step 5d) = {1679/2579/3569} (cannot be {3479} = [73]{49} which clashes with 17(3) cage), no 4
7d. 3 of {3569} must be in R3C78 (cannot be 3{569} which clashes with 17(3) cage), no 3 in R3C5
7e. 3 in R3 only in R3C678, CPE no 3 in R1C9
7f. 25(4) cage at R3C5 = {3589/3679/4579/4678}
7g. 7 of {3679/4678} must be in R3C5 -> no 6 in R3C5
7h. Consider combinations for 17(3) cage = {368/458/467}
17(3) cage = {368/467}, 6 locked for N3 => R3C5789 = {2579}
or 17(3) cage = {458}, 5 locked for N3, 30(7) cage at R1C5 must contain 5, locked for N2 => R3C5 = 7 => R3C5789 = {1679/2579}
-> R3C5789 = {1679/2579}, no 3
7i. R3C6 = 3 (hidden single in R3)
[Cracked, fairly straightforward from here; no more routine clean-ups.]
7j. 3 in N3 only in 17(3) cage = {368}, 6,8 locked for N3
7k. R3C5789 = {2579} (only remaining combination), 2,5 locked for R3, 2 locked for N3
7l. 4 in N3 only in R12C9, locked for C9
7m. 18(3) cage at R3C9 = {279} (only remaining combination), 2,7 locked for C9
7n. Naked triple {145} in R1C9 + R2C79, locked for 30(7) cage at R1C5, 5 locked for N3 -> R3C7 = 2
7o. R3C5 = 5 (hidden single in R3)
8a. R2C6 = 2 (hidden single in N2)
8b. R23C6 = [23] = 5, R1C9 + R2C79 = {145} = 10 -> R1C56 = 15 = {69}, locked for R1 and N2
8c. Naked triple {148} in R123C4, locked for C4 and N2 -> R2C5 = 7, R7C456 = [617]
8d. Naked pair {38} in R1C78, 8 locked for R1 and N3
8e. Naked pair {39} in R45C4, locked for N5, 3 locked for C4
8f. R3C5 = 5, R45C4 = {39} = 12 -> R4C5 = 8 (cage sum)
8g. 33(7) cage at R4C6 = {1245678}, no 9
8h. R3C5 = 5 -> R5C78 = 15 (step 2a) = [87], R1C7 = 3
8i. Naked pair {29} in R45C9, locked for N6
8j. R3C78 = [29] = 11 -> R4C78 = 6 = [15]
8k. R4C12 = [62] (only remaining permutation) -> R56C1 = [38] (only remaining permutation), R8C1 = 7 (hidden single in C1)
and the rest is naked singles.