Prelims
a) R1C67 = {16/25/34}
b) R23C5 = {69/78}
c) R34C9 = {19/28/37/46}, no 5
d) R5C23 = {49/58/67}, no 1,2,3
e) R5C78 = {15/24}
f) R67C1 = {29/38/47/56}, no 1
g) R78C5 = {18/27/36/45}, no 9
h) R9C34 = {16/25/34}
i) 19(3) cage at R8C9 = {289/379/469/478/568}, no 1
1a. 45 rule on N1 R3C3 = 8 -> R3C4 + R4C3 = 7 = {16/25/34}, no 7,9, clean-up: no 7 in R2C5, no 2 in R4C9, no 5 in R5C2
1b. 45 rule on C6789 4 innies R5689C6 = 30 = {6789}, locked for C6, clean-up: no 1 in R1C7
1c. Min R89C6 = 13 -> max R9C5 = 5
1d. 45 rule on N9 R7C7 = 3 -> R6C7 + R7C6 = 11 = [65/74/92], clean-up: no 4 in R1C6, no 6 in R8C5
1e. 45 rule on N3 2 innies R1C7 + R3C9 = 7 = [43/52/61], R1C6 = {123}, R4C9 = {789}
1f. 45 rule on N5 2 innies R4C6 + R6C4 = 7 = [16]/{25/34}
1g. 45 rule on N7 2 innies R7C1 + R9C3 = 6 = [24/42/51], R6C1 = {679}, R9C4 = {356}
2a. 12(3) cage at R5C9 = {138/156/237/246} (cannot be {129/147/345} which clash with R5C78), no 9
2b. Killer pair 1,2 in R5C78 and 12(3) cage, locked for N6
2c. Consider combinations for 12(3) cage
12(3) cage = {138} => R5C78 = {24}, 5 in N6 only in R4C78, locked for 21(4) cage at R2C6 => R7C6 = 5 (hidden single in C6) => R6C7 = 6 (cage sum)
or 12(3) cage = {156/246} => R4C8 = 3 (hidden single in N6) => R1C6 = 3 (hidden single in C6), R1C7 = 4, R3C9 = 3 (step 1e), R4C9 = 7
or 12(3) cage = {237}, 7 locked for N6
-> no 7 in R6C7, clean-up: no 4 in R7C6 (step 1d)
2d. 4 in C6 only in R234C6, locked for 21(4) cage at R2C6, no 4 in R4C78
2e. Consider permutations for R6C7 + R7C6 = [65/92]
R6C7 + R7C6 = [65]
or R6C7 + R7C6 = [92], no 9 in R4C9 => no 1 in R3C9, no 6 in R1C7, no 1 in R1C6 => R1C6 = 3, R234C6 = {145} = 10 => R4C78 = 11 = [83]
-> no 6 in R4C78
2f. 4 in N6 only in R5C78 = {24} or 12(3) cage = {246} -> 12(3) cage = {138/156/246} (cannot be {237}, locking-out cages), no 7
2g. 7 in N6 only in R4C789, locked for R4
2h. Consider combinations for 12(3) cage
12(3) cage = {138}, 8 locked for N6
or 12(3) cage = {156/246} => R4C8 = 3 (hidden single in N6) … => R4C9 = 7 (as in step 2c)
-> no 8 in R4C9, no 2 in R3C9, no 5 in R1C7 (step 1e), no 2 in R1C6
2i. 21(5) cage at R2C6 contains 4 for C6 but no 6 = {12459/13458/23457}
[With hindsight step 5 could have been found at this stage]
3a. 45 rule on N7 1 outie R9C4 = 1 innie R7C1 + 1
3b. R7C1 + R9C4 + R7C6 = [235/452/562], 2 locked for R7, clean-up: no 7 in R8C5
3c. R7C1 + R9C4 + R7C6 = [235/452/562] -> no 5 in R7C45, clean-up: no 4 in R8C5
3d. 2 in R7 only in R7C16, 2 in R7C1 => R6C1 = 9 or 2 in R7C6 => R6C7 = 9 (cage sum) -> 9 in R6C17 (locking cages), locked for R6
3e. 18(3) cage at R8C6 = {189/279/369/378/468} (cannot be {459} because 4,5 only in R9C5, cannot be {567} which clashes with R7C1 + R9C4 + R7C6), no 5
4a. 12(3) cage at R5C9 (step 2f) = {138/156/246}
4b. 45 rule on R6789 4 innies R6C5689 = 19 = {1378/1468/2368/2458} (cannot be {1567/2467} which clash with R6C16, ALS block, cannot be {3457} because 12(3) cage cannot contain two of 3,4,5), 8 locked for R6
[I was a bit slow to spot this, the first key step]
5a. R1C7 + R3C9 = 7 (step 1e)
5b. Consider placement for R6C7 = {69}
R6C7 = 6 => R1C7 = 4, R3C9 = 3 => R4C9 = 7
or R6C7 = 9 => R4C9 = 7
-> R4C9 = 7, R3C9 = 3, R1C7 = 4, R1C6 = 3, clean-up: no 4 in R4C3 (step 1a), no 2 in R5C8, no 4 in R6C4 (step 1f)
5c. 14(3) cage at R1C8 = {158/167/257}, no 9
5d. 9 in C9 only in R789C9, locked for N9
5e. 45 rule on N2 3 remaining innies R23C6 + R3C4 = 11 = {146/245}, 4 locked for N2
5f. 6 of {146} must be in R3C4 -> no 1 in R3C4, clean-up: no 6 in R4C3 (step 1a)
6a. 21(5) cage at R2C6 (step 2i) = {12459/13458} -> R4C78 = {59}/[83]
6b. 45 rule on R123 4 remaining outies R4C3678 = 17 = {12}{59}/{15}[83]/[24][83], no 3 in R4C3, clean-up: no 4 in R3C4 (step 1a)
6c. 4 in N2 only in R23C6, locked for C6, clean-up: no 3 in R6C4 (step 1f)
6d. R4C3678 = {12}{59}/{15}[83], 1,5 locked in R4
6e. 1 in R4 only in R4C36, R4C3 = 1 => R3C4 = 6 (cage sum) or R4C6 = 1 => R6C4 = 6 (step 1f) -> 6 in R36C4 (locking cages), locked for C4, clean-up: no 1 in R9C3, no 5 in R7C1 (step 1g), no 6 in R6C1
6f. Naked pair {24} in R7C1 + R9C3, locked for N7
6g. R7C1 + R9C4 + R7C6 (step 3c) = [235/452], 5 locked for N8, clean-up: no 4 in R7C5
7a. 18(3) cage at R8C6 (step 3e) = {189/279/369/468} (cannot be {378} which clashes with R78C5)
7b. Consider permutations for R67C1 = [74/92]
R67C1 = [74] => R6C6 = {68} => 18(3) cage = {189/279/369/378} (cannot be {468} which clashes with R6C6
or R67C1 = [92] => R9C3 = 4
-> 18(3) cage = {189/279/369}, no 4, 9 locked for C6 and N8
7c. 4 in N8 only in R78C4, locked for C4 and 21(5) cage at R6C2, no 4 in R6C23
7d. 45 rule on N8 4 innies R789C4 + R7C6 = 18 with 4 in R78C4 and 5 in R7C6 + R9C4 = {2457} (only possible combination) -> R7C6 = 2, R9C4 = 5, R7C1 = 4, R6C1 = 7, R9C3 = 2, R78C4 = [74], clean-up: no 6 in R5C23
[A lot easier from here]
7e. R78C4 = [74] = 11 -> R6C234 = 10 = {13}6/{35}2
7f. Naked triple {135} in R4C3 + R6C23, locked for N4, 3 locked for R6, clean-up: no 8 in R5C2
7g. Naked pair {49} in R5C23, locked for R5 and N4, clean-up: no 2 in R5C7
7h. Naked pair {15} in R5C78, locked for R5 and N6
7i. 12(3) cage at R5C9 = {246} (only remaining combination), 6 locked for N6 -> R6C9 = 9 (or cage total from step 7e), R4C78 = [83]
8a. Naked pair {26} in R4C12, locked for R4 and N4 -> R4C45 = [94], R5C1 = 8
8b. Naked pair {26} in R36C4, 2 locked for C4
8c. Naked pair {18} in R12C4, locked for N2, R1C5 = 7 (cage sum)
8d. Naked pair {69} in R23C5, 6 locked for C5 and N2, clean-up: no 3 in R8C5
8e. Naked pair {18} in R78C5, locked for C5 and N8, R9C5 = 3
8f. R3C4 = 2 -> R4C3 = 5 (cage sum)
[Now back to the corners]
9a. 14(3) cage at R1C8 = {158} (only remaining combination), locked for N3
9b. Naked triple {158} in R1C489, 1,5 locked for R1, 5 locked for N3
9c. 1 in R3 only in R3C12, locked for N1
9d. 2 in R1 only in 16(3) cage at R1C1 = {259} -> R1C12 = {29}, locked for N1, R2C1 = 5
9e. R1C3 = 6, R2C6 = 4, R2C23 = {37}, 7 locked for R2 and N1
9f. 17(3) cage at R8C1 = {368} (cannot be {179} which clashes with R7C3) -> R89C1 = [36], R9C2 = 8
9g. R9C6 = 9, R9C89 = [74] -> R8C9 = 8 (cage sum)
and the rest is naked singles.