Prelims
a) R23C6 = {59/68}
b) R34C5 = {29/38/47/56}, no 1
c) R34C8 = {18/27/36/45}, no 9
d) R56C1 = {49/58/67}, no 1,2,3
e) R56C5 = {19/28/37/46}, no 5
f) R67C3 = {39/48/57}, no 1,2,6
g) R67C8 = {19/28/37/46}, no 5
h) R78C5 = {79}
i) 20(3) cage at R1C1 = {389/479/569/578}, no 1,2
j) 7(3) cage at R1C5 = {124}
k) 21(3) cage at R2C7 = {489/579/678}, no 1,2,3
1a. Naked pair {79} in R78C5, locked for C5 and N8, clean-up: no 2,4 in R34C5, no 1,3 in R56C5
1b. Naked triple {124} in 7(3) cage at R1C5, locked for N2
1c. 45 rule on C1234 2 innies R89C4 = 3 = {12}, locked for C4 and N8
1d. 45 rule on N8 2 innies R7C46 = 10 = {46}, locked for R7 and N8, clean-up: no 8 in R6C3, no 4,6 in R6C8
1e. Killer pair 6,8 in R34C5 and R56C5, locked for C5
1f. 8 in N8 only in R89C6, locked for C6, clean-up: no 6 in R23C6
1g. Naked pair {59} in R23C6, locked for C6 and N2, clean-up: no 6 in R4C5
1h. Naked pair {38} in R89C6, 3 locked for C6 and N8 -> R9C5 = 5, clean-up: no 6 in R3C5
1i. Naked pair {38} in R34C5, 8 locked for C5, clean-up: no 2 in R56C5
1j. Naked pair {46} in R56C5, locked for N5, 4 locked for C5
1k. Naked triple {127} in R456C6, 1,2 locked for C6, 7 locked for N5
1l. R1C6 = 4 -> R7C46 = [46]
1m. 24(4) cage at R1C3 = {3678} (only remaining combination)
1n. 45 rule on N2 1 outie R1C3 = 1 innie R3C5 = {38}
1o. 20(3) cage at R1C1 = {479/569/578} (cannot be {389} which clashes with R1C3), no 3
2a. 45 rule on N3 2 outies R4C78 = 12 = {48/57}/[93], no 6, no 1,2 in R4C8, clean-up: no 3,7,8 in R3C8
2b. 45 rule on N7 2 outies R6C23 = 10 = [19/37/73] (cannot be [64] which clashes with R6C5), R6C2 = {137}, no 4,5 in R6C3, clean-up: no 7,8 in R7C3
2c. 45 rule on N9 2 outies R6C78 = 10 = {19/28/37}, no 4,5,6 in R6C7
3a. 45 rule on C123 3(1+2) innies R1C3 + R5C23 = 11 = 3+8/8+3 = 3{26}/8{12} (cannot be 3{17} which clashes with R6C23, cannot be 3[35] which clashes with R456C4 ALS block) -> R5C23 = {12/26}, 2 locked for R5 and N4
3b. 45 rule on C789 2 innies R5C78 = 7 = {34} (cannot be {16} which clashes with R5C23), locked for R5 and N6 -> R56C5 = [64], R5C23 = {12}, 1 locked for R5 and N4 -> R5C6 = 7
3c. R1C3 + R5C23 = 11, R5C23 = 3 -> R1C3 = 8, R123C4 = {367}, 3 locked for C4 and N2 -> R34C5 = [83]
3d. R6C23 (step 2b) = {37}, 7 locked for R6 and N4
3e. Naked pair {37} in R6C23, CPE no 3,7 in R8C3
3f. R4C78 (step 2a) = {57}, locked for N6, 5 locked for R4
3g. R56C1 = {58} (only remaining combination), locked for C1, 8 locked for N4
3h. Naked triple {469} in R4C123, 6,9 locked for R4 -> R4C4 = 8
3i. R6C9 = 6 (hidden single in R6)
3j. 21(3) cage at R2C7 = {579/678} (cannot be {489} because R4C7 only contains 5,7), no 4, 7 locked for C7
3k. 20(3) cage at R1C1 = {479/569}, 9 locked for N1
Clean-ups: R3C8 = {24}, no 3 in R7C3, no 3,7 in R7C8
[Plenty of progress from that easy routine start, now things start to get harder.]
4a. R4C12 = {469} -> 18(4) cage at R2C1 = {17}{46}/{23}{49}/{12}{69}, no 4,6 in R23C1
4b. 16(4) cage at R6C2 = {1267/1348/1357/2347/2356} (cannot be {1249/1258/1456} because R6C2 only contains 3,7), no 9
4c. 16(4) cage = {1267/1348/1357/2347} (cannot be {2356} = 3{256} which clashes with R67C3 = [75])
4d. 6 of {1267} must be in R8C2 (R78C2 cannot be {12} which clashes with R5C2), no 6 in R8C3
5a. 25(4) cage at R7C9 = {1789/3589/3679/4579/4678} (cannot be {2689} which clashes with R5C9), no 2
5b. 6 of {3679} must be in R8C8, 1,3 of {1789/3589} must be in R789C9 because R789C9 = {589/789} clashes with R5C9), no 1,3 in R8C8
5c. 45 rule on N9 1 outie R6C7 = 1 innie R7C8 -> R45C9 + R67C8 form naked quad {1289}
5d. 25(4) cage at R7C9 = {3589/3679/4579/4678} (cannot be {1789} which clashes with R45C9 + R67C8), no 1
5e. 25(4) cage = {3589/3679/4579/4678} -> R7C78 + R8C7 + R9C78 = {12359/12368/12458/12467}
5f. Consider placements for R34C8 = [27/45]
R34C8 = [27] => R4C7 = 5 => R7C78 + R8C7 + R9C78 = {12368}
or R34C8 = [45] => R5C7 = 4 => R7C78 + R8C7 + R9C78 = {12359/12368}
-> R7C78 + R8C7 + R9C78 = {12359/12368}, no 4,7, 3 locked for N9
[With hindsight, this step would also have deleted R7C78 + R8C7 + R9C78 = {23456} and therefore 25(4) cage = {1789}; I enjoyed finding the way I eliminated it.]
5g. 7 in N9 only in 25(4) cage = {4579/4678}
5h. 6 of {4678} must be in R8C8 -> no 8 in R8C8
5i. 3 in C9 only in R123C9, locked for N3
5j. R5C7 = 4 (hidden single in C7) -> R5C8 = 3
5k. 45 rule on C89 1 outie R1C7 = 1 remaining innie R9C8 = {1269}
6a. 5 in N7 only in R78C23
6b. 45 rule on N7 4 innies R78C23 = 18 = {1359/1458/2358/2457/3456}
6c. 16(4) cage at R6C2 (step 4c) and R78C23 share 3 cells -> 16(4) cage = {1348/1357/2347} (cannot be {1267} which doesn’t contain any of 3,4,5), no 6, 3 locked for C8
6d. R78C23 = {1359/1458/2457} (cannot be {2358} which only contains two numbers in 16(4) cage)
6e. Variable hidden killer triple 1,2,3 in R23C1 and R789C1 for C1, R23C1 (step 4a) = {12}/{17}/{23} -> R789C1 must contain at least 1 or 3
6f. R78C23 = {1458/2457} (cannot be {1359} which clashes with R789C1), no 3,9, 4 locked for R8 and N7
6g. R7C3 = 5 -> R6C23 = [37]
6h. R9C9 = 4 (hidden single in R9}
6i. R4C1 = 4 (hidden single in C1)
6j. Naked pair {69} in R4C23, CPE no 6 in R3C2
6k. Hidden killer pair 1,2 in R23C1 and R789C1 for C1, 18(4) cage at R23C1 = {17}[49]/{23}[46] -> R789C1 must contain one of 1,2
6l. Killer pair 1,2 in R789C1 and R78C23, locked for N7
6m. 18(4) cage at R2C3 = {1467/2349/3456} (cannot be {1269/1359/2367/2457} which clash with R23C1), 4 locked for N1
6n. 20(3) cage at R1C1 = {569} (only remaining combination), 5,6 locked for N1
6o. 18(4) cage at R2C3 = {2349} (cannot be {1467} because R234C3 = {146} clashes with R58C3, ALS block)
6p. R4C23 = [69], R2C3 + R3C23 = {234}, 2,3 locked for N1, 3 locked for C3 -> R9C3 = 6
6q. Naked pair {59} in R12C2, 9 locked for R12C2 and N1 -> R1C1 = 6
6r. Naked pair {17} in R23C1, locked for C1
6s. 2 in C1 only in R789C1, locked for N7
6t. R9C2 = 7 (hidden single in R9)
7a. Combined half cage R67C8 and R9C8 = {19}2/{28}1/{28}9, 2 locked for C8
7b. R3C8 = 4 -> R4C78 = [75], R3C23 = [23], R5C23 = [12], R78C2 = [84], R8C3 = 1, R89C4 = [21]
7c. Naked pair {79} in R7C59, 9 locked for R7
7d. R7C8 = {12} -> R6C8 = {89}, R6C7 = {12} (step 2c)
7e. Killer pair 2,9 in R67C8 and R9C8, 9 locked for C8
7f. R9C8 = {29} -> R1C7 (step 5k) = {29}
7g. 25(4) cage at R7C9 = {4579/4678}
7h. Killer pair 8,9 in R5C9 and R78C9, locked for C9
7i. 9 in N3 only in R123C7, locked for C7
8a. Consider placements for R1C8 = {17}
R1C8 = 1 => R4C9 = 1 (hidden single in C9), R5C9 = 9 (cage sum) => R7C9 = 7
or R1C8 = 7
-> R8C8 = 6
8b. R8C8 = 6, R9C9 = 4 -> R78C9 = 15 = [78] -> R5C9 = 9, R4C9 = 1 (cage sum), R6C8 = 8 -> R7C8 = 2
and the rest is naked singles.