Prelims
a) R1C12 = {89}
b) R12C7 = {29/38/47/56}, no 1
c) R4C12 = {17/26/35}, no 4,8,9
d) R4C45 = {17/26/35}, no 4,8,9
e) R6C23 = {29/38/47/56}, no 1
f) R7C23 = {89}
g) R89C2 = {13}
h) R89C3 = {29/38/47/56}, no 1
i) R8C56 = {16/25/34}, no 7,8,9
j) 8(3) cage at R1C3 = {125/134}
k) 22(3) cage at R9C4 = {589/679}
l) 40(8) cage at R1C6 = {12346789}, no 5
Steps resulting from Prelims
1a. Naked pair {89} in R1C12, locked for R1 and N1, clean-up: no 2,3 in R2C7
1b. Naked pair {89} in R7C23, locked for R7 and N7, clean-up: no 2,3 in R89C3
1c. Naked pair {13} in R89C2, locked for C2 and N7, clean-up: no 5,7 in R4C1, no 8 in R6C3
1d. Naked pair {89} in R17C2, locked for C2, clean-up: no 2,3 in R6C3
1e. 8(3) cage at R1C3 = {125/134}, 1 locked for R1
1f. 22(3) cage at R9C4 = {589/679}, 9 locked for R9 and N8
1g. 2 in N7 only in R789C1, locked for C1 and 29(6) cage at R5C1, clean-up: no 6 in R4C2
1h. 6,7 in R1 only in R1C6789, CPE no 6,7 in R3C7
2a. 45 rule on C12 2 outies R67C3 = 15 = [69/78], R6C3 = {67} -> R6C2 = {45}
2b. Killer pair 6,7 in R6C3 and R89C3, locked for C3
2c. 45 rule on N1 3 innies R123C3 = 9 = {135/234}, 3 locked for C3 and N1
2d. Killer pair 4,5 in R123C3 and R89C3, locked for C3
2e. Max R23C3 = 8 -> min R23C4 = 13, no 1,2,3 in R23C4
2f. 45 rule on N47 2 innies R45C3 = 10 = {19/28}
2g. 45 rule on N47 2 outies R5C45 = 8 = {17/26/35}, no 4,8,9
3. 45 rule on N7 3 innies R789C1 = 13 = {247/256}
3a. 29(6) cage at R5C1 contains 2 = {124679/234569/234578} (cannot be {123689/124589/125678} which clash with R45C3) -> R5C12 + R6C1 = {169/349/358}, no 7
[Note. R5C12 + R6C1 = 16.]
3b. R5C2 = {456} -> no 4,5,6 in R56C1
3c. 4 in N4 only in R56C2, locked for C2
4a. 45 rule on N12 3 outies R3C7 + R4C67 = 15 without 5 so must contain one of 1,2,3 -> R1C6 + R23C56 must contain two of 1,2,3 for 40(8) cage
4b. Hidden killer triple 1,2,3 in R1C45 and R1C6 + R23C56 for N2, R1C6 + R23C56 contain two of 1,2,3 -> R1C45 must contain one of 1,2,3
4c. 8(3) cage at R1C3 must contain two of 1,2,3 -> R1C3 = {123}
5. 45 rule on N1 4 outies R1C45 + R23C4 = 20 and must contain 5 for N2 = {1568/2459/2567/3458}
6. 45 rule on R6789 5(2+3) innies R6C19 + R789C1 = 23, R789C1 = 13 (step 3) -> R6C19 = 10 = [19/37/82/91]
7. 45 rule on R5 using R5C45 = 8 (step 2g) 4 innies R5C1239 = 23
7a. R5C123 cannot total 16, which clashes with R5C12 + R6C1 (step 3a) = 16 (combo crossover clash) -> no 7 in R5C9
[Inserting this step simplifies things a lot; I originally analysed all nine combinations for R5C1239 = 23]
7b. R5C1239 = {1589/2489/3569} (cannot be {4568} because 4,5,6 only in R5C29), 9 locked for R5
7c. 4,5 of {1589/2489} must be in R5C2, 9 of {3569} must be in R5C3 -> R5C12 + R6C1 (step 3a) = {349/358} (cannot be {169}), no 1,6, 3 locked for N4, clean-up: no 9 in R6C9 (step 6)
7d. Naked pair {45} in R56C2, locked for C2
7e. 6 in C2 only in R23C2, locked for N1
7f. R5C1239 = {1589/2489} (cannot be {3569} because R4C3 + R5C1239 = [13596] clashes with R5C45), no 3,6
7g. R6C1 = 3 (hidden single in N4) -> R6C9 = 7 (step 6), R6C3 = 6 -> R6C2 = 5, R5C2 = 4 -> R5C1 = 9 (cage sum), R4C1 = 1 -> R4C2 = 7, clean-up: no 7 in R789C1 (step 3)
7h. R5C12 = [94] -> R5C39 = {28}, locked for R5, clean-up: no 6 in R5C45 (step 2g)
7i. Naked pair {28} in R45C3, locked for C3, clean-up: no 4 in R23C3 (step 2c)
7j. Naked triple {135} in R123C3, 5 locked for C3, N1 and 21(4) cage at R2C3
7k. 5 in N2 only in R1C45 -> 8(3) cage at R1C3 = {125} -> R1C3 = 1, R1C45 = {25}, locked for R1 and N2, clean-up: no 6,9 in R2C7
7l. 7 in 40(8) cage only in R1C6 + R23C56, locked for N2
7m. R23C3 = {35} = 8 -> R23C4 = 13 = {49}, locked for C4 and N2
7n. 4 in R1 only in R1C789, locked for N3, clean-up: no 7 in R1C7
7o. 40(8) cage at R1C6 = {12346789} -> R3C7 = {29}, R4C67 = {249}, 4 locked for R4
7p. 45 rule on N3 3 innies R3C789 = 12 = {129} (only remaining combination, cannot be {138/156} because R3C7 only contains 2,9), locked for R3 and N3 -> R23C2 = [26], R23C4 =
[94], R23C1 = [47]
7q. Naked pair {38} in R3C56, locked for R3 and N2 -> R23C3 = [35]
[I really hadn’t expected to get so much from that 45 for R5.]
8. 1 in R3 only in 33(6) cage at R3C8 = {126789/135789}, 8 locked for N6
9. 14(3) cage at R5C6 = {167/356}
9a. 7 of {167} must be in R5C6 -> no 1 in R5C6
10. 17(4) cage at R6C4 = {1268/1358/2348} (cannot be {1259/2456} which clash with R1C4, cannot be {1349} because 4,9 only in R6C5, cannot be {1457} which clashes with 22(3) cage at R9C4, cannot be {1367/2357} because 3,5,6,7 only in R78C4), no 7,9
10a. R9C5 = 9 (hidden single in C5)
10b. 45 rule on N8 4 innies R7C456 + R8C4 = 16 = {1258/1267/1348/2347} (cannot be {1357/1456/2356} which clash with 22(3) cage at R9C4)
10c. 17(4) cage = {1268/2348} (cannot be {1358} = {18}{35} because R7C456 + R8C4 cannot contain both of 3,5), no 5
[Alternatively {1358} prevents 8 being placed in N8.]
10d. Consider placements for R1C45 = {25}
R1C4 = 2 => 17(4) cage = {1268}
or R1C4 = 5, R1C5 = 2, no 6 in R4C4 => R4C4 = {23} => 17(4) cage = {1268} (cannot be {2348} which clashes with R4C4)
-> 17(4) cage = {1268}, no 3,4, 6 locked for C4 and N8, clean-up: no 2 in R4C5, no 1 in R8C56, no 7 in 22(3) cage at R9C4
[Alternatively 17(4) cage = {2348} forces R1C45 = [52], R4C45 = [62].]
10e. Naked triple {589} in 22(3) cage at R9C4, 5,8 locked for R9 and N8, clean-up: no 2 in R8C56
10f. Naked pair {34} in R8C56, locked for R8 and N8 -> R89C2 = [13], R89C3 = [74]
10g. R5C4 = 7 (hidden single in C4) -> R5C5 = 1 (step 2g)
10h. R4C4 = 3 (hidden single in C4) -> R4C5 = 5, R5C6 = 6, R1C6 = 7, R2C56 = [61], R7C56 = [72], R78C4 = [16]
11. 1,7 in R9 only in R9C789 -> 19(4) cage at R8C7 = {1279} (cannot be {1567} which clashes with R7C789, ALS block) -> R8C7 = 9, R9C789 = {127}, locked for R9 and N9
11a. R2C7 = 8 (hidden single in C7) -> R1C7 = 3
and the rest is naked singles.