Prelims
a) R1C34 = {59/68}
b) R1C78 = {16/25/34}, no 7,8,9
c) R34C1 = {29/38/47/56}, no 1
d) R34C4 = {17/26/35}, no 4,8,9
e) R3C56 = {17/26/35}, no 4,8,9
f) R3C78 = {29/38/47/56}, no 1
g) R45C7 = {13}
h) R78C1 = {29/38/47/56}, no 1
i) R78C3 = {29/38/47/56}, no 1
j) 9(3) cage at R2C2 = {126/135/234}, no 7,8,9
k) 26(4) cage at R4C8 = {2789/3689/4589/4679/5678}, no 1
1. Naked pair {13} in R45C7, locked for C7 and N6, clean-up: no 4,6 in R1C8, no 8 in R3C8
2. 45 rule on C123 2 outies R19C4 = 13 = [58/67/85/94]
3. 45 rule on N7 1 outie R9C4 = 1 innie R7C2 + 1, R9C4 = {4578} -> R7C2 = {3467}
4. 45 rule on N8 2 innies R9C46 = 11 = [47/56/74/83]
5. 45 rule on N9 2 outies R6C9 + R9C6 = 9 = [27/54/63], clean-up: no 5 in R9C4 (step 4), no 8 in R1C4 (step 2), no 6 in R1C3, no 4 in R7C2 (step 3)
6. 45 rule on N5 1 outie R6C7 = 1 innie R4C4 + 1, no 2 in R4C4, no 5,9 in R6C7, clean-up: no 6 in R3C4
7. 45 rule on N3 1 innie R2C7 = 1 outie R4C9 + 3, no 2,4,6 in R2C7, no 7,8,9 in R4C9
8. 45 rule on C3 3 innies R129C3 = 12 = {129/138/156/345} (cannot be {147/237/246} because R1C3 only contains 5,8,9), no 7
8a. R1C3 = {589} -> no 5,8,9 in R29C3
9. 45 rule on R789 3 outies R6C129 = 14
9a. 15(3) cage at R6C1 = {168/267/348/357/456} (cannot be {159/249/258} because R7C2 only contains 3,6,7), no 9
9b. 6 of {168/267/456} must be in R7C2 (R6C12 cannot be {26} because R6C129 cannot be {26}6), no 6 in R6C12
10. 45 rule on N6 3 innies R4C9 + R6C79 = 15 = {258/267/456}
10a. 7,8 of {258/267} must be in R6C7 -> no 2 in R6C7, clean-up: no 1 in R4C4 (step 6), no 7 in R3C4
11. 45 rule on R12 3(2+1) outies R3C2 + R34C9 = 17
11a. Max R3C2 + R4C9 = 12 -> min R3C9 = 5
11b. Max R34C9 = 15 -> min R3C2 = 2
12. R2C7 = R4C9 + 3 (step 7), R6C7 = R4C4 + 1 (step 6), R2C7 cannot be the same as R6C7 -> R4C4 cannot be 2 more than R4C9
12a. 45 rule on N23 3(2+1) outies R1C3 + R4C49 = 17 = [836/854/872/935/962] (cannot be [575] because R4C4 cannot be 2 more than R4C9), no 5 in R1C3, clean-up: no 9 in R1C4, no 4 in R9C4 (step 2), no 7 in R9C6 (step 4), no 2 in R6C9 (step 5), no 3 in R7C2 (step 3)
12b. R129C3 (step 8) = {129/138} -> R29C3 = {12/13}, 1 locked for C3
13. 45 rule on N2356 2(1+1) innies R1C4 + R6C9 = 11 = {56}, CPE no 5,6 in R1C9 + R6C4
14. 18(3) cage at R1C1 = {279/369/378/459/468/567} (cannot be {189} which clashes with R1C3), no 1
14a. 1 in N1 only in R2C23, locked for R2
14b. 9(3) cage at R2C2 = {126/135}, no 4
14c. 1 in R3 only in R3C456, locked for N2
15. 45 rule on C123 1 innie R1C3 = 1 outie R9C4 + 1 -> R1C3 + R9C4 = [87/98]
15a. R1C3 + R4C49 (step 12a) = [836/854/935/962] (cannot be [872], IOD clash) -> no 7 in R4C4, clean-up: no 1 in R3C4, no 8 in R6C7 (step 6)
16. 1 in N2 only in R3C56 = {17}, locked for R3 and N2, clean-up: no 4 in R3C78, no 4 in R4C1
16a. 18(3) cage at R1C5 = {369/459/468}, no 2
16b. Killer pair 5,6 in R1C4 and 18(3) cage, locked for N2, clean-up: no 3 in R4C4, no 4 in R6C7 (step 6)
16c. Naked pair {56} in R14C4, locked for C4
17. R4C9 + R6C79 (step 10) = {267/456}
17a. 2,4 only in R4C9 -> R4C9 = {24}, clean-up: no 8,9 in R2C7 (step 7)
17b. R4C9 + R6C79 = {267/456}, 6 locked for R6 and N6
18. R3C2 + R34C9 = 17 (step 11)
18a. Max R3C2 + R4C9 = 10 -> no 5,6 in R3C9
19. 4 in R3 only in R3C13, locked for N1
19a. 45 rule on N1 3 innies R1C3 + R3C13 = 18 contains 4 = {459/468}
19b. R1C3 = {89} -> R3C13 = {45/46}, clean-up: R4C1 = {567}
19c. R3C78 = [29/83/92] (cannot be {56} which clashes with R3C13, ALS block), no 5,6 in R3C78
19d. Killer pair 2,3 in R3C4 and R3C78, locked for R3
19e. Killer pair 8,9 in R3C78 and R3C9, locked for N3
19f. Naked triple {456} in R3C123, locked for N1
[I wasn’t sure how much I would get from analysing this cage, but it seemed to be the right thing to try next.]
20. 24(5) cage at R1C9 = {12678/14568/23478} (cannot be {12579/13578/24567} which clash with R2C7, cannot be {12489} because 8,9 only in R3C9, cannot be {13569/23568} which clash with R1C78, cannot be {13479/23469} which clash with R3C78) -> R3C9 = 8, clean-up: no 3 in R3C8
20a. Naked pair {29} in R3C78, locked for R3 and N3 -> R3C4 = 3, R4C4 = 5, R1C4 = 6, R1C3 = 8, R9C4 = 7 (step 2), R9C6 = 4 (step 4), R6C9 = 5 (step 5), R7C2 = 6 (step 3), R3C2 = 5
20b. R1C78 = {34} (only remaining combination) -> R1C7 = 4, R1C8 = 3
20c. R34C1 = {47} (only remaining combination) -> R3C1 = 4, R4C1 = 7
20d. R1C2 = 7 (hidden single in N1) -> R12C1 = 11 = {29}, locked for C1 and N1
20e. R78C1 = {38} (only remaining combination), locked for C1 and N7 -> R6C1 = 1, R6C2 = 8 (cage sum), R9C1 = 5
21. R5C1 = 6 -> R45C2 = 7 = {34}, locked for C2 and N4 -> R2C23 = [13]
21a. Naked pair {29} in R89C2, locked for N7 -> R9C3 = 1
22. 17(3) cage at R1C6 = {278} (only possible combination) -> R1C6 = 2, R2C6 = 8, R2C7 = 7, R12C1 = [92], R1C5 = 5, R6C7 = 6, R4C9 = 4 (step 17), R1C9 = 1, R2C89 = [56]
23. R78C8 = {14} (hidden pair in C8) = 5 -> R78C7 = 14 = {59}, locked for C7 and N9
23a. Naked triple {237} in R789C9, locked for C9 and N9 -> R5C9 = 9, R9C78 = [86]
24. 19(4) cage at R5C6 contains 6 = {1369} (only possible combination, cannot be {2467} because 2,4 only in R6C5) -> R5C6 = 1, R6C56 = {39}, locked for R6 and N5
25. 14(3) cage at R7C4 = {158} (only possible combination, cannot be {239} which clashes with R9C5), locked for R7 and N8 -> R7C4 = 1 (hidden single in C4), R7C5 = 8, R7C6 = 5
and the rest is naked singles.