Prelims
a) R2C45 = {17/26/35}, no 4,8,9
b) R7C12 = {16/25/34}, no 7,8,9
c) R78C5 = {79}
d) R89C9 = {29/38/47/56}, no 1
e) R9C45 = {16/25/34}, no 7,8,9
f) 10(3) cage at R1C1 = {127/136/145/235}, no 8,9
g) 11(3) cage at R1C7 = {128/137/146/236/245}, no 9
h) 23(3) cage at R8C1 = {689}
i) 26(4) at R2C7 = {2789/3689/4589/4679/5678}, no 1
Steps resulting from Prelims
1a. Naked pair {79} in R78C5, locked for C5 and N8, clean-up: no 1 in R2C4
1b. Naked triple {689} in 23(3) cage at R8C1, locked for N7, clean-up: no 1 in R7C12
2. 45 rule on N7 2 outies R78C4 = 14 = {68}, locked for C4 and N8, clean-up: no 2 in R2C5, no 1 in R9C45
2a. Naked triple {689} in R8C124, locked for R8, 9 also locked for N7 -> R78C5 = [97], clean-up: no 2,3,4,5 in R9C9
2b. 1 in N8 only in R789C6, locked for C6
3. 35(6) cage at R3C5 = {146789/236789/245789/345689}, 9 locked for R6 and N5
4. 45 rule on N3 2 innies R23C9 = 8 = {17/26/35}, no 4,8,9
5. 45 rule on C6789 2 innies R16C6 = 16 = {79}, locked for C6
6. 45 rule on R1 1 innie R1C1 = 1 outie R2C2 + 4 -> R1C1 = {567}, R2C2 = {123}
6a. 10(3) cage at R1C1 = {127/136/145/235}
6b. R1C1 = {567} -> no 5,6,7 in R23C1
6c. 10(3) cage = {127/136/235} (cannot be {145} which clashes with R1C1 + R2C2 = [51], IOD clash), no 4 in R23C1
6d. Naked triple {123} in R2C12 + R3C1, locked for N1
6e. Killer triple 1,2,3 in R2C12 and R2C45, locked for R2, clean-up: no 5,6,7 in R3C9 (step 4)
7. 45 rule on N1245 1 innie R5C6 = 6
7a. R5C6 = 6 -> R45C7 = 8 = {17/35}
8. 35(6) cage at R3C5 = {146789/236789/245789/345689}, 8 locked for C5
8a. 7,9 of {146789/236789/245789} only in R6C46 -> no 1,2 in R6C46
8b. 6 of {146789/236789/345689} must be in R3C5 -> no 1,3 in R3C5
9. 45 rule on C9, using R23C9 = 8, 2 remaining innies R14C9 = 11 = [29]/{38/47/56}, no 1 in R1C9, no 1,2 in R4C9
10. 45 rule on N78 1 outie R7C7 = 1 innie R9C6 + 5 -> R7C7 = {678}, R9C6 = {123}
11. 45 rule on R789 3 innies R7C89 + R8C8 = 13 must contain one of 6,7,8 in R7C89
11a. Killer triple 6,7,8 in R7C4, R7C7 and R7C89, locked for R7
11b. 7 in R7 only in R7C789, locked for N9, clean-up: no 4 in R8C9
[25(5) cage at R2C9 and 15(3) cage at R5C9 almost form a combined cage, but the value in R7C9 can be repeated in R45C8 so I can’t see any way to use this observation.]
12. 45 rule on N4 2 outies R2C3 + R3C2 = 1 innie R5C3 + 6, IOU no 6 in R3C2
12a. Min R2C3 + R3C2 = 9 -> min R5C3 = 3
13. 4 in C9 only in R14C9 (step 9) or in 15(3) cage at R5C9 -> 15(3) cage = {159/168/249/348/456} (cannot be {258} which clashes with R89C9, cannot be {267/357}, locking out cages), no 7
[Alternatively 1 in R23C9 and 15(3) cage gives the same result.]
13a. 7 in R7 only in R7C78, CPE no 7 in R6C7
[I can see a contradiction move to eliminate one combination from 35(6) cage at R3C5, but there must be something better. Yes, there is, but it took me a long time to spot part of this step…]
14. 9 in R1 only in 30(6) cage at R2C1 = {124689/134589/134679/234579} (cannot be {125679} which clashes with R1C1, cannot be {123789} which clashes with R2C45), 4 locked for R1, clean-up: no 7 in R4C9 (step 9)
[Cracked. The rest is fairly straightforward.]
15. 7 in C9 only in R12C9, locked for N3
15a. 26(4) cage at R2C7 = {3689/4589}, no 2, 8 locked for N3, clean-up: no 3 in R4C9 (step 9)
15b. 8 in R1 only in R1C23, locked for N1
15c. 30(6) cage at R2C1 (step 14) contains 8 = {124689/134589}, no 7 -> R16C6 = [97]
15c. 30(6) cage = {124689/134589}, CPE no 1 in R2C5, clean-up: no 7 in R2C4
15d. 9 in N1 only in R2C3 + R3C23, CPE no 9 in R4C3
16. R3C4 = 7 (hidden single in N2)
16a. R6C4 = 9 (hidden single in C4)
16b. 1 in R2 only in R2C12, locked for N1
16c. R3C9 = 1 (hidden single in R3), R2C9 = 7 (step 4)
16d. 11(3) cage at R1C7 = {236} (only remaining combination), locked for R1 and N3
16e. R1C1 = 7 (hidden single in R1), R2C2 = 3 (step 6), R23C1 = [12]
16f. R2C4 = 2 (hidden single in R2) -> R2C5 = 6
16g. Clean-up: no 4,6 in R4C9 (step 9), no 4 in R7C1, no 5 in R7C2, no 5 in R9C5
17. R3C6 = 3 (hidden single in N2) -> R24C6 = 12 = {48}, locked for C6, clean-up: no 8 in R7C7 (step 10)
18. R3C3 = 6 (hidden single in N1), R3C4 = 7 -> 25(5) cage at R3C3 = {13678/34567}, no 9
18a. 25(5) cage = {13678/34567}, CPE no 3 in R5C5
19. 45 rule on N4 2 outies R2C3 + R3C2 = R5C3 + 6 (step 12)
19a. R2C3 + R3C2 from {459} = {45/59} (cannot be {49} because no 7 in R5C3), 5 locked for N1 and 36(7) cage at R2C3, no 5 in R4C123 + R5C2 + R6C3
19b. Naked pair {48} in R1C23, locked for R1 and N1
19c. Naked pair {59} in R2C3 + R3C2, locked for N1 and 36(7) cage at R2C3, no 9 in R4C12 + R5C2
19d. R2C3 + R3C2 = {59} = 14 -> R5C3 = 8, R1C23 = [84]
19e. 25(5) cage at R3C3 (step 18) = {13678} (only remaining combination) -> R45C4 = {13}, locked for C4 and N5 -> R1C45 = [51], R9C4 = 4 -> R9C5 = 3
20. 36(7) cage at R2C3 = {2345679} (only remaining combination), no 1
20a. R5C1 = 9, R6C2 = 1, R6C1 = 5 (hidden triple in N4), R7C1 = 3 -> R7C2 = 4
20b. Naked pair {68} in R89C1, locked for C1 and N7 -> R4C1 = 4, R8C2 = 9, R3C2 = 5, R2C3 = 9, R24C6 = [48], R3C5 = 8
20c. R4C2 = 6 (hidden single in N4)
21. R45C7 = {17} (hidden pair in N6), locked for C7 -> R7C7 = 6, R9C6 = 1 (step 10), R7C4 = 8
21a. Naked pair {25} in R7C69, locked for R7 -> R7C3 = 1, R7C8 = 7
21b. Naked pair {25} in R8C36, locked for R8 -> R8C9 = 3 -> R9C9 = 8, R8C78 = [41], R3C78 = [94]
21c. R8C7 = 4, R9C6 = 1 -> R9C78 = 11 = [29], R7C9 = 5, R4C9 = 9 -> R1C9 = 2 (step 9)
and the rest is naked singles.