Prelims
a) 21(3) cage at R1C1 = {489/579/678}, no 1,2,3
b) 20(3) cage at R1C2 = {389/479/569/578}, no 1,2
c) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
d) 21(3) cage at R6C5 = {489/579/678}, no 1,2,3
e) 21(3) cage at R7C8 = {489/579/678}, no 1,2,3
f) 13(4) cage at R1C5 = {1237/1246/1345}, no 8,9
g) 12(4) cage at R3C9 = {1236/1245}, no 7,8,9
h) 27(4) cage at R5C2 = {3789/4689/5679}, no 1,2
Steps resulting from Prelims
1. 12(4) cage at R3C9 = {1236/1245}, CPE no 1,2 in R6C9
1a. 27(4) cage at R5C2 = {3789/4689/5679}, CPE no 9 in R4C3
1b. 1,2 in N1 only in R2C3 + R3C123, CPE no 1,2 in R4C3
2. 45 rule on R89 1 outie R7C8 = 1 innie R8C7 + 4, no 4 in R7C8, no 6,7,8,9 in R8C7
3. 45 rule on R9 2 outies R8C56 = 1 innie R9C1 + 9
3a. Max R8C56 = 17 -> max R9C1 = 8
4. 45 rule on N1 4 innies R2C3 + R3C123 = 1 outie R1C4 + 4
4a. Min R2C3 + R3C123 = 10 -> min R1C4 = 6
4b. Max R2C3 + R3C123 = 13, no 8,9 in R3C123
[1,2 in N1 only in R2C3 + R3C123 = 10,11,12,13 = {1234/1235/1236/1245/1237/1246} contains at least one of 3,4 may be useful later.]
5. 13(4) cage at R1C5 and 19(5) cage at R1C6 both contain 1, Caged X-Wing for 1 in R12, no other 1 in R12
6. 19(5) cage at R7C4 contains 1, CPE no 1 in R7C9
7. 45 rule on N5 4(2+2) outies R3C34 + R56C7 = 1 innie R4C6 + 29
7a. Max R3C34 + R56C7 = 16 + 17 = 33 -> max R4C6 = 4
7b. Min R3C34 + R56C7 = 30, max R3C34 = 16 -> min R56C7 = 14, no 1,2,3,4
7c. Min R3C34 + R56C7 = 30, max R56C7 = 17 -> min R3C34 = 13, no 1,2,3, no 4,5 in R3C4
7d. 36(6) cage at R2C7 contains 7,8,9 in R23C78 + R4C7, CPE no 7,8,9 in R1C7
8. Max R2C3 + R3C123 = 13 (step 4b), min R3C3 = 4 -> max R2C3 + R4C12 = 9, no 7 in R2C3 + R3C12
9. 45 rule on C1234 3 outies R8C56 + R9C5 = 3(2+1) innies R2C34 + R7C4 + 14
9a. Max R8C56 + R9C5 = {789} = 24 -> max R2C34 + R7C4 = 10 -> max R7C4 = 6 (R2C34 + R7C4 cannot be {12}7 which clashes with {789})
9b. Min R2C34 + R7C4 = {12}1 = 4 -> min R8C56 + R9C5 = 18 cannot be {189} which clashes with R2C34 + R7C4 = {12}1), no 1 in R8C56 + R9C5
10. 19(5) cage at R7C4 and 19(5) cage at R8C1 both contain 1
10a. Consider placement of 1 in 19(5) cage at R8C1
1 in R8C123 + R9C1 => no 1 in R7C12
or 1 in R8C4 => 1 in 19(5) cage at R7C4 must be in R7C7 => no 1 in R7C12
-> no 1 in R7C12
10b. 1 in R7 only in R7C4567, locked for 19(5) cage at R7C4, no 1 in R8C7, clean-up: no 5 in R7C8 (step 2)
[This is simpler with the contradiction move R8C7 cannot be 1 because 1 in 19(5) cage at R8C1 would have to be in R9C1 and then cannot place 1 in R7. However I prefer to use forcing chains, rather than contradiction moves, these days.]
10c. 1 in R8 only in R8C1234, locked for 19(5) cage at R8C1, no 1 in R9C1
11. 45 rule on N9 3 innies R7C79 + R8C7 = 1 outie R9C6 + 3
11a. Min R7C79 + R8C7 = 6 -> min R9C6 = 3
11b. Max R7C79 + R8C7 = 12, min R7C9 + R8C7 = 5 -> max R7C7 = 7
12. 45 rule on N6 3 innies R456C7 = 2 outies R37C9 + 17
12a. Min R37C9 = 3 -> min R456C7 = 20, no 1,2 in R4C7
12b. Max R456C7 = 24 -> max R37C9 = 7, no 7,8,9 in R7C9, no 6 in R3C9
[I should have spotted the next step earlier …
However I would probably still have needed the earlier harder steps.]
13. 9(3) cage at R2C6 = {126/135} (cannot be {234} which clashes with 13(4) cage at R1C5, ALS block), no 4, 1 locked for R3 and N2
14. R2C3 = 1 (hidden single in N1)
14a. 2 in N1 only in R3C12, locked for R3 and 22(5) cage at R3C1, no 2 in R4C12
14b. 9(3) cage at R2C6 (step 13) = {126/135}
14c. 2 of {126} must be in R2C6 -> no 6 in R2C6
14d. 2 in C3 only in R89C3, locked for N7, CPE no 2 in R8C56
15. 13(4) cage at R1C5 = {1246/1345} (cannot be {1237} which clashes with 9(3) cage at R2C6), no 7, 4 locked for N2
15a. R1C46 + R3C4 = {789} (hidden triple for N2)
16. 19(5) cage at R1C6 = {12349/12358/12367/12457} (cannot be {13456} because R1C6 only contains 7,8,9), 2 locked for N3
16a. R1C6 = {789} -> no 7,8,9 in R1C89 + R2C9
17. 7,8,9 in N3 only in R23C78, locked for 36(6) cage at R2C7, no 7,8,9 in R4C7
18. 12(4) cage at R3C9 = {1236/1245}, 1,2 locked for N6
19. 16(4) cage at R5C8 = {2347/2356}, no 8,9 -> R7C9 = 2, clean-up: no 6 in R7C8 (step 2)
19a. R4C8 = 2 (hidden single in N6)
19b. 1 in N6 only in R45C9, locked for C9
19c. R1C78 = [21] (hidden pair in N3)
20. 16(4) cage at R5C8 = {2347/2356}, 3 locked for N6
20a. R56C7 = {89} (hidden pair in N6), locked for C7
21. R89C9 = {89} (hidden pair in C9), locked for N9 -> R7C8 = 7, R8C7 = 3 (step 2)
21a. 21(3) cage at R7C8 contains 7 = {579/678}, no 4
21b. 3 in R7 only in R7C123, locked for N7
22. R6C9 = 7 (hidden single in C9)
22a. 16(4) cage at R5C8 (step 20) = {2347} (only remaining combination) -> R56C8 = {34}, locked for C8 and N6
22b. Naked pair {56} in R89C8, locked for C8 and N9
22c. Naked pair {14} in R79C7, locked for C7
23. R234C7 = {567} = 18, R23C8 = {89} = 17 -> R4C6 = 1
23a. Naked pair {56} in R4C79, locked for R4 and N6 -> R5C9 = 1
23b. R3C5 = 1 (hidden single in N2)
24. 21(3) cage at R6C5 = {489} (only remaining combination), locked for R6, 4 also locked for N5 -> R56C8 = [43]
25. 19(5) cage at R7C4 = {13456} (only remaining combination), no 8,9, 4,5,6 locked for R7, 5,6 also locked for N8
26. Naked triple {389} in R7C123, locked for N7
27. R3C48 = {89} (hidden pair in R3)
28. R4C6 = 1, R56C7 = {89} = 17 -> R3C34 = 13 (step 7) = [49/58], no 6,7 in R3C3
28a. R3C7 = 7 (hidden single in R3)
29. 27(4) cage at R5C2 = {5679} (cannot be {3789} because R6C3 only contains 5,6) -> R7C3 = 9, R5C23 + R6C3 = {567}, locked for N4, 7 also locked for R5
30. R6C12 = {12} = 3, R7C12 = {38} = 11 -> R5C1 = 9 (cage sum), R56C7 = [89]
31. R4C45 = {79} (hidden pair in R4), R6C34 = {56} (hidden pair in R6)
32. R3C34 = 13 = [49/58] (step 28) -> R456C4 = 16 = {367} (only remaining combination, cannot be {259} which clashes with R3C4) -> [736], R4C5 = 9, R6C3 = 5, R3C3 = 4, R3C4 = 9, R1C4 = 8, R1C6 = 7, R23C8 = [98]
33. 21(3) cage at R1C1 = {678} (only remaining combination) -> R1C1 = 6, R2C12 = {78}, R1C3 = 3, R1C2 = 9 (hidden single in N1), R4C3 = 8
34. R1C678 = [721] = 10 -> R12C9 = 9 = {45} (only remaining combination, cannot be {36} because 3,6 only in R2C9), locked for C9 and N3 -> R24C7 = [65], R34C9 = [36]
35. R3C6 = 6 (hidden single in R3), R2C6 = 2 (cage sum), R5C56 = [25], R7C67 = [41], R7C45 = [56], R6C56 = [48], R8C9 = 8
36. 21(3) cage at R7C8 (step 21a) = {678} (only remaining combination) -> R8C8 = 6
and the rest is naked singles.