Prelims
a) R34C2 = {17/26/35}, no 4,8,9
b) R6C78 = {16/25/34}, no 7,8,9
c) R89C5 = {18/27/36/45}, no 9
d) R8C67 = {16/25/34}, no 7,8,9
e) 21(3) cage at R7C7 = {489/579/678}, no 1,2,3
1. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 6 = {123}, locked for N5 and D\
2. 45 rule on R5 1 innie R5C4 = 1 outie R6C4 + 2, no 4,5 in R5C4, no 8,9 in R6C4
3. 45 rule on D/ 1 outie R5C6 = 1 innie R4C6 + 2, no 8,9 in R4C6, no 4,5 in R5C6
4. 45 rule on R1 3 innies R1C159 = 21 = {489/579/678}, no 1,2,3
4a. Min R1C5 = 4 -> max R23C5 = 8, no 8,9 in R23C5
5. 45 rule on C9 3 innies R159C9 = 19 = {289/379/469/478/568}, no 1
6. 45 rule on R5 3 innies R5C456 = 16 = {169/178/367} (cannot be {268} which clashes with R4C6 + R5C5 + R6C4 = {246} using steps 2 and 3, or alternatively with 3 innies for D/ = 12), no 2 in R5C5
[Alternatively 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 12 = {147/156/246/345}
45 rule on R5 3 innies R5C456 = 16 = {169/178/367} (cannot be {268} which clashes with R4C6 + R5C5 + R6C4 = {246}, no 2 in R5C5]
6a. R5C456 = {169/178/367}, R4C6 + R5C5 + R6C4 = {147/156/345} using steps 2 and 3 -> R456C5 = {158/149/389} = 14,20
6b. 45 rule on C5 4 innies R4567C5 = 24, R456C5 cannot total 14 -> R7C5 = 4, R456C5 = 20 = {389} -> R5C5 = 3, placed for D/, R46C5 = {89}, locked for C5, clean-up: no 1,5,6 in R89C5, no 3 in R8C7
7. Naked pair {67} in R5C46, locked for R5
7a. Naked pair {45} in R4C6 + R6C4, locked for D/
8. Naked pair {27} in R89C5, locked for C5 and N8, clean-up: no 5 in R8C7
8a. Naked triple {156} in 12(3) cage at R1C5, locked for N2
9. 18(3) cage at R1C9 = {189/279}, no 6, 9 locked for N3 and D/
10. 15(3) cage at R7C3 = {168/267}, 6 locked for N7
11. R1C159 (step 4) = {579/678} (cannot be {489} because R1C5 only contains 5,6), no 4
11a. R1C5 = {56} -> no 5,6 in R1C1
12. 18(3) cage at R1C1 = {459/468/567}
12a. R1C1 = {789} -> no 7,8,9 in R2C2 + R3C3
13. 45 rule on R9 3 innies R9C159 = 16 = {178/268} (cannot be {169} because R9C5 only contains 2,7, cannot be {259/457} because 4,5,9 only in R9C9), no 4,5,9, 8 locked for R9
13a. R9C5 = {27} -> no 2,7 in R9C19
13b. R9C159 = {178/268}, CPE no 8 in R1C19 using diagonals
14. Naked pair {79} in R1C19, locked for R1, R1C5 = 5 (step 11)
15. R159C9 (step 5) = {289/469/478} (cannot be {568} because R1C9 only contains 7,9), no 5
15a. 2,4 only in R5C9 -> R5C9 = {24}
16. 14(3) cage at R5C7 = {149/248} (cannot be {158} because R5C9 only contains 2,4), no 5, 4 locked for R5 and N6, clean-up: no 3 in R6C78
16a. Killer pair 1,2 in 14(3) cage and R6C78, locked for N6
16b. Killer pair 1,2 in R6C6 and R6C78, locked for R6
17. 15(3) cage at R5C1 = {159/258}, 5 locked for R5 and N4, clean-up: no 3 in R3C2
18. 18(3) cage at R1C1 = {459/567}, 5 locked for N1 and D\, clean-up: no 3 in R4C2
18a. 21(3) cage at R7C7 = {489/678}, 8 locked for N9
18b. 4 of {489} must be in R8C8 -> no 9 in R8C8
18c. 9 on D\ only in R1C1 + R7C7, CPE no 9 in R7C1
19. 45 rule on C1 3 innies R159C1 = 18 = {189/567} (cannot be {279} because R9C1 only contains 1,6,8), no 2
19a. R1C1 = {79} -> no 9 in R5C1
20. 15(3) cage at R6C1 = {249/267/348/357/456} (cannot be {159/168/258} which clash with R159C1), no 1
20a. 5 in C1 only in R159C1 = {567} or in 15(3) cage -> 15(3) cage = {249/348/357/456} (cannot be {267} which contains 7 but not 5, locking-out cages)
21. 12(3) cage at R2C9 = {138/147/156/237/345} (cannot be {129} which clashes with 18(3) cage at R1C9, cannot be {246} which clashes with R5C9), no 9
22. 14(3) cage at R9C6 = {149/239/356} (cannot be {167/257} which clash with R9C159, cannot be {347} which clashes with 21(3) cage at R7C7), no 7
23. 15(3) cage at R9C2 = {249/357/456} (cannot be {267} which clashes with R9C5, cannot be {159} which clashes with 14(3) cage at R9C6), no 1
23a. 9 of {249} must be in R9C4 -> no 9 in R9C23
23b. 15(3) cage at R7C3 (step 10) = {168/267}
23c. Consider combinations for 15(3) cage at R9C2
15(3) cage at R9C2 = {249/357} => 15(3) cage at R7C3 = {168}
or 15(3) cage at R9C2 = {456}, locked for R9 => R9C9 = 8, R9C1 = 1 => 15(3) cage at R7C3 = {168}
-> 15(3) cage at R7C3 = {168}, locked for N7 and D/
24. Naked triple {279} in 18(3) cage at R1C9, locked for N3
25. 12(3) cage at R1C6 = {138/246}
25a. 2 of {246} must be in R1C6 -> no 4 in R1C6
26. 12(3) cage at R2C9 (step 21) = {138/156/345} (cannot be {147} which clashes with 12(3) cage at R1C6), no 7
26a. 6 of {156} must be in R4C9 (R23C9 cannot be {16} which clashes with 12(3) cage at R1C6), no 6 in R23C9
26b. Killer pair 1,4 in 12(3) cage at R1C6 and 12(3) cage at R2C9, locked for N3
27. 14(3) cage at R6C9 = {167/239/257/347} (cannot be {149/248} which clash with R159C9, cannot be {158/356} which clash with 12(3) cage at R2C9), no 8
28. 12(3) cage at R1C2 = {138/246}
28a. 4 of {246} must be in R1C4 (R1C23 cannot be {46} which clashes with 18(3) cage at R1C1), no 4 in R1C23, no 2 in R1C4
29. 12(3) cage at R2C1 = {129/138/237/246} (cannot be {147} which clashes with R159C1)
29a. 9 of {129} must be in R23C1 (R23C1 cannot be {12} which clashes with 12(3) cage at R1C2), no 9 in R4C1
30. R159C1 (step 9) = {189/567}
30a. 12(3) cage at R2C1 (step 29) = {129/138/237/246}
Consider combinations for 12(3) cage
12(3) cage at R2C1 = {129}, 9 locked for N1
or 12(3) cage at R2C1 = {138} => 12(3) cage at R1C2 = {246} (cannot be {138} which clashes with 12(3) cage at R2C1, ALS block), 6 locked for N1 => R2C2 + R3C3 = {45} => R1C1 = 9 (step 18)
or 12(3) cage at R2C1 = {237/246} => R159C1 = {189} => R1C1 = 9
-> 9 in R1C1 + 12(3) cage at R2C1, locked for C1 and N1
30b. 9 in C1 only in 18(3) cage at R1C1 = {459} or in 12(3) cage at R2C1 -> 12(3) cage cannot be {24}6/{46}2 (locking-out cages), no 4 in R23C1, no 6 in R4C1
31. 15(3) cage at R6C1 (step 20a) = {348/357/456}, no 2
31a. 6 of {456} must be in R6C1 -> no 4 in R6C1
31b. Killer pair 5,8 in R159C1 and 15(3) cage, locked for C1
32. 15(3) cage at R9C2 (step 23) = {249/357} (cannot be {456} which clashes with 15(3) cage at R6C1), no 6
32a. Killer pair 2,7 in 15(3) cage at R9C2 and R9C5, locked for R9
[Just spotted a trickier step which provides the final breakthrough.]
33. Grouped hidden killer pair for 1,2 in 12(3) cage at R1C2, 12(3) cage at R2C1, R34C2 and 15(3) cage at R5C1 for N14, 12(3) cage at R1C2, R34C2 and 15(3) cage at R5C1 each contain one of 1,2, 2 in C1 only in 12(3) cage at R2C1 -> 12(3) cage at R2C1 cannot also contain 1 -> 12(3) cage at R2C1 (step 29a) = {237/246}, no 1,9
[Edit. I've changed Double to Grouped, which seems to be the standard term, even though double feels right to me.]
34. R1C1 = 9 (hidden single in C1), placed for D\, 18(3) cage at R1C1 (step 18) = {459} (only remaining combination), 4 locked for N1 and D\, R1C9 = 7
35. R159C9 (step 15) = {478} (only remaining combination) -> R5C9 = 4, R9C9 = 8
36. R159C1 (step 19) = {189} (only remaining combination) -> R5C1 = 8, R9C1 = 1, R9C5 = 7 (step 13), R8C5 = 2, clean-up: no 5 in R8C6
37. 15(3) cage at R5C1 (step 17) = {258} (only remaining combination), 2 locked for R5 and N4, clean-up: no 6 in R3C2
38. Naked pair {19} in R5C78, locked for N6, clean-up: no 6 in R6C78
38a. Naked pair {25} in R6C78, locked for R6 and N6 -> R6C4 = 4, R4C6 = 5, R5C6 = 7 (cage sum), R5C4 = 6
39. Naked pair {36} in R46C9, locked for C9
39a. Naked pair {15} in R23C9, locked for C9 and N3, R4C9 = 6 (cage sum), 14(3) cage at R6C9 = [329], clean-up: no 2 in R3C2
40. Naked pair {67} in R7C7 + R8C8, locked for N9, clean-up: no 1 in R8C6
40a. 14(3) cage at R9C6 (step 22) = {356} (only remaining combination) -> R9C6 = 6, R9C78 = {35}, locked for R9 and N9, R8C6 = 3, R8C7 = 4
41. 12(3) cage at R1C6 (step 25) = {246} (only remaining combination) = [264]
and the rest is naked singles, without using the diagonals.