Prelims
a) R1C67 = {19/28/37/46}, no 5
b) R1C89 = {16/25/34}, no 7,8,9
c) R34C2 = {13}
d) R3C89 = {16/25/34}, no 7,8,9
e) R45C6 = {69/78}
f) R45C8 = {17/26/35}, no 4,8,9
g) R5C12 = {18/27/36/45}, no 9
h) R6C12 = {59/68}
i) R78C2 = {89}
j) R7C45 = {59/68}
k) R78C9 = {49/58/67}, no 1,2,3
l) R8C56 = {14/23}
m) 19(3) cage at R1C2 = {289/379/469/478/568}, no 1
n) 9(3) cage at R5C3 = {126/135/234}, no 7,8,9
Steps resulting from Prelims
1a. Naked pair {13} in R34C2, locked for C2, clean-up: no 6,8 in R5C1
1b. Naked pair {89} in R78C2, locked for C2 and N7, clean-up: no 1 in R5C1, no 5,6 in R6C1
1c. Killer pair 8,9 in R7C2 and R7C45, locked for R7, clean-up: no 4,5 in R8C9
2. 19(3) cage at R1C2 = {469/478} (cannot be {289/379} because 3,8,9 only in R1C3, cannot be {568} which clashes with R6C2), no 2,3,5
2a. 8,9 only in R1C3 -> R1C3 = {89}
2b. R12C2 = {46/47}, 4 locked for C2 and N1, clean-up: no 5 in R5C1
3. 45 rule on R12 2 innies R12C1 = 7 = {16/25}
3a. Hidden killer pair 8,9 in R34C1 and R6C1 for C1, R6C1 = {89} -> R34C1 must contain one of 8,9
3b. 45 rule on R12 2 outies R34C1 = 13 = {58}/[94] (cannot be {67} because R34C1 must contain one of 8,9)
3c. R34C1 = [58/94] (cannot be [85] which clashes with R6C12 = [95])
3d. 8 in C1 only in R46C1, locked for N4
4. 39(7) cage = {1356789/2346789}
4a. Caged X-Wing for 3 in R34C2 and 39(7) cage for R34, no other 3 in R34, clean-up: no 4 in R3C89, no 5 in R5C8
4b. 4 in R3 only in 39(7) cage = {2346789}, no 1,5
4c 4,7,8 in R3 only in R3C34567, locked for 39(7) cage at R3C3, no 4,7,8 in R4C37
5. 45 rule on C12 4(1+3 or 2+2) outies R1C3 + R9C345 = 28
5a. R1C3 = {89} -> R9C345 = 19,20, no 1 in R9C345
5b. 2 of 19 = {289} must be in R9C3 -> no 2 in R9C45
6. 45 rule on R9 3(2+1) outies R78C1 + R8C8 = 11
6a. Min R78C1 = 3 -> max R8C8 = 8
7. 35(7) cage at R7C1 is missing two numbers which total 10
7a. 3,7 in C1 only in R5C1 and R789C1 -> 35(7) cage must contain both of 3,7 = {1235789/1345679/2345678}
8. 45 rule on C89 2 innies R2C89 = 2 outies R9C67 + 8
8a. Max R2C89 = 17 -> max R9C67 = 9, no 9 in R9C67
8b. Min R9C67 = 3 -> min R2C89 = 11, no 1 in R2C89
9. Hidden killer pair 1,5 in R3C12 and R3C89 for R3, R3C89 contains one of 1,5 -> R3C12 must contain one of 1,5
9a. Killer pair 1,5 in R12C1 and R3C12, locked for N1
9b. 5 in N1 only in R123C1, locked for C1
9c. 35(7) cage at R7C1 must contain 5, locked for R9
10. 35(6) cage at R1C5 must contain both of 8,9, CPE no 8,9 in R2C4
11. 5 in C1 only in R123C1 -> 20(4) cage at R1C1 = {16}[58]/{25}[94]
11a. 19(3) cage at R1C2 (step 2) = {478} (only remaining combination, cannot be {469} which clashes with 20(4) cage) -> R1C3 = 8, R12C2 = {47}, locked for C2 and N1, clean-up: no 2 in R1C67, no 2 in R5C1
12. 45 rule on N5 1 innie R6C6 = 1 outie R5C3 + 3, no 1,2,3 in R6C6
13. 45 rule on N14 5 innies R23456C3 = 24 = {13569/23469} (other combinations don’t contain at least three of 2,3,6,9 for R234C3), no 7, 3,6,9 locked for C3
14. R5C1 = 7 (hidden single in N4), R5C2 = 2, clean-up: no 8 in R4C6, no 1,6 in R4C8, no 5 in R6C6 (step 12)
14a. 3 in C1 only in R789C1, locked for 35(7) cage at R7C1, no 3 in R9C45
14b. 35(7) cage must contain both of 3,7 (step 7a), 7 locked for R9
14c. 6 in N7 only in R789C1 + R9C2, locked for 35(7) cage, no 6 in R9C45
14d. 35(7) cage contains 6 -> 35(7) cage (step 7a) must contain both of 4,6 (step 7) = {1345679/2345678}
15. R1C3 + R9C345 = 28
15a. R1C3 = 8 -> R9C345 = 20 = {479/578}, no 2
15b. Killer pair 8,9 in R7C45 and R9C45, locked for N8
16. Hidden killer pair 8,9 in R9C45 and R9C789 for R9, R9C45 contains one of 8,9 -> R9C789 must contain one of 8,9
16a. 21(5) cage cannot contain both of 8,9 -> no 8 in R8C8
17. 3 in N1 only in R2C3 + R3C23
17a. 45 rule on N1 using R12C1 = 7 (step 3), 4 remaining innies R3C12 + R23C3 = 19 = {1369/2359} = [91]{36}/[53]{29} -> R23C3 = {29/36}
17b. R23456C3 (step 13) = {13569/23469} = {36}9{15}/{29}{346} (cannot be {36}{249} which clashes with R46C1, ALS block), no 9 in R6C3
18. Hidden killer pair 1,2 in R12C1 and R789C1 for C1, R12C1 contains one of 1,2 -> R789C1 must contain one of 1,2
18a. Hidden killer pair 1,2 in R789C1 and R78C3 for N7, R789C1 contains one of 1,2 -> R78C3 must contain one of 1,2
18b. 17(4) cage at R6C3 = {1367/1457/2357/2456} contains one of 1,2 which must be in R78C3 -> no 1,2 in R6C3 + R8C4
18c. R23456C3 (step 13) = {13569/23469}
18d. 1 of {13569} must be in R5C3 -> no 5 in R5C3, clean-up: no 8 in R6C6 (step 12)
18e. 5 in N4 only in R6C23, locked for R6
19. 45 rule on R123 4 outies R4C1237 = 18 = {1368/2349} (cannot be {1269} because R4C1 only contains 4,8)
19a. 2 of {2349} must be in R4C7 -> no 9 in R4C7
20. Deleted, replaced by step 25e.
21. 45 rule on N5 3 innies R56C4 + R6C6 = 12 = {129/156/237/246/345} (cannot be {147} because 9(3) cage at R5C3 cannot be 4{14})
21a. 2 of {246} must be in R6C4, 4 of {345} must be in R6C6 -> no 4 in R6C4
22. R9C345 (step 15a) = {479/578}, 17(4) cage at R6C3 (step 18b) = {1367/1457/2357/2456}
22a. Consider combinations for R7C45 = {59/68}
R7C45 = {59}, locked for N8 => R9C345 = 5{78} => 17(4) cage = {1367}
or R7C45 = {68}, locked for N8 -> R9C345 = {479}, R9C2 = 5 (hidden single in R9), R6C2 = 6, R6C3 = 5 (hidden single in R6) => 17(4) cage = {1457/2357}
-> 17(4) cage = {1367/1457/2357}, also no 5 in R8C4 + R9C45
22b. Caged X-Wing for 7 in 35(7) cage at R7C1 and 17(4) cage, no other 7 in N78
22c. 5 in R9 only in R9C23, locked for N7
23. R23456C3 (step 17b) = {13569/23469} = {36}9{15}/{29}{346}
23a. 17(4) cage at R6C3 (step 22a) = {1367/2357} (cannot be {1457} = 5{147} which clashes with R23456C3, a sort of CCC), no 4
23b. 17(4) cage at R6C3 = {1367/2357}, CPE no 3 in R6C4
24. 4 in N7 only in R789C1 + R9C3, locked for 35(7) cage at R7C1, no 4 in R9C45
24a. Killer triple 7,8,9 in R7C45 and R9C45, locked for N8, 7 also locked for R9
25. R23456C3 (step 17b) = {13569/23469} = {36}9{15}/{29}{346}
25a. 1,4 only in R5C3 -> R5C3 = {14}
25b. 9(3) cage at R5C3 = {126/234} (cannot be {135} because 3,5 only in R5C4), no 5
25c. 2 only in R6C4 -> R6C4 = 2
25d. 3,6 only in R5C4 -> R5C4 = {36}
25e. 9(3) cage = {126} = {16}2 => R5C8 = 3 or {234} = [432] -> 3 in R5C48, locked for R5
25f. Naked pair {36} in R58C4, locked for C4, clean-up: no 8 in R7C5
[Thanks Ed for pointing out that I'd overlooked clean-up: R6C6 = {47} (step 12) in step 25a.
This leads to R5C3 + R6C7 = [14/47], CPE no 4 in R5C57, which is a bit simpler than step 27a.]
26. 1 in N5 only in 18(4) cage at R4C4 = {1359/1458} (cannot be {1368/1467} which clash with R45C6), no 6,7
26a. 3 of {1359} must be in R6C5 -> no 9 in R6C5
26b. 7 in N5 only in R46C6, locked for C6, clean-up: no 3 in R1C7
27. 45 rule on N5 2 remaining innies R5C4 + R6C6 = 10 = [37/64]
27a. 9(3) cage at R5C3 (step 25b) = {126/234} = [16]2/[43]2 -> R5C34 + R6C6 = [16]4/[43]7, CPE no 4 in R5C57
28. 12(3) cage at R1C4 = {129/156/345} (cannot be {147} because no 1,4,7 in R2C3, cannot be {237/246} because 2,3,6 only in R2C3), no 7
28a. 2 of {129} must be in R2C3 -> no 9 in R2C3, clean-up: no 2 in R3C3 (step 17a)
28b. 9 in N1 only in R3C13, locked for R3
29. 5 in R8 only in R8C789, locked for N9, clean-up: no 8 in R8C9
30. Hidden killer pair 2,5 in R1C145 and R1C89 for R1, R1C89 must contain both or neither of 2,5 -> R1C145 must contain both or neither of 2,5
30a. 45 rule on R1 4 remaining innies R1C1245 = 20 = {1379/1469/2459/2567/3467} (cannot be {2369} which contains 2 but not 5)
30b. 4,7 of {1379/1469/2459/2567} must be in R1C2, 4,7 of {3467} must be in R1C24 -> no 4,7 in R1C5
31. 35(6) cage at R1C5 = {236789/345689} (cannot be {146789/245789} which clash with R2C2), no 1
31a. Killer pair 4,7 in R2C2 and 35(6) cage, locked for R2
32. R1C1245 (step 30a) = {1379/1469/2459/2567/3467}
32a. 12(3) cage at R1C4 (step 28a) = {129/156} (cannot be {345} = [435] because R1245 = {3467} = [6743] clashes with R23C3 = [36], step 17a), no 4 in R1C4, no 3 in R2C3, clean-up: no 6 in R3C3 (step 17a)
32b. 12(3) cage = {129/156}, 1 locked for C4 and N2, clean-up: no 9 in R1C7
32c. R1C1245 = {1379/1469/2459/2567} (cannot be {3467} because 4,7 only in R1C2)
32d. 1 in N5 only in R456C5, locked for C5, clean-up: no 4 in R8C6
32e. 9 in N3 only in R2C789, locked for 35(6) cage at R1C5, no 9 in R1C5 + R2C56
33. 3 in R2 only in R2C56789, locked for 35(6) cage at R1C5, no 3 in R1C5
33a. 3 in R1 only in R1C67 = [37] or in R1C89 = {34} -> R1C67 = [37/91] (cannot be {46}, locking-out cages)
33b. R1C67 = [37] -> R1C4 = 9 (hidden single in R1) or R1C67 = [91] -> no 1 in R1C4 (locking-out cages)
33c. R2C4 = 1 (hidden single in C4), clean-up: no 6 in R1C1 (step 3)
34. 6 in N1 only in R2C13, locked for R2
34a. 35(6) cage at R1C5 (step 31) = {236789/345689} -> R1C5 = 6, clean-up: no 1 in R1C89, no 8 in R7C4
35. Naked pair {59} in R7C45, locked for R7 and N8 -> R78C2 = [89], clean-up: no 4 in R7C9
35a. Naked pair {67} in R78C9, locked for C9 and N9, clean-up: no 1 in R3C8
35b. Naked pair {78} in R9C45, locked for R9
35c. R8C7 = 8 (hidden single in N9)
35d. 8 in R3 only in R3C456, locked for N2
36. 35(7) cage at R7C1 is missing two numbers which total 10 (step 7)
36a. No 9 in 35(7) cage -> no 1 in 35(7) cage
[Cracked. The rest is fairly straightforward.]
37. R1C1 = 1 (hidden single in C1), R1C2 = 6 (step 3), R1C7 = 7, R1C6 = 3, R12C2 = [47], R2C3 = 2, R1C4 = 9 (cage sum), R34C2 = [31], R3C3 = 9, R3C1 = 5, R4C1 = 8 (step 3b), R6C1 = 9, R6C2 = 5, R5C3 = 4, R5C4 = 3 (cage sum), R9C23 = [65], R7C45 = [59], R45C4 = [45], R6C6 = 7, R8C4 = 6, R46C3 = [63], R4C6 = 9, R5C6 = 6, R4C789 = [372], R3C9 = 1, R3C8 = 6, R5C8 = 1, R78C9 = [67], R78C3 = [71], R8C6 = 2, R8C5 = 3, R56C5 = [81], R9C45 = [87], R2C56 = [45], R3C5 = 2, R3C7 = 4, R2C7 = 9
38. R5C7 = 5, R6C67 = [76], R8C7 = 8 total 26 -> R7C67 = 5 = [41]
and the rest is naked singles.