Prelims
a) R12C1 = {19/28/37/46}, no 5
b) R23C7 = {17/26/35}, no 4,8,9
c) R4C67 = {18/27/36/45}, no 9
d) R6C34 = {17/26/35}, no 4,8,9
e) R78C3 = {39/48/57}, no 1,2,6
f) R89C9 = {15/24}
g) 23(3) cage at R1C5 = {689}
h) 19(3) cage at R1C7 = {289/379/469/478/568}, no 1
i) 7(3) cage at R3C8 = {124}
j) 11(3) cage at R4C1 = {128/137/146/236/245}, no 9
k) 27(4) cage at R8C8 = {3789/4689/5679}, no 1,2
Steps resulting from Prelims
1a. Naked triple {689} in 23(3) cage at R1C5, locked for N2
1b. Naked triple {124} in 7(3) cage at R3C8, locked for C8
2. 19(3) cage at R1C7 = {379/478} (cannot be {289/469/568} which clash with 23(3) cage at R1C5, ALS block), no 2,5,6, 7 locked for R1 and N3, clean-up: no 3 in R2C1, no 1 in R23C7
2a. Killer triple 6,8,9 in 23(3) cage at R1C5 and 19(3) cage, locked for R1, clean-up: no 1,2,4 in R2C1
2b. 6 in R1 only in R1C56, locked for N2
3. 45 rule on N3 3 outies R4C89 + R5C8 = 10 = {127/145} (cannot be {136/235} because 3,5,6 only in R4C9) -> R4C9 = {57}, 1 in R45C8, locked for C8 and N6, clean-up: no 8 in R4C6
4. 45 rule on R1 3 outies R2C126 = 24 = {789}, locked for R2, 7 also locked for N1, clean-up: no 4 in R1C1
5. Hidden killer pair 8,9 in 19(3) cage at R1C7 and R3C9 for N3, 19(3) cage contains one of 8,9 -> R3C9 = {89}
5a. R2C9 = 1 (hidden single in N3), clean-up: no 5 in R89C9
5b. Naked pair {24} in R89C9, locked for C9 and N9
5c. Hidden killer pair 5,6 in R23C7 and R2C8 for N3, R23C7 contains one of 5,6 -> R2C8 = {56}
6. 17(3) cage at R5C9 = {359/368}, no 7, 3 locked for N6, clean-up: no 6 in R4C6
7. 5 in R1 only in 17(4) cage at R1C2 = {1259/1358/1457/2357}
7a. 7 in N1 only in R12C1 = [37] or in 17(4) cage = {1457/2357} -> 17(4) cage = {1259/1457/2357} (cannot be {1358}, locking-out cages), no 8 in R2C2
8. 45 rule on N7 1 outie R6C1 = 1 innie R7C2 + 2, no 1,2 in R6C1, no 8,9 in R7C2
9. 45 rule on N47 3 innies R456C3 = 18 = {189/279/369/468/567} (cannot be {378/459} which clash with R78C3)
9a. 1 of {189} must be in R6C3 -> no 1 in R45C3
10. 4 in R2 only in R2C345, locked for 22(5) cage at R2C3, no 4 in R34C4
10a. 22(5) cage = {12469/13459/13468/23458/23467} (cannot be {12478} because 1,7,8 only in R34C4)
10b. 8,9 of {12469/13459/13468/23458} must be in R4C4 -> no 1,5 in R4C4
11. 45 rule on R9 3 outies R8C489 = 17 = {269/278/458/467} (cannot be {179/359/368} because R8C9 only contains 2,4), no 1,3
11a. R8C9 = {24} -> no 2,4 in R8C4
12. 36(6) cage at R3C1 must contain 7 in R45C3 + R5C4, CPE no 7 in R5C12
12a. Hidden killer pair 8,9 in R3C123 and R3C9 for R3, R3C9 = {89} -> R3C123 must contain one of 8,9
12b. R456C3 (step 9) = {279/369/468/567} (cannot be {189} because R45C3 cannot contain both of 8,9 since one of 8,9 must be in R3C123), no 1, clean-up: no 7 in R6C4
[I originally saw this elimination as R456C3 (step 9) = {279/369/468/567} (cannot be {189} because R6C34 = [17] clashes with the only remaining position for 7 in the 36(6) cage, killer combo clash), no 1, clean-up: no 7 in R6C4]
12c. 11(3) cage at R4C1 = {128/137/146/245} (cannot be {236} which clashes with R456C3)
13. 45 rule on C1234 2 outies R29C5 = 1 innie R7C4 + 1
13a. Max R29C5 = 10, no 9 in R9C5
14. 45 rule on N36 3 innies R456C7 = 18 = {279/459/468} (cannot be {567} which clashes with R4C9)
14a. 2 in C7 only in R23C7 = {26} or in R456C7 = {279} -> R456C7 = {279/459} (cannot be {468}, locking-out cages), no 6,8 in R456C7, 9 locked for C7 and N6, clean-up: no 1,3 in R4C6
14b. Killer pair 5,7 in R456C7 and R4C9, locked for N6
15. Killer pair 5,7 in R4C67 and R4C9, locked for R4
15a. 11(3) cage at R4C1 (step 12c) = {128/146} (cannot be {245} which clashes with R4C67), no 3,5, 1 locked for N4
15b. R456C3 (step 12b) = {279/369/567} (cannot be {468} which clashes with 11(3) cage), no 4,8
15c. Killer pair 2,6 in 11(3) cage and R456C3, locked for N4, clean-up: no 4 in R7C2 (step 8)
16. 45 rule on R12 4 remaining innies R2C3458 = 1 outie R3C7 + 12, IOU R2C345 cannot total 12
[I originally saw this are 45 rule on R12 5 remaining innies R2C34578 = 20
R2C78 cannot total 8, which would clash with R23C7 (CCC) -> R2C345 cannot total 12
Where there’s a CCC there’s usually an IOU, and vice versa.]
16a. 22(5) cage at R2C3 (step 10a) = {13468/23458/23467} (cannot be {12469/13459} because R2C345 = {246/345} would total 12), no 9 in R4C4
16b. 8 of {13468/23458} must be in R4C4, 3 of {23467} must be in R2C345 (R2C345 cannot be {246} which totals 12) -> no 3 in R4C4
16c. 1,7 of {13468/23467} must be in R3C4, 2 of {23458} must be in R2C345 (R2C345 cannot be {345} which totals 12) -> no 2 in R3C4
[This CCC/IOU seems to crack the puzzle; the rest is a lot easier with step 20b giving the final breakthrough.]
17. R4C35 = {39} (hidden pair in R4)
17a. Naked pair {39} in R4C35, CPE no 3,9 in R5C4
17b. R456C3 (step 15b) = {279/369} (cannot be {567} because R4C3 only contains 3,9), no 5, 9 locked for C3, N4 and 36(6) cage at R3C1, clean-up: no 3 in R6C4, no 7 in R7C2 (step 8), no 3 in R78C3
17c. 36(6) cage at R3C1 must contain 7 in R5C34, locked for R5
18. R3C9 = 9 (hidden single in R3)
18a. 19(3) cage at R1C7 (step 2) = {478} (only remaining combination) -> R1C7 = 4, R1C89 = {78}, locked for R1, R3C8 = 2, clean-up: no 6 in R23C7
18b. Naked pair {35} in R23C7, locked for C7 and N3 -> R2C8 = 6
18c. Naked triple {279} in R456C7, locked for C7 and N6 -> R4C9 = 5
18d. Naked triple {168} in R789C7, locked for N9
18e. 7 in R4 only in R4C67 = {27}, locked for R4
19. 6,8 in R3 only in R3C123, locked for N1 and 36(6) cage at R3C1, no 6,8 in R5C34, clean-up: no 2 in R1C1
20. 9 in C7 only in R56C7, locked for 27(6) cage at R5C6, no 9 in R5C6 + R7C8
20a. 27(6) cage contains 1,9 = {123489/123579/124569}
20b. Caged X-Wing for 2 in 27(6) cage and R4C67, no other 2 in C6 and N5, clean-up: no 6 in R6C3
21. R456C3 (step 17b) = {279} (only remaining combination) -> R4C3 = 9, R56C3 = {27}, locked for C3 and N4, clean-up: no 5 in R6C4, no 5 in R7C2 (step 8), no 5 in R78C3
21a. Naked pair {48} in R78C3, locked for C3 and N7
21b. 11(3) cage at R4C1 (step 15a) = {146} (only remaining combination), locked for N4, clean-up: no 2 in R7C2 (step 8)
22. 14(3) cage at R5C2 = {158/356}, 5 locked for C2 and N4
22a. 1,6 only in R7C2 -> R7C2 = {16}
23. Naked pair {38} in R6C18, locked for R6 -> R6C2 = 5, R6C9 = 6, R6C4 = 1 -> R6C3 = 7, R5C3 = 2, R56C7 = [92], R4C67 = [27]
24. R4C4 = 8 (hidden single in R4) -> 22(5) cage at R2C3 (step 16a) = {23458} (only remaining combination) -> R2C45 = {24}, locked for N2
24a. Naked pair {35} in R13C4, locked for C4 and N2
25. R3C12 = {48} (hidden pair in R3), locked for 36(6) cage at R3C1 -> R5C4 = 7
26. R5C56 = {56} (hidden pair in N5), locked for R5
27. 27(6) cage at R5C6 (step 20a) = {123579} (only remaining combination) -> R5C6 = 5, R7C7 = 1, R7C89 = {37}, locked for R7 and N9
27a. R7C2 = 6, R5C2 = 3 (cage sum), R6C1 = 8, R3C1 = 4, R5C1 = 1, R1C1 = 3 -> R2C1 = 7, R2C2 = 9
28. R9C3 = 3 (hidden single in C3) -> R9C12 = 10 = [91]
and the rest is naked singles.