Prelims
a) R1C12 = {49/58/67}, no 1,2,3
b) R3C12 = {19/28/37/46}, no 5
c) R5C45 = {16/25/34}, no 7,8,9
d) R56C6 = {19/28/37/46}, no 5
e) R7C67 = {39/48/57}, no 1,2,6
f) R7C89 = {16/25/34}, no 7,8,9
g) R8C89 = {49/58/67}, no 1,2,3
h) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9
i) 11(3) cage at R8C3 = {128/137/146/236/245}, no 9
j) 15(5) cage at R4C8 = {12345}
k) 32(5) cage at R5C3 = {26789/35789/45689}, no 1
1. 15(5) cage at R4C8 = {12345}, locked for N6
2. 45 rule on N1 2 innies R13C3 = 7 = {16/25/34}, no 7,8,9
3. 45 rule on N9 2 innies R78C7 = 8 = [35/53/71], R7C7 = {357}, R8C7 = {135}, clean-up: R7C6 = {579}
3a. Combined cage R78C7 + R7C89 = {35}{16}/[71]{25}/[71]{34}, 1 locked for N9
4. 39(7) cage at R1C8 = {1356789/2346789}, 3 locked for N3
5. 45 rule on N36 2 outies R13C6 = 10 = {19/28/37/46}, no 5
6. 45 rule on N4789 2 outies R6C45 = 14 = {59/68}
6a. 45 rule on N4789 3 innies R7C5 + R8C45 = 8 = {125/134}, 1 locked for N8
6b. 1 in R9 only in R9C123, locked for N7
7. 45 rule on N47 2 outies R79C4 = 11 = {38/47/56}/[92], no 2 in R7C4
8. Hidden killer pair 8,9 in R8C89 and 17(3) cage at R9C7 for N9, each can only contain one of 8,9 -> R8C89 = {49/58}, no 6,7, 17(3) cage = {269/278/368/458} (cannot be {359} which clashes with R8C89, cannot be {467} which doesn’t contain 8 or 9)
9. Killer triple 4,5,6 in R1C12, R13C3 and 15(3) cage at R2C1, locked for N1, because 15(3) cage must contain at least one of 4,5,6
10. 45 rule on N1236 3 outies R4C456 = 14 = {149/167/239/248/257/347} (cannot be {158/356} which clash with R6C45)
10a. 7 in N5 only in R4C456 or in R56C6 = {37} -> R4C456 = {149/167/248/257/347} (cannot be {239} which contains 3 but not 7, locking-out cages)
10b. Killer triple 4,5,6 in R4C456, R5C45 and R6C45 for N5, locked for N5
11. 45 rule on C6789 4 outies R2349C5 = 26 = {2789/3689/4589/4679/5678}, no 1
[Just spotted …]
12. 45 rule on N9 1 outie R7C6 = 1 innie R8C7 + 4
12a. R7C67 = {57} or R7C6 + R8C7 = [95], killer CPE no 5 in R7C89, clean-up: no 2 in R7C89
[Hope it’s acceptable to call this step a killer CPE, rather than writing it as a forcing chain. Maybe it’s some sort of killer Wing?]
13. 2 in N9 only in 17(3) cage at R9C7, locked for R9, clean-up: no 9 in R7C4 (step 7)
13a. 17(3) cage (step 8) = {269/278}, no 3,4,5
14. 9 in N8 only in R78C6 + R9C56
14a. 45 rule on N9 4 outies R78C6 + R9C56 = 26 = {2789/3689/4679} (cannot be {4589} which clashes with R7C5 + R8C45), no 5, clean-up: no 7 in R7C7, no 1 in R8C7 (step 3)
15. Naked pair {35} in R78C7, locked for C7 and N9, clean-up: no 4 in R7C89, no 8 in R8C89
15a. Naked pair {16} in R7C89, locked for R7 and N9, clean-up: no 5 in R9C4 (step 7)
15b. Naked pair {49} in R8C89, locked for R8 and N9
15c. Naked triple {278} in 17(3) cage at R9C7, locked for R9, clean-up: no 3,4 in R7C4 (step 7)
16. R78C6 + R9C56 (step 14a) = {3689/4679} (cannot be {2789} because 2,8 only in R8C6), no 2
16a. 8 of {3689} must be in R8C6 -> no 3 in R8C6
17. 2 in N8 only in R7C5 + R8C45 (step 6a) = {125} (only remaining combination), locked for N8 and 22(5) cage at R6C4, no 5 in R6C45, clean-up: no 9 in R6C45 (step 6), 6 in R9C4 (step 7)
17a. 3,4 in N8 only in R9C456, locked for R9
17b. 5 in R9 only in R9C123, locked for N7
18. Naked pair {68} in R6C45, locked for R6 and N5, clean-up: no 1 in R5C45, no 2 in R56C6
18a. Killer pair 7,9 in R56C6 and R7C6, locked for C6, clean-up: no 1,3 in R13C6 (step 5)
18b. Killer pair 6,8 in R13C6 and R8C6, locked for C6
18c. Naked pair 3,4 in R9C46, locked for N8
19. R78C6 + R9C56 (step 6) = {4679} (only remaining combination, cannot be {3689} which clashes with R56C6) -> R7C6 = 7, R8C6 = 6, R9C45 = [94], R79C4 = [83], R7C7 = 5, R8C7 = 3, R6C45 = [68], R7C5 = 2, clean-up: no 5 in R5C4, no 4 in R5C5, no 3 in R56C6
20. Naked pair {28} in R13C6, locked for C6 and N2
20a. Naked pair {19} in R56C6, locked for C6 and N5
20b. 7 in N5 only in R4C45, locked for R4
20c. 7 in R6 only in R6C123, locked for N4
21. 25(5) cage at R2C5 = {34567}, R24C6 = {35} -> R234C5 = {467}, locked for C5
22. R9C4 = 3 -> R89C3 = 8 = [26/71], no 8 in R8C3, no 5 in R9C3
22. R13C3 (step 2) = {25/34} (cannot be {16} which clashes with R9C3), no 1,6
23. 8 in R8 only in R8C12, locked for 34(7) cage at R5C1, no 8 in R5C1
23a. 34(7) cage with 8 must also contain 3, locked for C1, clean-up: no 7 in R3C2
23b. 5 in R9 only in R9C12, locked for 34(7) cage at R5C1, no 5 in R56C1
23c. 34(7) cage with 5 must also contain 6
23d. 34(7) cage = {1235689/1345678}, the missing pair of numbers must be 2,9 or 4,7
24. R4C456 (step 10) = {257/347}
24a. 2 of {257} must be in R4C4 -> no 5 in R4C4
25. 10(3) cage at R3C3 = {127/145/235}
25a. 1 of {127} must be in R3C4 -> no 7 in R3C4
25b. 4 of {145} must be in R4C4 -> no 4 in R3C34, clean-up: no 3 in R1C3 (step 2)
25c. Naked pair {15} in R38C4, locked for C4
26. 16(3) cage at R1C6 = {178/268} (cannot be {169} because R1C6 only contains 2,8), no 4,9, CPE no 8 in R1C89
27. 9 in N2 only in 21(4) cage at R1C3 = {1479/2379/3459}
27a. 1,3 only in R1C5 -> R1C5 = {13}
28. 32(5) cage at R5C3 = {35789/45689} (cannot be {26789} because 2,6,7 only in R56C3), no 2
28a. 32(5) cage only contains two of 3,4,9, R7C23 = {349} -> no 3,4,9 in R56C3
28b. 32(5) cage = {35789/45689}, 5 locked for C3 and N4, 9 locked for R7, clean-up: no 2 in R13C3 (step 2)
28c. Killer pair 6,7 in 32(5) cage and R89C3, locked for C3
29. R13C3 = [43], R7C3 = 9, clean-up: no 9 in R1C12, no 7 in R3C1
29a. Naked pair {79} in R12C4, locked for C4 and N2
30. 21(4) cage at R1C3 (step 27) = {1479} (only remaining combination) -> R1C5 = 1, R8C34 = [15], R5C5 = 3, R5C4 = 4, R34C4 = [52], R24C6 = [35], R4C5 = 7
31. 16(3) cage at R1C6 = {268} (only remaining combination, cannot be {178} which clashes with R1C12), no 1,7
31a. 16(3) cage at R1C6 = {268}, 6 locked for C7 and N3, CPE no 2 in R1C89
31b. 2 in R1 only in R1C67, locked for 16(3) cage, no 2 in R2C7
32. R3C12 = {19} (only remaining combination, cannot be {28} which clashes with R3C6), locked for R3 and N1
32a. 15(3) cage at R2C1 = {258/267}, 2 locked for R2
32b. Killer pair 6,8 in 15(3) cage and R2C7, locked for R2 -> R23C5 = [46]
32c. 7 in R3 only in R3C789, locked for N3
33. Naked triple {24789} in 30(5) cage at R3C6, 2 locked for R3, 9 locked for N6
34. 5 in N4 only in R56C3, locked for C3
34a. 45 rule on N4 4 innies R56C13 = 21 = {3567} (only remaining combination, cannot be {1569} which clashes with R3C1, cannot be {3459} because 3,4 only in R6C1) -> R6C1 = 3, R56C3 = [57], R5C1 = 6, R8C3 = 2, R2C3 = 8, clean-up: no 5 in R1C1, no 5,7 in R1C2
35. R1C12 = [76]
and the rest is naked singles.
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