Prelims
a) R34C5 = {29/38/47/56}, no 1
b) R5C12 = {15/24}
c) R5C89 = {16/25/34}, no 7,8,9
d) R67C5 = {59/68}
e) 19(3) cage at R5C4 = {289/379/469/478/568}, no 1
f) 10(3) cage at R9C2 = {127/136/145/235}, no 8,9
g) 23(3) cage at R9C6 = {689}
h) 22(6) cage at R6C8 = {123457}, no 6,8,9
Steps resulting from Prelims
1a. R34C5 = {29/38/47} (cannot be {56} which clashes with R67C5), no 5,6
1b. R5C89 = {16/34} (cannot be {25} which clashes with R5C12), no 2,5
1c. Killer pair 1,4 in R5C12 and R5C89, locked for R5
1d. Naked triple {689} in 23(3) cage at R9C6, locked for R9
1e. Naked triple {689} in 23(3) cage at R9C6, no 6,8,9 in R8C7 (anti-King)
2. 45 rule on R5 2 innies R5C37 = 13 = {58/67}, no 1,2,3,9
2a. 9 in R5 only in 19(3) cage at R5C4, locked for N5, clean-up: no 2 in R3C5, no 5 in R7C5
2b. 19(3) cage = {289/379}, no 5,6
3. 45 rule on R1234 2 innies R4C37 = 9 = {18/27/36/45}, no 9
4. 45 rule on R6789 2 innies R6C37 = 9 = {18/27/36/45}, no 9
5. 45 rule on C1234 3 innies R258C4 = 21 = {489/579/678}, no 1,2,3
6. 9 in N6 only in R4C89, locked for R4 and 35(6) cage at R1C9, no 9 in R12C9 + R3C89
6a. R4C9 = 9 (hidden single in C9)
6b. 8 in C9 only in R123C9, locked for N3 and 35(6) cage at R1C9, no 8 in R4C8
6c. 6 in C9 only in R1235C9, CPE no 6 in R4C8
7. 9 in N4 only in R6C12, locked for 34(6) cage at R6C1, no 9 in R7C12 + R8C1
8. 9 in N7 only in R7C3 + R8C23, locked for 25(5) cage at R6C4, no 9 in R7C4
9. 8 in N6 only in 16(3) cage at R4C7, locked for C7
9a. 6 in 16(3) cage = {268} or in R5C89 = {16} -> 16(3) cage = {268/358} (cannot be {178}, locking-out cages), no 1,4,7, clean-up: no 2,5,8 in R4C3 (step 3), no 6 in R5C3 (step 2), no 2,5,8, in R6C3 (step 4)
9b. Killer pair 3,6 in 16(3) cage and R5C89, locked for N6
10. 22(6) cage at R6C8 = {123457}, 3 locked for N9
11. 8,9 in C8 only in R1289C8
11a. 45 rule on C89 4 innies R1289C8 = 26 = {2789/3689/4589}, no 1
12. 15(3) cage at R4C3 = {168/348/357} (cannot be {456) which clashes with R5C12)
12a. R4C37 = 9 (step 3), R6C37 = 9 (step 4)
12b. 15(3) cage = {168/357} (cannot be {348} because 16(3) cage at R4C7 cannot be {565}), no 4, clean-up: no 5 in R4C7 (step 3), no 5 in R6C7 (step 4)
12c. 5 of {357} must be in R5C3 -> no 7 in R5C3, clean-up: no 6 in R5C7 (step 2)
13. Naked pair {58} in R5C37, locked for R5, clean-up: no 1 in R5C12
13a. Naked pair {24} in R5C12, locked for R5 and N4, clean-up: no 3 in R5C89
13b. Naked pair {16} in R5C89, locked for N6, clean-up: no 2 in 16(3) cage at R4C7 (step 9a), no 3,7 in R4C3 (step 3), no 3,7 in R4C7 (step 4)
13c. Naked pair {38} in R46C7, locked for C7 -> R5C7 = 5, R5C3 = 8
13d. Naked pair {16} in R46C3, locked for C3 and N4
13e. Naked triple {379} in 19(3) cage at R5C4, locked for N5, clean-up: no 4,8 in R3C5
14. 22(6) cage at R6C8 = {123457}, 1,5 locked for N9
14a. 1 in C7 only in R123C7, locked for N3
15. R5C3 = 8, no 8 in R46C4 (anti-King)
15a. R5C7 = 5, no 5 in R46C6 (anti-King)
15b. Naked pair {38} in R46C7, no 3 in R5C6 (anti-King)
15c. 5 in N5 only in R46C4 + R6C5, CPE no 5 in R7C4 using anti-King
15d. Naked triple {247} in R4C8 + R6C89, CPE no 2,4,7 in R7C8
16. R5C5 = 3 (hidden single in R5), clean-up: no 8 in R4C5
16a. 8 in N5 only in R46C6 + R6C5, CPE no 8 in R7C6 using anti-King
16b. 8 in R4 only in R4C67, no 8 in R3C6 (anti-King)
17. 45 rule on C6789 3 innies R258C6 = 16 = {169/178/259/349/367/457} (cannot be {268/358} because R5C6 only contains 7,9)
17a. R5C6 = {79} -> no 7,9 in R28C6
18. R1289C8 (step 11a) = {3689/4589} (cannot be {2789} which clashes with R46C8, ALS block), no 2,7
18a. R1289C8 = {3689/4589} -> R34567C8 = {12367/12457}
18b. 1,5 of {12457} must be in R57C8 -> no 5 in R3C8
19. 20(5) cage at R2C7 = {12359/12368/12458/12467/13457/23456}
19a. 3,5 of {12359} must be in R2C8 + R3C6 -> no 9 in R2C8 + R3C6
20. R4C37 = 9 (step 3)
20a. Consider combinations for R4C12 = {35/37/57}
R4C12 = {35/37} => R4C7 = 8, R4C3 = 1 => no 1 in R4C46
or R4C12 = {57} => R4C7 = 3 (hidden single in R4), R4C6 = 8 (hidden single in R4) => no 1 in R4C6
-> no 1 in R4C6
20b. 1 in R4 only in R4C34, no 1 in R3C4 (anti-King)
21. 24(5) cage at R6C6 must contain two of 2,4,7 and one of 6,8,9 in N9 = {12489/12579/12678/13479/23469/23478/24567} (cannot be {13569/13578/14568/23568} which don’t contain two of 2,4,7)
21a. 2,4,7 of {12489/12678} must be in N9, 3,5 of {12579/13479/23469/23478/24567} must be in R7C6 -> no 2,4,7 in R7C6
21b. 24(5) cage = {12489/12579/12678/13479/23478/24567} (cannot be {23469} = [63]{249} because [63]{249} + R9C678 = [968] clashes with R67C5, using anti-King)
22. R258C6 (step 17) = {169/178/259/349/367/457}
22a. Consider combinations for 24(5) cage at R6C6 (step 21b) = {12489/12579/12678/13479/23478/24567}
24(5) cage = {12489/12579/12678/13479}, 1 locked for C6 => R258C6 = {259/349/367/457}
or 24(5) cage = {23478} => R6C6 = {24} (because cannot have all three of 2,4,7 in N9) => R8C8 = 8, R9C6 = 8, R4C6 = 6 => R258C6 = {259/349/457} (cannot be {169/178/367} which clash with R49C6)
or 24(5) cage = {24567} => R6C6 = {24} (because cannot have all three of 2,4,7 in N9) => 6 of {24567} must be in N9 => R9C6 = 6, R4C6 = 8 => R258C6 = {259/349/457} (cannot be {169/178/367} which clash with R49C6)
-> R258C6 = {259/349/367/457}, no 1,8
23. Consider combinations for R67C5 = [59]/{68}
R67C5 = [59], 8 in N5 only in R46C6, locked for C6 => R9C6 = 6
or R67C5 = {68} => no 6,8 in R67C6 (anti-King)
-> no 6 in R67C6
23a. 6 in R6 only in R6C345, no 6 in R7C4 (anti-King)
[It took me a very, very long time to find the next step.]
24. Consider placements for 9 in C6
R1C6 = 9 => R1C78 = 8 = [26] (deleted as unnecessary), R89C8 = {89} (hidden pair in C8), locked for N9, R9C7 = 6, R9C6 = 8 => R6C5 = 8 (hidden single in N5) => R67C5 = [86]
or R5C6 = 9, 7 in C6 only in R13C6, R3C5 = 9 => R67C5 = {68}
or 9 in R79C6 => R67C5 = {68}
-> R67C5 = {68}, locked for C5
24b. Naked pair {68} in R67C5, no 6 in R6C4, no 8 in R6C6, no 8 in R7C4 (all anti-King)
25. 5 in N5 only in R46C4, locked for C4
25a. R258C4 (step 5) = {489/678}, 8 locked for C4
25b. R5C4 = {79} -> no 7,9 in R28C4
26. 24(5) cage at R6C6 (step 21b) = {12489/12579/13479/23478/24567} (cannot be {12678} which clashes with R9C78, ALS block)
26a. 3,5 of {12579/13479} must be in R7C6, 9 of {12489} must be in R7C6 (because {12489} cannot have both of 8,9 in N9 which would clash with R9C78, ALS block) -> no 1 in R7C6
27. Consider placements for R3C5 = {79}
R3C5 = 7
or R3C5 = 9, R5C4 = 9 (hidden single in C4), R5C6 = 7
-> no 7 in R1C6
28. 17(3) cage at R1C6 = {179/269/359/368/458/467} (cannot be {278} because no 2,7,8 in R1C8)
28a. 7 of {179} must be in R1C7 -> no 1 in R1C7
28b. 1 in N3 only in R23C7, locked for 20(5) cage at R2C8, no 1 in R3C6
29. Consider placements for R5C4 = {79}
R5C4 = 7 => R5C6 = 9, R3C6 = 7 (hidden single in C6), R3C4 = 9
or R5C4 = 9
-> no 9 in R13C4
30. R5C4 = 9 (hidden single in C4), R5C6 = 7
30a. R258C4 (step 25a) = {489} (only remaining combination), locked for C4
30b. R258C6 (step 22a) = {367/457}, no 2 in R28C6
30c. 4 in N5 only in R4C56 + R6C5, CPE no 4 in R3C6 using anti-King
[Cracked. The rest is fairly straightforward.]
31. 4 in N8 only in 19(4) cage at R8C4 = {1459/2458/3457} (cannot be {1468} which clashes with R7C5, cannot be {2467} which clashes with R34C5), no 6, 5 locked for N8, clean-up: no 3 in R2C6 (step 30b)
31a. 9 of {1459} must be in R8C5 -> no 1 in R8C5
31b. 4 of {1459/3457} must be in R8C4, 4 of {2458} must be in R8C6 (R89C5 cannot be {24} which clashes with R4C5) -> 4 in R8C46, locked for R8 and N8
32. 24(5) cage at R6C6 (step 26) = {12489/13479/23478}, no 6
32a. 9 of {12489} must be in R7C6, 9 of {13479} must be in R8C8 -> no 9 in R7C7
33. 6 in N9 only in R9C78, locked for R9C6
34. R7C5 = 6 (hidden single in N8), R6C5 = 8, R6C7 = 3, R4C7 = 8, R4C3 = 1 (step 3), R6C3 = 6
34a. R4C3 = 1 -> no 1 in R3C2 (anti-King)
34b. R6C7 = 3 -> no 3 in R7C68 (anti-King)
34c. R7C6 = 9, R8C8 = 8, R9C6 = 8, R8C4 = 4, R2C4 = 8
34d. R8C4 = 4 -> no 4 in R79C3 (anti-King)
35. 19(4) cage at R8C4 (step 31) = {3457} (only remaining combination) -> R8C6 = 3, R2C6 = 6 (step 30b), R89C5 = {57}, locked for C5 and N8 -> R3C5 = 9, R4C5 = 2, R4C68 = [47], R6C6 = 1, R46C4 = [65]
35a. R13C4 = {37} (hidden pair in N2), no 3,7 in R2C3 (anti-King)
35b. Naked pair {25} in R13C6, no 2 in R2C7 (anti-King)
35c. R2C6 = 6 -> no 1,3 in R13C7 (anti-King)
35d. R4C8 = 7 -> no 7 in R3C7 (anti-King)
36. 7 in N3 only in R12C7, locked for C7 -> R78C7 = [42], R3C7 = 1
36a. Naked pair {79} in R12C7, locked for C7 and N3 -> R9C78 = [69]
36b. R7C7 = 4 -> R6C8 = 2 (anti-King)
37. Naked pair {35} in R4C12, locked for 32(6) cage at R1C1, no 3,5 in R123C1 + R3C2
37a. 32(6) cage = {135689/235679} (only combinations containing both of 3,5), no 4, 9 locked for C1 and N1, 6 also locked for N1 -> R6C12 = [79]
37b. R8C3 = 9 (hidden single in N7)
38. 22(5) cage at R2C2 = {13567/23467}
38a. 1 of {13567} only in R2C2 -> no 5 in R2C2
39. 15(3) cage at R1C2 = {348/357} (cannot be {258} because R1C4 only contains 3,7), no 1,2, 3 locked for R1
39a. 8 of {348} must be in R1C2 -> no 4 in R1C2
40. 17(3) cage at R1C6 (step 28) = {269} (only remaining combination, cannot be {467} because 4,6 only in R1C8) -> R1C6 = 2, R1C78 = [96], R3C6 = 5, R2C78 = [73], R5C89 = [16], R7C8 = 5, R3C8 = 4
41. 32(6) cage at R1C1 (step 37a) = {135689} (only remaining combination, cannot be {235679} because R1C1 only contains 1,8), no 2,7
41a. Naked pair {68} in R3C12, locked for R3 and N1 -> R12C1 = [19]
42. 15(3) cage at R1C2 (step 39) = {357} (only remaining combination), locked for R1 -> R1C9 = 8, R23C9 = [52]
43. R6C4 = 5, R8C3 = 9 -> 25(5) cage at R6C4 = {13579/23569} -> R7C3 = 3, R8C2 = {67}
44. R6C12 = [79] -> 34(6) cage at R6C1 = {145789} (only remaining combination) -> R7C12 = [81], R89C1 = [54]
and the rest is naked singles, without using anti-King eliminations.
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