AK = Anti-King, AN = Anti-kNight, AKAN = Anti-King+Anti-kNight
a) R12C6 = {18/27/36/45}, no 9
b) R4C23 = {69/78}
c) R6C78 = {12}
d) R89C4 = {49/58/67}, no 1,2,3
e) 9(3) cage at R3C4 = {126/135/234}, no 7,8,9
f) 21(3) cage at R6C6 = {489/579/678}, no 1,2,3
1. Naked pair {12} in R6C78, locked for R6 and N6, no 1,2 in R7C78 (AK), no 1,2 in R8C78 (AN), no 1,2 in R5C6 + R7C9 (AKAN)
2. 45 rule on N2 1 innie R3C6 = 1 outie R4C4 + 5, R3C6 = {6789}, R4C4 = {1234}
3. 45 rule on N8 1 outie R6C6 = 1 innie R7C4 + 7, R6C6 = {89}, R7C4 = {12}
4. 9(3) cage at R3C4 = {126/135/234}
4a. 6 of {126} must be in R3C4 (R34C4 cannot be {12} which clashes with R7C4), no 6 in R3C5
5. 1,2 in R7 only in R7C1234, no 1,2 in R8C2 + R9C3 (AN), no 1,2 in R8C3 (AK)
6. 3 in N8 only in 18(4) cage, no 3 in R89C7 (AKAN)
7. 1,2 in N4 only in R4C1 + R5C123, no 1,2 in R3C2 (AKAN)
8. 45 rule on N2 3 innies R3C456 = 14
8a. 9(3) cage at R3C4 = {126/135/234} cannot be {126}, here’s how
{126} can only be [612] (cannot be [621] because R3C456 cannot be [626] => no 1 in R4C45 + R5C5, no 1 in R5C4 (AN), no 1 in R4C6 (AK) so cannot place 1 in N5
-> 9(3) cage at R3C4 = {135/234}, no 6, CPE no 3 in R12C4, no 3 in R2C3 + R4C6 (AKAN), no 3 in R2C5 + R3C2 + R5C45 (AN), no 3 in R3C3 + R4C5 (AK)
8b. 3 in C4 only in R346C4, no 3 in R5C3 (AKAN)
9. 9(3) cage at R3C4 = {135/234}
9a. R3C45 cannot contain 1 or 2 because these cells “see” all 1 and 2 in N5 using AKAN -> 1,2 of 9(3) cage only in R4C4 -> R4C4 = {12}, R3C45 = {345}, 3 locked for R3 and N3, clean-up: no 6 in R12C6, no 8,9 in R3C6 (step 2)
9b. R12C6 = {18/27} (cannot be {45} which clashes with R3C45), no 4,5
[With hindsight the key part of steps 8 and 9 can be seen more directly as
1,2 in N5 only in R4C456 + R5C45, no 1,2 in R3C45 (AKAN)
This became available after step 1. Maybe someone writing an optimised walkthrough would do steps 1, 2 and 3 followed by this step.]
10. Naked pair {12} in R47C4, locked for C4, no 1,2 in R5C35 (AKAN)
10a. 1,2 in N5 only in R4C456, locked for R4, no 1,2 in R2C5 (AN)
10b. Hidden killer pair 1,2 in R1C5 and R12C6 for N2, R12C6 contains one of 1,2 -> R1C5 = {12}
11. 9 in N2 only in 22(4) cage, no 9 in R13C3 (AKAN), no 9 in R2C2 (AN), no 9 in R2C3 (AK)
11a. 4,5 in N2 only in R1C4 + R23C45, no 4,5 in R23C3 (AKAN)
12. R5C12 = {12} (hidden pair in N4), no 1,2 in R3C1 + R7C12 (AN)
12a. R7C34 = {12} (hidden pair in R7), no 1,2 in R8C5 (AN)
12b. 23(5) cage at R5C4 = {12389/12479/12569/12578}
12c. 3 of {12389} must be in R6C4 (R56C4 cannot be {89} which clashes with R6C6), no 3 in R7C2
13. 3 in N4 only in R4C1 + R6C123, no 3 in R7C1 (AKAN)
13a. 3 in R7 only in R7C789, locked for N9
13b. 3 in C6 only in R589C6, no 3 in R7C7 (AKAN)
13c. 3 in R7 only in R7C89, no 3 in R5C89 (AN), no 3 in R6C9 (AK)
13d. 3 in R5 only in R5C67, no 3 in R4C7 (AK), no 3 in R4C8 + R6C5 (AN)
[There’s also a forcing chain which eliminates 3 from R2C2 but I’ll leave that; there’s a simpler way to do this later.]
14. 3 in N6 only in R4C9 + R5C7, 3 in R7 only in R7C89
If R5C7 = 3 => no 3 in R7C8 (AN) => R7C9 = 3
-> 3 in R4C9 or R7C9, locked for C9
15. Naked pair {12} in R5C2 + R7C3 (they cannot have the same value because of AN), no 1,2 in R3C3 (AN)
15a. 1,2 in R3 only in R3C789, locked for N3
16. 6 in N2 only in R1C4 + R2C45 + R3C6, no 6 in R3C3 (AKAN)
17. R4C23 = {69} (cannot be {78} which clashes with R3C3 using AK), locked for R4 and N4, no 6 in R2C23 (AN), no 6,9 in R3C2 (AK), no 6,9 in R3C1 + R5C4 (AKAN)
18. 6 in N5 only in R5C56 + R6C45, no 6 in R7C56 (AKAN)
19. 21(3) cage at R6C6 = {489/579}, CPE no 9 in R89C6, no 9 in R5C56 + R7C8 (AN), no 9 in R6C5 + R7C7 (AK), no 9 in R6C4 + R8C7 (AKAN)
20. R6C6 = 9 (hidden single in N5)
20a. R7C56 = {48/57}
20b. R89C4 = {49/67} (cannot be {58} which clashes with R7C56), no 5,8
20c. Combined cage R7C56 + R89C4 = {48}{67}/{57}{49}, 4,7 locked for N8
21. R7C4 = 2 (step 3), R4C4 = 1, R3C6 = 6 (step 2), R7C3 = 1, no 6 in R1C7 + R2C8 + R5C57 (AN), no 6 in R2C7 (AK), no 1 in R5C2 (AN)
21a. R5C12 = [12]
22. 9(3) cage at R3C4 (step 9) = {135} (only remaining combination), no 4, 5 locked for R3 and N2, no 5 in R4C5 (AK), no 5 in R4C6 (AKAN), no 5 in R5C45 (AN)
23. Naked triple {478} in R3C123, locked for R3 and N1 -> R2C3 = 2, no 4,7,8 in R4C1 (AKAN), clean-up: no 7 in R1C6
23a. Naked triple {129} in R3C789, 9 locked for N3
24. 6 in N5 only in R6C45, locked for R6, no 6 in R8C45 (AN), clean-up: no 7 in R9C4
24a. 6 in N8 only in R9C45, locked for R9
25. 2 in C6 only in R14C6, no 2 in R3C7 (AKAN)
26. 5 in N5 only in R5C6 + R6C45, no 5 in R5C3 (AKAN)
26a. Naked triple {478} in R5C345, locked for R5, no 4,7,8 in R6C3 (AKAN), no 4,7,8 in R6C4 (AK), no 4,7,8 in R6C5 (AN)
26b. 5 in R5 only in R5C6789, no 5 in R4C7 (AK), no 5 in R4C8 (AN)
26c. R4C19 = {35} (hidden pair in R4)
26d. Naked triple {356} in R6C456, locked for R6, no 5 in R7C5 (AKAN), no 3 in R8C3 (AN), clean-up: no 7 in R8C6 (step 20a)
27. 18(4) cage at R4C1 = {1368/1458} (cannot be {1359} because 3,5,9 only in R47C1, cannot be {1467} because R4C1 only contains 3,5), no 7,9, 8 locked for C1
27a. R4C1 = {35} -> no 5 in R7C1
27b. 8 in C1 only in R67C1, no 8 in R67C2 (AK), no 8 in R8C2 (AN)
27c. 7 in N4 only in R5C3 + R6C2, no 7 in R37C2 + R5C4 (AN)
28. 23(5) cage at R5C4 (step 12b) = {12389/12569}, no 4
28a. R5C4 = 8 -> 23(5) cage = {12389} -> R6C4 = 3, R7C2 = 9, R3C45 = [53], R6C3 = 5, R4C19 = [35], R4C23 = [69], R5C6 = 5, R6C5 = 6, no 3,5 in R2C2 (AN), no 5 in R2C8 + R7C7 (AN), no 9 in R8C4 (AN), clean-up: no 7 in R7C5 (step 20a), no 4 in R9C4
29. Naked pair {48} in R7C56, locked for R7 and N8 -> R7C1 = 6, R6C1 = 8 (step 27), R7C789 = [753], R8C4 = 7, R9C4 = 6, no 7 in R6C9 (AN)
30. Naked pair {59} in R12C1, locked for C1 and N1, R12C2 = [31], R1C3 = 6, clean-up: no 8 in R1C6
31. R6C9 = 4, R6C2 = 7, R5C3 = 4, R5C5 = 7, R4C78 = [87], R8C3 = 8, R3C123 = [487], R8C1 = 2, R9C13 = [73], R89C6 = [31], R1C56 = [12], R4C56 = [24], R7C56 = [48], R2C6 = 7, no 4 in R2C7 + R8C8 (AN)
32. R2C5 = 8 (hidden single in N2), R2C9 = 6, R5C9 = 9, R8C9 = 1, no 1 in R6C8 (AN)
and the rest is naked singles.
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