Prelims
a) R12C5 = {69/78}
b) R23C1 = {69/78}
c) R3C23 = {19/28/37/46}, no 5
d) R5C12 = {17/26/35}, no 4,8,9
e) R67C3 = {19/28/37/46}, no 5
f) R6C67 = {16/25/34}, no 7,8,9
g) R67C9 = {15/24}
h) R8C67 = {18/27/36/45}, no 9
i) 10(3) cage at R1C1 = {127/136/145/235}, no 8,9
j) 10(3) cage at R1C6 = {127/136/145/235}, no 8,9
k) 24(3) cage at R1C9 = {789}
l) 22(3) cage at R6C1 = {589/679}
m) 19(3) cage at R6C8 = {289/379/469/478/568}, no 1
n) 9(3) cage at R7C1 = {126/135/234}, no 7,8,9
1. Naked triple {789} in 24(3) cage at R1C9, locked for C9 and N3
1a. Max R3C78 = 11 -> min R4C78 = 11, no 1 in R4C78
2. 45 rule on N1 2 outies R12C4 = 8 = {17/26/35}, no 4,8,9
3. 45 rule on N1 2 innies R12C3 = 10 = {19/28/37/46}, no 5
3a. 5 in N1 only in 10(3) cage at R1C1 = {145/235}, no 6,7
4. 45 rule on N4 3 innies R6C123 = 22 = {589/679}, 9 locked for R6 and N4, clean-up: no 6,7,8,9 in R7C3
4a. 45 rule on N4 1 outie R7C2 = 1 innie R6C3, no 5 in R7C2
5. R5C12 = {17/26/35}, R6C123 (step 4) = {589/679} -> combined cage R5C12 + R6C123 = {17/26}{589}/{35}{679}, 5 locked for N4
6. 45 rule on R5 1 outie R4C9 = 1 innie R5C3 + 1, R4C9 = {2345}, R5C3 = {1234}
7. Max R45C9 = 11 -> min R5C8 = 7
7a. Min R45C9 = 9, no 2 in R4C9, no 1,2,3 in R5C9, clean-up: no 1 in R5C3 (step 6)
7b. 18(3) cage at R4C9 = {369/468/567} (cannot be {459} which clashes with R67C9) -> R5C9 = 6, clean-up: no 2 in R5C12, no 1 in R6C6
7c. Max R89C9 = 9 -> min R9C78 = 11, no 1 in R9C78
7d. 1 in C8 only in R123C8, locked for N3
8. Killer pair 5,7 in R5C12 and R6C123, locked for N4
9. 45 rule on N4 2 outies R7C23 = 10 = [64/73/82/91]
9a. 45 rule on N47 4 innies R89C23 = 26 = {2789/4589/4679/5678} (cannot be {3689} which clashes with R7C23), no 1,3
10. 45 rule on N47 2 innies R8C23 = 1 outie R9C4 + 10
10a. Max R8C23 = 17 -> max R9C4 = 7
11. 15(4) cage at R4C1 = {1248/2346}
11a. 45 rule on R1234 4 innies R4C1239 = 16 must contain two or four odd numbers (cannot all be even because {2468} = 20), 15(4) cage only contains one odd number -> R4C9 must be an odd number = {35}, no 4, clean-up: no 3 in R5C3 (step 6), no 8 in R5C8 (step 7b)
12. 45 rule on C9 3 remaining innies R489C9 = 9 = {135/234}
12a. 45 rule on C9 2 outies R9C78 = 1 innie R4C9 + 11, R4C9 = {35} -> R9C78 = 14,16 = {59/68/79}, no 2,3,4
12b. 19(3) cage at R6C8 = {289/478/568} (cannot be {379} which clashes with R5C8, cannot be {469} which clashes with R9C78), no 3, 8 locked for C8
12c. 2 of {289} must be in R78C8 (R78C8 cannot be {89} which clashes with R9C78), no 2 in R6C8
12d. 8 of R9C78 = {68} must be in R9C7 -> no 6 in R9C7
13. Consider combinations for 18(3) cage at R4C9 (step 7b) = {369/567}
13a. 18(3) cage = {369} => R4C9 = 3 => 3 in N4 only in R5C12 = {35}
or 18(3) cage = {567} => R5C8 = 7, locked for R5 => R5C12 = {35}
-> R5C12 = {35}, locked for R5 and N4
14. R6C123 (step 4) = {679} (only remaining combination), locked for R6 and N4, clean-up: no 1 in R6C7, no 2 in R7C3
14a. 22(3) cage at R6C1 = {679} (only remaining combination), no 8
15. Naked quad {1248} in 15(4) cage at R4C1, 1,8 locked for R4
16. 9(3) cage at R7C1 = {126/234} (cannot be {135} which clashes with R5C1), no 5, 2 locked for C1 and N7
17. R89C23 (step 9a) = {4589/5678} (cannot be {4679} which clashes with R7C2)
17a. Hidden killer pair 7,9 in R7C2 and R89C23, R89C23 contains one of 7,9 -> R7C2 = {79}, clean-up: no 4 in R7C3 (step 9), no 6 in R6C3
[Alternatively hidden killer pair 1,3 in 9(3) cage at R7C1 and R7C3 …]
18. 22(4) cage at R3C7 = {2569/3469/4567} (cannot be {1579/2479} which clash with R5C8), no 1, 6 locked for R3 and N3, clean-up: no 9 in R2C1, no 4 in R3C23
18a. Killer pair 7,9 in 22(4) cage and R5C8, locked for N6
19. Killer triple 7,8,9 in R3C1, R3C23 and R3C9, locked for R3
19a. Hidden killer pair 8,9 in R12C5 and R2C6 for N2, R12C5 contains one of 8,9 -> R2C6 = {89}
19b. Naked quad {6789} in R2C1569, locked for R2, clean-up: no 1,2,3,4 in R1C3 (step 3), no 1,2 in R1C4 (step 2)
20. 45 rule on R5 2 remaining innies R5C38 = 11 = [29/47]
20a. Consider combinations for R5C38 = [29/47]
R5C38 = [29] => R4C123 = {148}, no 9 in R4C78
or R5C38 = [47] => R4C123 = {128}, no 7 in R4C78
-> R4C78 cannot be {29/47} -> R4C78 cannot total 11 -> R3C78 cannot total 11, but contains 6 (step 18) -> no 5 in R3C78
21. 5 in R3 only in R3C456, locked for N2, clean-up: no 3 in R12C4 (step 2)
21a. Killer pair 6,7 in R1C4 and R12C5, locked for N2
22. 10(3) cage at R1C6 = {145/235}, 5 locked for R1 and N3
22a. R2C2 = 5 (hidden single in N1), R5C12 = [53], clean-up: no 7 in R3C3
23. 12(3) cage at R3C4 = {147/246/345} (cannot be {129} which clashes with R2C4, cannot be {156/237} which clash with R12C4), no 9
23a. 6 of {246} must be in R4C4 -> no 2 in R4C4
24. 22(4) cage at R3C7 (step 18a) = {3469/4567} (cannot be {2569} = {26}{59} which clashes with 18(3) cage at R4C9), no 2
25. Hidden killer pair 1,2 in R3C23 and R3C456 for R3, R3C456 must contain one of 1,2 (cannot be both which would clash with R2C4) -> R3C23 must contain one of 1,2 = {19/28}, no 3,7
25a. Killer pair 1,2 in R2C4 and R3C456, locked for N2
25b. Killer pair 1,2 in 10(3) cage at R1C1 and R3C23, locked for N1, clean-up: no 8,9 in R1C3 (step 3)
25c. Killer pair 3,4 in R1C12 and R1C6, locked for R1
26. Naked pair {67} in R1C34, locked for R1, clean-up: no 8,9 in R2C5
27. Cage X-Wing for 4 in 12(3) cage at R3C4 and 22(4) cage at R3C7, no other 4 in R34
[Cracked. The rest is fairly straightforward. Routine clean-ups omitted from here.]
28. Naked triple {128} in R4C123, locked for R4 and N4 -> R5C3 = 4, R5C8 = 7 (step 20), R4C9 = 5 (cage sum), R2C3 = 3, R1C3 = 7 (step 3), R1C4 = 6, R2C4 = 2 (step 2), R2C5 = 7, R1C5 = 8, 24(3) cage at R1C9 = [987], R2C1 = 6, R3C1 = 9, R6C123 = [769], R7C23 = [91]
29. Naked triple {234} in 9(3) cage at R7C1, locked for C1 and N7 -> R1C1 = 1, R1C2 = 4 (cage sum)
30. R67C9 = {24} (only remaining combination), locked for C9 -> R89C9 = {13}, locked for N9
30a. R89C9 = {13} = 4 -> R9C78 = 16 = [79]
31. 19(3) cage at R6C8 (step 12b) = {568} (only remaining combination) -> R6C8 = 8, R78C8 = {56}, locked for C8 and N9
32. R2C678 = [941] = 14 -> R3C6 + R4C56 = 14 = {356} (only remaining combination) -> R3C6 = 5, R4C56 = {36} (locked for R4 and N5)
33. R9C2 = 8, R8C2 = 7, R7C6 = 7 (hidden single in R7), R7C57 = 7 = [52]
33a. R9C2 = 8 -> R9C34 = 8 = [53]
and the rest is naked singles.
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