Prelims
a) R1C34 = {19/28/37/46}, no 5
b) R2C45 = {79}
c) R34C1 = {16/25/34}, no 7,8,9
d) R45C2 = {18/27/36/45}, no 9
e) R56C8 = {12}
f) R67C9 = {15/24}
g) R8C56 = {29/38/47/56}, no 1
h) R9C67 = {16/25/34}, no 7,8,9
i) 10(3) cage at R1C5 = {127/136/145/235}, no 8,9
j) 7(3) cage at R3C4 = {124}
k) 20(3) cage at R5C1 = {389/479/569/578}, no 1,2
l) 24(3) cage at R6C6 = {789}
Steps resulting from Prelims
1a. Naked pair {79} in R2C45, locked for R2 and N2, clean-up: no 1,3 in R1C3
1b. Naked pair {12} in R56C8, locked for C8 and N6, clean-up: no 4,5 in R7C9
2. 45 rule on N1 2 innies R1C3 + R3C1 = 6 = {24}, locked for N1, R1C4 = {68}, R4C1 = {35}
3. 45 rule on N9 2 innies R7C9 + R9C7 = 7 = [16/25], R9C6 = {23}
4. 10(3) cage at R1C5 = {136/145/235}
4a. Hidden killer pair 3,4 in 10(3) cage and R3C56 for N2, 10(3) cage contains one of 3,4 -> R3C56 must contain one of 3,4 -> 31(5) cage at R3C5 = {34789/35689/45679} (cannot be {16789/25789} which don’t contain 3 or 4), no 1,2
4b. 31(5) cage = {34789/35689/45679}, CPE no 9 in R4C89
4c. Caged X-Wing for 9 in 31(5) cage and 24(3) cage at R6C6, no other 9 in C67, clean-up: no 2 in R8C5
5. Hidden killer pair 1,2 in 10(3) cage at R1C5 and R3C4 for N2, 10(3) cage contains one of 1,2 -> R3C4 = {12}
5a. 7(3) cage at R3C4 = {124}, 4 locked for R4, clean-up: no 5 in R5C2
6. 45 rule on N12 4 innies R3C1456 = 15 = {1248/2346} (cannot be {1356} because R3C1 only contains 2,4), no 5, 2 locked for R3
6a. 5 in N2 only in 10(3) cage at R1C5 = {145/235}, no 6
6b. R3C56 = {36/48} -> 31(5) cage at R3C5 = {34789/35689} (cannot be {45679} which doesn’t contain one of 3,8)
6c. 3,6 of {35689} must be in R3C56 -> no 6 in R4C67 + R5C7
6d. 4,8 of {34789} must be in R3C56 -> no 4 in R5C7
6e. 4 in N6 only in R56C9, locked for C9
7. 6 in N6 only in 18(3) cage at R4C8 = {369/468}, no 5,7
7a. 4,9 only in R5C9 -> R5C9 = {49}
7b. R4C89 = {36/68}, 6 locked for R4, clean-up: no 3 in R5C2
7c. 7 in N6 only in R456C7, locked for C7
8. 45 rule on N89 4 innies R7C4569 = 21 = {1389/1479/1569/1578/2379/2469/2478/2568} (cannot be {3459/3468/3567} because R7C9 only contains 1,2)
8a. R7C9 = {12} -> no 1,2 in R7C45
[Just spotted this; it’s been available for a while but it’s powerful now and cracks this puzzle.]
9. 45 rule on N9 1 innie R9C7 = 1 outie R6C9 + 1 -> R6C9 + R9C7 = [45/56], CPE no 5 in R45C7
9a. R6C9 = 5 (hidden single in N6), R7C9 = 1, R9C7 = 6, R9C6 = 1
9b. R5C9 = 4 (hidden single in N6) -> 18(3) cage at R4C8 (step 7) = {468} (only remaining combination), 8 locked for R4 and N6, clean-up: no 5 in R4C2, no 1 in R5C2
10. 31(5) cage at R3C5 (step 6b) = {34789} (cannot be {35689} which clashes with R1C4), no 5 -> R3C56 = {48}, locked for R3 and N2 -> R3C1 = 2, R4C1 = 5, R1C34 = [46], R3C4 = 1, R4C34 = [24], clean-up: no 7 in R45C2
11. 4 in N4 only in 20(4) cage at R5C1 = {479} (only remaining combination), locked for N4
11a. Naked triple {479} in R6C127, locked for R6 -> R6C6 = 8, R3C56 = [84], clean-up: no 3,7 in R8C5, no 3 in R8C6
12. 45 rule on N5 2 remaining innies R4C6 + R6C4 = 10 = [73], R7C6 = 9, R6C7 = 7, clean-up: no 4 in R8C5, no 2 in R8C6
12a. R5C1 = 7 (hidden single in N4)
12b. Naked pair {56} in R8C56, locked for R8 and N8
12c. Naked triple {278} in R789C4, locked for C4 and N8 -> R2C45 = [97], R5C4 = 5, R7C5 = 4, R9C5 = 3, R89C4 = 10 = {28}, locked for C4 -> R7C4 = 7
13. R4C2 = 3 (hidden single in N4), R5C2 = 6, R56C3 = [81], R45C7 = [93], R56C8 = [12], R5C6 = 2, R456C5 = [196], R8C56 = [56], R1C5 = 2
14. R3C7 = 5 -> 16(3) cage at R2C7 = {259/358/457}, no 1,6
15. 18(3) cage at R2C3 = {369/567}, 6 locked for C3 and N1
15a. R3C2 = {79} -> no 7,9 in R2C3
16. R89C3 = {79} (hidden pair in C3), locked for N7
16a. R7C1 = 6 (hidden single in N7)
16b. 16(3) cage at R7C2 = {259} (only remaining combination, cannot be {358} because R8C3 only contains 7,9) -> R7C2 = 2, R78C3 = [59], R7C78 = [83], R8C7 = 2 (cage sum), R2C7 = 4, R3C8 = 7 (cage sum)
and the rest is naked singles.
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