Prelims
a) R45C7 = {19/28/37/46}, no 5
b) R45C9 = {19/28/37/46}, no 5
c) R56C2 = {18/27/36/45}, no 9
d) R5C34 = {89}
e) R6C34 = {29/38/47/56}, no 1
f) R67C7 = {79}
g) R6C89 = {15/24}
h) R9C78 = {18/27/36/45}, no 9
i) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
j) 13(4) cage at R1C6 = {1237/1246/1345}, no 8,9
k) 26(4) cage at R2C3 = {2789/3689/4589/4679/5678}, no 1
l) 11(4) cage at R6C5 = {1235}
m) 27(4) cage at R8C1 = {3789/4689/5679}, no 1,2
n) And, of course, both 45(9) cages = {123456789}
1. 45 rule on N9 2 innies R78C7 = 9 = [72], R6C7 = 9, clean-up: no 1,3,8 in R45C7, no 1 in R45C9, no 2 in R6C34
1a. Naked pair {46} in R45C7, locked for C7 and N6, clean-up: no 2 in R6C89, no 3,5 in R9C8
1b. Naked pair {15} in R6C89, locked for R6 and N6, clean-up: no 4,8 in R5C2, no 6 in R6C34
1c. 11(4) cage at R6C5 = {1235}, 1,5 locked for R7
2. 45 rule on N6 1 remaining outie R3C8 = 8, R45C8 = 10 = {37}, locked for C8 and N6, clean-up: no 1 in R9C7
2a. Naked pair {28} in R45C9, locked for C9
2b. Naked triple {135} in R123C7, locked for C7 and N3 -> R9C7 = 8, R9C8 = 1, R6C89 = [51]
3. 45 rule on N3 1 remaining innie R3C7 = 1 outie R1C6 + 3 -> R3C7 = 5, R1C6 = 2
3a. R3C7 = 5 -> R23C6 = 4 = {13}, locked for C6 and N2
3b. 45(9) cage at R5C6 = {123456789}, 3 locked for R8
4. R1C6 = 2 -> 13(4) cage at R1C6 = {1246} (only remaining combination, cannot be {1237} because R1C8 only contains 4,6) -> R1C7 = 1, R1C89 = {46}, locked for R1 and N3, R2C7 = 3, R23C6 = [13]
4a. Naked pair {79} in R23C9, locked for C9 and N3 -> R2C8 = 2
4b. 1 in R3 only in R3C12, locked for 21(5) cage at R1C1, no 1 in R4C1
5. 2 in R9 only in 12(3) cage at R9C3 = {237/246}, no 5,9
6. Naked pair {89} in R5C34, locked for R5 -> R45C9 = [82], clean-up: no 7 in R6C2
6a. 45(9) cage at R1C5 = {123456789}, 2 locked for R4
6b. 45(9) cage at R1C5 = {123456789}, 8 locked for C5 and N2
6c. 8 in C6 only in R678C6, locked for 45(9) cage at R5C6, no 8 in R8C34
7. 27(4) cage at R8C1 = {3789/4689/5679}, 9 locked for N7
8. 45 rule on R6789 3 outies R5C126 = 10 = {136/145}, no 7, 1 locked for R5 and N4, clean-up: no 2 in R6C2
8a. 6 of {136} must be in R5C6 -> no 6 in R5C12, clean-up: no 3 in R6C2
8b. Killer pair 4,6 in R5C126 and R5C7, locked for R5
9. 26(4) cage at R2C3 = {2789/4589/4679/5678}
9a. 8 of {4589/5678} must be in R2C3 -> no 5 in R2C3
10. Caged X-Wing for 3 in 45(9) cage at R1C5 and R45C8, no other 3 in R45, clean-up: no 6 in R6C2
10a. R5C126 (step 8) = {145} (only remaining combination), locked for R5 -> R45C7 = [46]
10b. 45(9) cage at R1C5 = {123456789}, 4 locked for C5 and N2
10c. 4 in C6 only in R56789C6, locked for 45(9) cage at R5C6, no 4 in R8C34
11. Killer pair 4,8 in R6C2 and R6C34, locked for R6
12. 2 in R67 only in 17(4) cage at R5C1 and 11(4) cage at R6C5 -> 17(4) cage must contain 2
= {1268/2348/2456} (cannot be {2357} = [57]{23} which clashes with 11(4) cage), no 7
13. 45 rule on N1 4(3+1) outies R123C4 + R4C1 = 27 but the only remaining candidates are 5,6,7,9 which total 27 -> R123C4 + R4C1 must form naked quad {5679}, CPE no 5,6,7,9 in R4C4
14. 25(4) cage at R1C2 = {3589/3679/4579} (cannot be {4678} because 4,6 only in R2C2), CPE no 9 in R1C1
14a. 4,6 of {3679/4579} must be in R2C2 -> no 7 in R2C2
15. 45 rule on C12 2 outies R1C34 = 1 innie R4C2 + 9
15a. R1C34 cannot total 11 -> no 2 in R4C2
16. 26(4) cage at R2C3 = {2789/4589/4679/5678}
16a. Caged X-Wing for 7 in 26(4) cage = {2789/4679/5678} and R23C9, no other 7 in R23 or 26(4) cage = {4589} = [8549], R23C9 = [97]
-> no other 7 in R3
16b. Caged X-Wing for 9 in 26(4) cage = {2789/4589/4679} and R23C9, no other 9 in R23 or 26(4) cage = {5678} = [85]{67}, R23C9 = [79]
-> no other 9 in R3
16c. Caged X-Wing for 9 in 26(4) cage = {2789/4589/4679} and R5C34, no other 9 in C34 or 26(4) cage = {5678} = [85]{67}, R5C34 = [98]
-> no other 9 in C3
17. 45 rule on R1 2 innies R1C15 = 1 outie R2C2, IOU no 7 in R1C5
[I was a bit slow in spotting …]
18. 2 in C12 only in 21(5) cage at R1C1 and 17(4) cage at R5C1 -> 21(5) cage must contain 2 -> R3C12 = {12}, locked for R3
19. 26(4) cage at R2C3 = {4589/4679/5678}, 25(4) cage at R1C2 (step 14) = {3589/3679/4579}
19a. R3C12 = {12} = 3 -> R124C1 = 18 = {369/378/459/567} (cannot be {468} which clashes with 26(4) cage at R2C3)
19b. Consider combinations for 26(4) cage
26(4) cage = {4589/5678} => R2C3 = 8, no 8 in R2C1
or 26(4) cage = {4679} => 25(4) cage = {3589/3679}, 3 locked for N1, no 3 in R1C1
-> R124C1 = {369/459/567} (cannot be {378} because no 3 in R1C1 or no 8 in R2C1), no 8
20. 8 in C1 only in R789C1, locked for N7
21. 17(4) cage at R5C1 (step 12) = {1268/2348} (cannot be {2456} = [56]{24} which clashes with R124C1) -> R7C1 = 8
21a. R5C1 = {14} -> no 4 in R7C2
21b. Killer pair 1,4 in R5C1 and R56C2, locked for N4, clean-up: no 7 in R6C4
22. 27(4) cage at R8C1 = {5679} (only remaining combination), locked for N7
[Cracked. The rest is straightforward.]
23. R9C3 = 4 (hidden single in N7), R9C45 = 8 = {26}, locked for R9 and N8
23a. R9C9 = 3 (hidden single in R9)
24. 45(9) cage at R5C6 = {123456789} -> R6C6 = 6
24a. R6C3 = 7 (hidden single in R6), R6C4 = 4
25. 26(4) cage at R2C3 = {5678} (only remaining combination) -> R2C34 = [85], R3C34 = [67]
and the rest is naked singles.
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