Prelims
a) R1C12 = {79}
b) R12C9 = {18/27/36/45}, no 9
c) R89C1 = {19/28/37/46}, no 5
d) R9C89 = {19/28/37/46}, no 5
e) 22(3) cage at R1C3 = {589/679}
f) 11(3) cage at R1C5 = {128/137/146/236/245}, no 9
g) 7(3) cage at R2C3 = {124}
h) 8(3) cage at R4C2 = {125/134}
i) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
j) 13(4) cage at R1C6 = {1237/1246/1345}, no 8,9
k) 26(4) cage at R3C4 = {2789/3689/4589/4679/5678}, no 1
1. Naked pair {79} in R1C12, locked for R1 and N1, clean-up: no 2 in R2C9
1a. 22(3) cage at R1C3 contains 9 -> R2C4 = 9, R1C34 = 13 = {58}, locked for R1, clean-up: no 1,4 in R2C9
2. Naked triple {124} in 7(3) cage at R2C3, locked for N1
2a. 45 rule on N1 1 innie R1C3 = 1 outie R4C1 + 1, R1C3 = {58} -> R4C1 = {47}
2b. 9 in N4 only in R456C3, locked for C3
3. 8(3) cage at R4C2 = {125/134}, 1 locked for N4
4. 45 rule on N3 1 innie R3C9 = 1 outie R1C6 + 5, R1C6 = {1234}, R3C9 = {6789}
5. 45 rule on N7 1 outie R9C4 = 1 innie R7C1 + 2, no 7,8 in R7C1, no 1,2 in R9C4
6. 11(3) cage at R6C1 = {128/137/146/236/245}
6a. 5,6 of {236/245} must be in R6C12 (R6C12 cannot be {23/24} which clashes with 8(3) cage at R4C2), no 5,6 in R7C1, clean-up: no 7,8 in R9C4 (step 5)
7. 45 rule on C12 4 innies R3789C2 = 24, max R3C2 = 4 -> min R789C2 = 20, no 1,2 in R789C2
8. 13(4) cage at R1C6 = {1237/1345} (cannot be {1246} which clashes with R12C9 = [36] because 13(4) cage “sees” R1C9), no 6
8a. 5,7 only in R2C8 -> R2C8 = {57}
8b. 13(4) cage at R1C6 = {1237/1345}, 1,3 locked for R1, clean-up: no 6,8 in R2C9
9. 11(3) cage at R1C5 = {128/146/236/245} (cannot be {137} because R1C5 only contains 2,4,6), no 7
9a. 7 in R2 only in R2C789, locked for N3, clean-up: no 2 in R1C6 (step 4)
10. 45 rule on C89 4 innies R1237C8 = 15 = {1257/1347/1356} (cannot be {1239/1248/2346} because R2C8 only contains 5,7), no 8,9, 1 locked for C8, clean-up: no 9 in R9C9
11. 18(3) cage at R2C7 = {189/279/369/459/468} (cannot be {567} which clashes with R2C8, cannot be {378} which clashes with R2C89, ALS block)
11a. 1,2 of {189/279} must be in R3C8 -> no 1,2 in R23C7
12. 11(3) cage at R1C5 (step 9) = {128/146/236/245}
12a. Hidden killer pair 1,2 in R2C3 and R2C56 for R2, R2C56 cannot contain more than one of 1,2 -> R2C3 = {12}, R2C56 must contain one of 1,2
12b. 4 in N1 only in R3C23, locked for R3
13. 18(3) cage at R2C7 (step 11) = {189/279/369/459/468}
13a. 8 of {468} must be in R3C7, 9 of {189/279/369/459} must be in R3C7 -> R3C7 = {89}
13b. 4 of {459} must be in R2C7 -> no 5 in R2C7
14. 45 rule on C1234 1 outie R7C5 = 2 innies R58C4 + 3
14a. Min R58C4 = 3 -> min R7C5 = 6
14b. Max R58C4 = 6, no 6,7,8 in R58C4
15. 45 rule on C6789 2 innies R25C6 = 1 outie R3C5 + 4, IOU no 4 in R5C6
16. 45 rule on R1234 2 innies R4C25 = 1 outie R5C3 + 3, IOU no 3 in R4C5
17. 45 rule on R6789 2 innies R6C58 = 1 outie R5C7
17a. Min R6C58 = 3 -> min R5C7 = 3
17b. Max R6C58 = 9, no 8,9 in R6C5, no 9 in R6C8
18. 11(3) cage at R6C1 = {128/137/146/236/245}
18a. Consider placements for R4C1 = {47}
R4C1 = 4
or R4C1 = 7 => R1C3 = 8 (step 2a) => 8 in N4 only in 11(3) cage = {128}
-> no 4 in R6C12 + R7C1, clean-up: no 6 in R9C4 (step 5)
18b. 11(3) cage = {128/137/236}, no 5
[Going a bit further]
18c. R4C1 = 4 => 8(3) cage at R4C2 = {125}, locked for N4 => 11(3) cage = {36}2/{37}1
or R4C1 = 7 => R1C3 = 8 (step 2a) => 8 in N4 only in 11(3) cage = {128}, 2 locked for N4
-> no 2 in R456C3, no 3 in R7C1, clean-up: no 5 in R9C4 (step 5)
[And further still]
18d. R4C1 = 4 => R1C3 = 5 (step 2a)
or R4C1 = 7 => R1C3 = 8 (step 2a) => 8 in N4 only in 11(3) cage = {128}, 2 locked for N4 => 8(3) cage = {134} => 5 in N4 only in R456C3
-> 5 in R1C3 + R456C3, locked for C3, 4 in R4C1 + 8(3) cage, locked for N4
18e. R4C1 = 4 => 8(3) cage at R4C2 = {125}, locked for N4 => 11(3) cage = {36}2/{37}1
or R4C1 = 7 => R1C3 = 8 (step 2a) => 8 in N4 only in 11(3) cage = {128}, 2 locked for N4 => 8(3) cage = {134}
-> 3 in 8(3) cage + R6C12, locked for N4
[The above are really just one forcing chain, I’ve split it into sub-steps for clarity; they were also how I saw this step developing.]
18f. 5 in N7 only in R789C2, locked for C2
18g. 5 in 8(3) cage only in R5C1 -> no 2 in R5C1
19. 3 in C3 only in R789C3, locked for N7, clean-up: no 7 in R89C1
19a. R3C789C2 (step 7) = 24 = 2{589}/4{569/578} (cannot be 1{689} which clashes with R89C1, cannot be 2{679} which clashes with R1C2) -> no 1 in R3C2
[I overlooked no 4 in R789C2, but it probably didn’t make any difference to step 21 and it’s eliminated by step 21e.]
19b. 1 in N1 only in R23C3, locked for C3
19c. 1 in N7 only R789C1, locked for C1
20. Deleted. I originally used a somewhat “chainy” locking-out cages step, based on the position of 3 in N7, to eliminate 2 from the 17(3) cage at R8C2. However step 21, based on the position of 4 in N7, is more direct and more powerful.
21. 17(3) cage at R7C2 = {269/278/368/458/467} (cannot be {359} because 5,9 only in R8C2)
21a. 4 in N7 only in 17(3) cage = {458/467} or R89C1 = {46} or 20(4) cage at R8C2 = {458}3/{467}3 -> 17(3) cage = {269/278/458/467} (cannot be {368} which clashes with R89C1 = {46} and with 20(4) cage = {458}3/{467}3, locking-out cages), no 3
[Cracked, but there’s still a fairly long ending.]
21b. R9C3 = 3 (hidden single in N7), R9C4 = 4, R7C1 = 2 (step 5), R6C12 = 9 = {36}, locked for R6 and N4, clean-up: no 4 in 8(3) cage at R4C2, no 6,8 in R8C1, no 8 in R9C1, no 6,7 in R9C89
21c. R5C1 = 5, R45C2 = {12}, locked for C2 -> R3C2 = 4
21d. 1 in C1 only in R89C1 = {19}, locked for C1 and N7 -> R1C12 = [79], R4C1 = 4, R1C3 = 5 (step 2a), R1C4 = 8
21e. R78C3 = {46} (hidden pair in C3), locked for N7, R7C2 = 7 (cage sum)
21f. R9C89 = {28} (only remaining combination, cannot be [91] which clashes with R9C1), locked for R9 and N9 -> R89C2 = [85]
22. 16(3) cage at R8C6 = {169/367} (cannot be {259} because 2,5 only in R8C6), no 2,5
22a. 3 of {367} must be in R8C6 -> no 7 in R8C6
22b. 16(3) cage = {169/367}, CPE no 6 in R9C5
22c. 6 in R9 only in R9C67, locked for 16(3) cage, no 6 in R8C6
23. 2 in N8 only in 12(3) cage at R8C4 = {129/237}, no 5,6
23a. 5 in N8 only in R7C46, locked for R7
24. 12(3) cage at R8C4 (step 23) = {129/237} and 16(3) cage at R8C6 (step 22) = {169/367} form combined 28(6) cage R8C456 + R9C567 = {123679}, 3 locked for R8 and N8
24a. R89C1 and combined cage form grouped X-Wing in 1,9 for R89, no other 1,9 in R8
25. 12(3) cage at R7C7 = {147/156/345}, no 9
26. 45 rule on N9 1 outie R6C9 = 1 innie R9C7 -> R6C9 = {179}, R9C7 = {179}
26a. R9C6 = 6 (hidden single in R9)
26b. 9 in N9 only in R7C9 + R9C7, R6C9 = R9C7 -> 9 in R67C9, locked for C9, clean-up: no 4 in R1C6 (step 4)
26c. R3C7 = 9 (hidden single in N3), R7C9 = 9 (hidden single in N9)
26d. 16(3) cage at R8C6 (step 22) = {169/367}
26e. R9C7 = {17} -> no 1 in R8C6
27. R7C5 = 8 -> R58C4 = 5 (step 14) = {23}, locked for C4
28. 26(4) cage at R3C4 = {5678} (only remaining combination) -> R34C4 = {56}, locked for C4, R45C3 = {78}, locked for C3 -> R6C3 = 9, R67C4 = [71], R6C9 = 1, R9C7 = 1 (step 26), R8C6 = 9 (cage sum), R9C5 = 7, R7C6 = 5 (hidden single in N8), R3C6 = 7 (hidden single in C6)
29. R7C6 = 5 -> 24(4) cage at R5C7 = {4578} (only remaining combination) -> R5C7 = 7, R6C67 = {48}, locked for R6
30. R67C9 = [19] -> R8C89 = 13 = {67}, locked for R8 and N9 -> R78C3 = [64], R8C7 = 5
31. 45 rule on N5 3 remaining innies R4C46 + R6C6 = 15 = {168} (only remaining combination, cannot be {258} which clashes with R6C5, cannot be {348} because R4C4 only contains 5,6, cannot be {456} because no 4,5,6 in R4C6) -> R4C46 = [61], R6C6 = 8
32. R1C6 = 3, R3C9 = 8 (step 4)
33. R34C6 = [71] = 8 -> R2C5 + R3C7 = 10 = [28]
and the rest is naked singles.
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A nice challenging killer. I look forward to more of them.
