Prelims
a) R23C3 = {39/48/57}, no 1,2,6
b) R5C12 = {19/28/37/46}, no 5
c) R5C67 = {14/23}
d) R78C3 = {69/78}
e) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
f) 9(3) cage at R2C2 = {126/135/234}, no 7,8,9
g) 8(3) cage at R8C4 = {125/134}
1. 8(3) cage at R8C4 = {125/134}, 1 locked for N8
2. 45 rule on N1 2 outies R4C12 = 8 = {26/35}/[71], no 4,8,9, no 1 in R4C1
3. 45 rule on N7 2 outies R6C12 = 17 = {89}, locked for R6 and N4, clean-up: no 1,2 in R5C12
3a. Min R6C2 = 8 -> max R78C2 = 8, no 8,9 in R78C2
4. Killer pair 3,4 in R5C12 and R5C67, locked for R5
5. 45 rule on R5 2 innies R5C89 = 14 = {59/68}
6. 45 rule on C6789 2 outies R46C5 = 9 = {27/36/45}/[81], no 1,9 in R4C5
7. 45 rule on C123 3 innies R456C3 = 10 = {127/145/235} (cannot be {136} which clashes with R5C12), no 6
8. 45 rule on C12 2 outies R19C3 = 8 = {26/35}/[71], no 4,8,9, no 7 in R9C3
9. Hidden killer pair 8,9 in R23C3 and R67C3, R67C3 contains one of 8,9 -> R23C3 must contain one of 8,9 = {39/48}, no 5,7
10. 19(3) cage at R1C1 = {379/478/568} (cannot be {289/469} which clash with R23C3), no 2, clean-up: no 6 in R9C3 (step 8)
10a. Killer pair 8,9 in 19(3) cage and R23C3, locked for N1
11. 13(3) cage at R2C1 = {157/247/256/346}
11a. 45 rule on C1 3 innies R159C1 = 14
11b. Min R15C1 = 7 -> max R9C1 = 7
11c. 18(3) cage at R6C1 = {189/279/369/378/468} (cannot be {459/567} which clash with 13(3) cage), no 5
12. 45 rule on R9 1 innie R9C6 = 2 outies R8C49 + 5
12a. Min R8C49 = 3 -> min R9C6 = 8
12b. R9C89 = {89} -> R8C49 = 3,4 = {12/13}, 1 locked for R8
12c. Min R68C2 = 10 -> max R7C2 = 6
13. 13(3) cage at R9C1 = {139/148/157/238/247/256/346}
13a. 8,9 of {139/238} must be in R9C2, 3 of {346} must be in R9C3 -> no 3 in R9C2
14. 45 rule on R1 2 outies R2C49 = 1 innie R1C6 + 9, IOU no 9 in R2C9
14a. Max R2C49 = 17 -> max R1C6 = 8
15. Killer quad 6,7,8,9 in 18(3) cage at R6C1, R78C3 and 13(3) cage at R9C1, locked for N7
16. 16(3) cage at R6C2 = {259/349/358}, no 1
16a. 9(3) cage at R2C2 = {126} (only remaining combination, cannot be {135/234} which clash with 16(3) cage), locked for C2, clean-up: no 3,5 in R4C1 (step 2), no 4 in R5C1
16b. 16(3) cage = {349/358}, 3 locked for C2 and N7, clean-up: no 5 in R1C3 (step 8), no 7 in R5C1
17. 13(3) cage at R2C1 (step 11) = {157/247/256} (cannot be {346} which clashes with R5C1), no 3
17a. 18(3) cage at R6C1 (step 11c) = {189/468} (cannot be {279} which clashes with the 13(3) cage), no 2,7, 8 locked for C1
18. R456C3 (step 7) = {145/235} (cannot be {127} which clashes with R4C12), no 7, 5 locked for C3, clean-up: no 3 in R1C3 (step 8)
19. 2 in N7 only in 13(3) cage at R9C1, locked for R9
19a. 13(3) cage = {247/256} -> R9C3 = 2, R9C12 = [47/65/74], R1C3 = 6 (step 8), clean-up: no 9 in R78C3
19b. Naked pair {12} in R23C2, locked for C2 and N1 -> R4C2 = 6, R5C1 = 3, R5C2 = 7, R4C1 = 2, clean-up: no 2 in R5C67, no 3,7 in R6C5 (step 6)
19c. Naked pair {14} in R5C67, locked for R5 -> R5C3 = 5, clean-up: no 9 in R5C89 (step 5)
19d. Naked pair {68} in R5C89, locked for R5 and N6
[I didn’t spot the CPE until step 25.]
19e. Naked pair {29} in R5C45, locked for N5, clean-up: no 7 in R4C5 (step 6)
20. R23C3 = {39} (hidden pair in C2), locked for N1
20a. R4C1 = 2 -> R23C1 = 11 = {47}, locked for C1 and N1 -> R1C12 = [58], R9C1 = 6, R9C2 = 5 (cage sum), R6C2 = 9, R6C1 = 8, R78C1 = [19]
21. 8(3) cage at R8C4 = {134} (only remaining combination), locked for N8, 4 also locked for R9
22. 7 in R9 only in 19(4) cage at R8C9, locked for N9
22a. 19(4) cage = {1279/1378}
23. 31(5) cage at R7C6 = {25789/35689/45679} (cannot be {34789} which clashes with R8C2), 9 locked for C6 and N8, CPE no 5 in R8C5
23a. 5 in R8 only in R8C678, locked for 31(5) cage, no 5 in R7C6
24. 18(3) cage at R1C4 = {189/279/369/378/459} (cannot be {468/567} because 5,6,8 only in R2C4)
24a. 5,6,8 of {189/369/378/459} must be in R2C4 -> no 1,3,4 in R2C4
25. Naked pair {68} in R5C89, CPE no 6,8 in R37C8
25a. 37(7) cage at R3C8 = {1246789/1345789/2345689}, 8 locked for C9
26. 17(3) cage at R4C8 = {269/278/458/467} (cannot be {179/359} because R5C8 only contains 6,8, cannot be {368} because 6,8 only in R5C8), no 1,3
27. 17(4) cage at R1C7 = {1259/1349/1367/1457/2357} (cannot be {2456} because 5,6 only in R2C9)
27a. 5,6 of {1259/1367/1457/2357} only in R2C9 -> no 2,7 in R2C9
28. R2C49 = R1C6 + 9 (step 14)
28a. Max R2C49 = 15 -> no 7 in R1C6
28b. Min R2C49 = 10 -> no 2 in R2C4
29. 25(4) cage at R3C7 = {1789/3589/3679/4579/4678} (cannot be {2689} because 2,6 only in R3C7), no 2
30. 8 in R9 only in R9C6789, CPE no 8 in R8C78
30a. 31(5) cage at R7C6 (step 23) = {25789/35689/45679}
30b. 2,5 of {25789} must be in R8C78 -> no 2 in R78C6
[I ought to have spotted these steps earlier.]
31. 2 in C6 only in R123C6, locked for N2 and 21(5) cage at R1C6, no 2 in R2C78
32. 2 in N8 only in 24(5) cage at R6C3 = {12678/23478/24567} (cannot be {23568} because R6C3 only contains 1,4)
31a. R6C3 = {14} -> no 1,4 in R6C4
31b. 24(5) cage = {12678/24567} (cannot be {23478} which clashes with 31(5) cage at R7C6), no 3
[Since this puzzle has a high SS score, I think it’s now time to use a short forcing chain.]
32. 31(5) cage at R7C6 (step 23) = {25789/35689/45679}
32a. Consider combinations for 19(4) cage at R8C9 (step 22a) = {1279/1378}
19(4) cage = {1279} => R9C6 = 8 (hidden single in R9) => 31(5) cage = {35689}
or 19(4) cage = {1378} => R9C6 = 9 (hidden single in R9) => R8C49 = {13} (step 12b), locked for R8 => R8C2 = 4 => 31(5) cage = {25789}
-> 31(5) cage = {25789/35689}, no 4, 8 locked for C6 and N8
[This seems to be the key which cracks this puzzle; that’s why step 30, which I was slow to spot, was very important. 8 locked in R789C6 leads to step 34 and step 35.]
33. R8C2 = 4 (hidden single in R8), R7C2 = 3
34. 24(5) cage at R6C3 (step 31b) = {24567} (only remaining combination) -> R6C3 = 4, R4C3 = 1
35. 8 in R4 only in R4C45, CPE no 8 in R23C5
35a. R4C5 = 8 (hidden single in C5), R6C5 = 1 (step 6), R5C67 = [41]
35b. 1 in N8 only in R89C4, locked for C4
35c. 25(4) cage at R3C7 (step 29) = {3589/4678}
35d. 6,7 of {4678} must be in R3C7 + R4C6 -> no 4,7 in R3C7, no 7 in R4C7
36. 17(3) cage at R4C8 = {269/278/458/467}
36a. 4 of {458} must be in R4C8 -> no 5 in R4C8
37. 18(3) cage at R1C4 (step 24) = {369/378/459}
37a. 5,6,8 only in R2C4 -> R2C4 = {568}
38. R4C3 = 1 -> 25(5) cage at R2C5 = {13678/14569/14578} (cannot be {13489/13579} which clash with 18(3) cage at R1C4)
38a. 8 of {13678} must be in R3C4 -> no 3 in R3C4
39. R6C5 = 1 -> 17(4) cage at R6C5 = {1259/1268/1358/1367/1457} (cannot be {1349} because 4,9 only in R7C7)
39a. 8,9 of {1259/1268} must be in R7C7 -> no 2 in R7C7
39b. 4,8 of {1358/1457} must be in R7C7 -> no 5 in R7C7
40. Killer triple 1,2,3 in R8C49 and 31(5) cage at R7C6, locked for R8
40a. 2 in N8 only in 24(5) cage at R6C3, locked for R7
41. R2C49 = R1C6 + 9 (step 14)
41a. Min R2C49 = 10, no 1 in R4C9
42. 18(3) cage at R1C4 (step 37) = {369/378/459}, 17(4) cage at R1C7 (step 27) = {1259/1349/1367/1457/2357}
42a. Hidden killer pair 4,7 in 18(3) cage and 17(4) cage at R1C7 for R1, 18(3) cage cannot contain both of 4,7 -> 17(4) cage must contain at least one of 4,7 = {1349/1367/1457/2357} (cannot be {1259} which doesn’t contain 4 or 7)
42b. Hidden killer pair 1,2 in R1C6 and 17(4) cage for R1, 17(4) cage contains one of 1,2 -> R1C6 = {12}
43. 1,2 in C6 only in R123C6, locked for 21(5) cage at R1C7, no 1 in R2C8
43a. R2C26 = {12} (hidden pair in R2)
43b. Naked pair {12} in R12C6, locked for C6
44. 17(4) cage at R1C7 (step 42a) = {1349/1367/1457/2357}
44a. 4 of {1349} must be in R1C789 (R1C789 cannot be {139} which clashes with 18(4) cage at R1C4), 5 of {1457} must be in R2C9 -> no 4 in R2C9
45. R2C49 = R1C6 + 9 (step 14)
45a. R2C49 = [56/65/83] (cannot total 10) = 11 -> R1C6 = 2, R2C6 = 1, R23C2 = [21]
46. 2 in R3 only in R3C89, locked for 37(7) cage at R3C8, no 2 in R6C9
46a. 37(7) cage at R3C8 = {2345689} (only remaining combination), no 7
46b. 37(7) cage at R3C8 = {2345689}, CPE no 3 in R12C9, clean-up: no 8 in R2C4 (step 45a)
46c. R3C4 = 8 (hidden single in N2)
47. Naked pair {56} in R2C49, locked for R2
48. 18(3) cage at R1C4 (step 37) = {369/459}, no 7, 9 locked for R1 and N2
48a. 7 in R1 only in 17(4) cage at R1C7, locked for N3
49. 8 in R2 only in R2C78, R12C6 = {12} -> 21(5) cage at R1C6 = {12378/12468}, no 5,9
49a. 6,7 only in R3C6 -> R3C6 = {67}
50. R2C3 = 9 (hidden single in R2), R3C3 = 3
51. 37(7) cage at R3C8 (step 46b) = {2345689}, 3 locked for C9 and N6, 6 also locked for C9 -> R2C9 = 5, R2C4 = 6, R6C9 = 3, R3C6 = 7, R23C1 = [74], R3C5 = 5
51a. 37(7) cage = {2345689} -> R7C8 = 5
52. 18(3) cage at R1C4 (step 48) = {369} (only remaining combination), 3 locked for R1 and N2 -> R2C5 = 4, R4C4 = 7 (cage sum), R67C4 = [52]
53. R4C56 = [83] = 11 -> R34C7 = 14 = [95]
54. R6C56 = [16] = 7 -> R67C7 = 10 = [28]
and the rest is naked singles.
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