Prelims
a) R1C45 = {69/78}
b) R12C6 = {12}
c) R23C8 = {59/68}
d) R3C67 = {19/28/37/46}, no 5
e) R56C2 = {19/28/37/46}, no 5
f) R5C89 = {18/27/36/45}, no 9
g) R9C56 = {29/38/47/56}, no 1
h) 10(3) cage at R2C9 = {127/136/145/235}, no 8,9
i) 21(3) cage at R7C3 = {489/579/678}, no 1,2,3
j) 10(3) cage at R8C4 = {127/136/145/235}, no 8,9
k) 14(4) cage at R6C8 = {1238/1247/1256/1346/2345}, no 9
l) 37(6) cage at R7C7 = {256789/346789}, no 1
Steps resulting from Prelims
1a. Naked pair {12} in R12C6, locked for C6 and N2, clean-up: no 8,9 in R3C7, no 9 in R9C5
1b. 37(6) cage at R7C7 = {256789/346789}, 6,7,8,9 locked for N9
2. 45 rule on R1234 1 outie R5C1 = 2, clean-up: no 8 in R56C2, no 7 in R5C89
2a. R5C1 = 2 -> R234C1 = 20 = {389/479/569/578}, no 1
3. 45 rule on R4 2 innies R4C19 = 10 = {37/46}/[82/91], no 5
4. 45 rule on C12 2 innies R34C2 = 8 = {17/35}/[26], no 4,8,9, no 6 in R3C2
5. 45 rule on N1 2(1+1) remaining outies R2C4 + R4C1 = 17 = {89}, clean-up: R4C9 = {12} (step 3)
5a. Naked pair {89} in R2C4 + R4C1, CPE no 8,9 in R2C1 + R4C4
5b. R234C1 (step 2a) = {389/479/569/578}
5c. 3 of {389} must be in R2C1 -> no 3 in R3C1
6. 45 rule on N3 1 innie R3C7 = 1 outie R4C9 + 2 -> R3C7 = {34}, clean-up: R3C6 = {67}
6a. 12(3) cage at R2C5 = {345} (hidden triple in N2)
6b. Naked triple {345} in R3C457, locked for R3, 5 also locked for N2, clean-up: no 9 in R2C8, no 3,5 in R4C2 (step 4)
6c. 3,4,5 in R234C1 (step 2a) = {389/479/569/578} only in R2C1 -> R2C1 = {345}
7. 14(4) cage at R6C8 = {1346/2345} (cannot be {1238/1247/1256} which clash with R4C9), no 7,8
7a. 14(4) cage at R6C8 = {1346/2345}, CPE no 3,4 in R5C9, clean-up: no 5,6 in R5C8
7b. Killer pair 1,2 in R4C9 and 14(4) cage, no 1 in R5C9, clean-up: no 8 in R5C8
8. 45 rule on C9 2 innies R19C9 = 2 outies R56C8 + 12
8a. Max R19C9 = 17 -> max R56C8 = 5, no 5,6 in R6C8
8b. Min R56C8 = 4 (cannot be {12} which clashes with R4C9) -> min R19C9 = 16 -> R19C9 = {79/89}
8c. R56C8 = 4,5 = {13/14/23}
8d. Killer pair 1,2 in R4C9 and R56C8, locked for N6
8e. 2 in N6 only in R4C9 + R6C8, CPE no 2 in R78C9
9. 45 rule on N9 3 innies R78C9 + R8C7 = 8 = {134} (only remaining combination, cannot be {125} because R78C9 = {15} clashes with 10(3) cage at R2C9), locked for N9
9a. 4 must be in R78C9 (R78C9 cannot be {13} which clashes with 10(3) cage at R2C9), no 4 in R8C7
9b. 4 in N9 only in R78C9, locked for C9 and 14(4) cage at R6C8, no 4 in R6C8
[Sorry, my step numbering has got mixed up. I must have simplified some earlier steps and forgotten to adjust the step numbers. Don’t think any steps are missing.]
12. 14(4) cage at R6C8 (step 7a) = {1346/2345}
12a. 5,6 only in R6C9 -> R6C9 = {56}
13. 2 in C9 only in 10(3) cage at R2C9 (step 8) = {127} (only remaining combination, cannot be {235} because 3,5 only in R2C9), locked for C9, 7 also locked for N3
13a. Naked pair {89} in R19C9, locked for C9, clean-up: no 1 in R5C8
13b. Naked pair {56} in R56C9, locked for N6
13c. Killer pair 8,9 in R1C45 and R1C9, locked for R1
13d. Killer pair 8,9 in R1C9 and R23C8, locked for N3
13e. R4C9 + R5C8 = {12} (hidden pair in N6)
13f. R8C7 = 1 (hidden single in N9)
14. 19(4) cage at R1C7 = {1459/1468/2359/2368} (cannot be {1369/2458} which clash with R23C8)
14a. 1 of {1459/1468} must be in R1C8 -> no 4 in R1C8
14b. 4 in N3 only in R123C7, locked for C7
15. Max R1C12 = 11 (cannot be {67} which clashes with R1C45, cannot be {57} because 17(3) cage cannot be {57}5) -> min R2C2 = 6
16. R34C2 (step 4) = {17}/[26], R56C2 = {19/37/46} -> combined cage R3456C2 = {17}{46}/[26]{19}/[26]{37}, 6 locked for C2 and N4
17. R1C45 = {69/78}, R2C4 = {89} -> variable combined cage R1C45 + R2C4 = {69}8/{78}9
17a. 17(3) cage at R1C1 = {179/359/368/458/467} (cannot be {269/278} which clash with R1C45 + R2C4), no 2
17b. 7 of {179} must be in R1C1 (R12C2 cannot be [79] which clashes with combined cage R3456C2), 7 of {467} must be in R2C2 -> no 1 in R1C1, no 7 in R1C2
18. 2 in N1 only in 25(5) cage at R1C3 = {12589/12679/23479/23569/23578/24568}
18a. 1,2 in N1 only in 17(3) cage at R1C1 = {179} or 25(5) cage -> 25(5) cage = {12589/12679/24568} (cannot be {23479/23578}, locking-out cages, cannot be {23569} which clashes with 17(3) cage = {179} = {17}9), no 3
18b. Killer pair 5,6 in 17(3) cage and 25(5) cage, locked for N1
[Note. This works because 25(5) cage contains both of 5,6 when 17(3) cage = {179}. Alternatively the step can be considered to be a combined cage, since any 5,6 in the 25(5) cage must be in N1.]
19. R234C1 (step 2a) = {389/479}, 9 locked for C1
20. Naked pair {34} in R2C15, locked for R2
21. 1 in C1 only in R679C1, CPE no 1 in R7C2
22. 25(5) cage at R1C3 (step 18a) = {12589/12679/24568}
22a. 17(3) cage at R1C1 = {179/359/458/467} (cannot be {368} which clashes with 25(5) cage because 8 of {368} can only be in R2C2)
22b. Hidden triple killer 1,6,8 in 25(5) cage, 17(3) cage and R234C1 for N1, 25(5) cage contains two of 1,6,8 (not necessarily both in N1) -> 17(3) cage at R1C1 and R234C1 must contain at least one of 1,6,8
22c. R234C1 = {step 19) = {389/479} -> 17(3) cage at R1C1 (step 17a) = {179/458/467} (cannot be {359} because R234C1 = {479} doesn’t contain 8), no 3
[I suppose that’s a variant on locking-out cages using the hidden killer triple.]
[Even though this seems to be a key breakthrough, there’s still a lot of work to do.]
23. R2C1 = 3 (hidden single in N1), R34C1 (step 19) = {89}, locked for C1, R2C5 = 4, clean-up: no 7 in R9C6
23a. Deleted
24. R3C7 = 4 (hidden single in R3), R3C6 = 6, R4C9 = 2 (step 6), R4C1 = 8 (step 3), R3C1 = 9, R3C8 = 8, R2C8 = 6, R1C9 = 9, R9C9 = 8, clean-up: no 3,5 in R9C5, no 3 in R9C6
24a. Naked pair {17} in R23C9, locked for N3
25. Naked pair {78} in R1C45, locked for R1 and N2 -> R2C4 = 9
26. R6C8 = 1, R78C9 = {34} = 7 -> R6C9 = 6, R5C9 = 5, R5C8 = 4, clean-up: no 9 in R5C2
27. 17(3) cage at R1C1 (step 22a) = {458/467}, no 1, 4 locked for N1
28. 2 in R3 only in R3C23, locked for N1
29. 25(5) cage at R1C3 (step 18a) = {12589/12679}
29a. 8 of {12589} must be in R2C3 -> no 5 in R2C3
29b. 5,6 only in R1C3 -> R1C3 = {56}
30. R1C6 = 1 (hidden single in R1), R2C6 = 2, R2C7 = 5
31. 8 in N6 only in R56C7, locked for 27(5) cage at R5C7, no 8 in R6C6 + R7C56
31a. 27(5) cage = {13689/14589/15678/23589/24678/34578} (cannot be {12789} because 1,2 only in R7C5)
31b. 3,5,7 of {34578} must be in R7C5, 1,2 of other combinations must be in R7C5 -> no 6,9 in R7C5
31c. 27(5) cage = {14589/23589/34578}
32. 25(4) cage at R5C4 = {1789/2689/3589/3679}, 9 locked for N5
33. 27(5) cage at R5C7 (step 31c) = {14589/23589/34578}
33a. 45 rule on N78 2 remaining innies R7C56 = 1 outie R6C1 + 4
33b. R6C1 = {457} -> R7C56 = 8,9,11 -> {45/29/38/47} (cannot be [17/27] because 27(5) cage only contains one of 1,2,7, cannot be {35} because R6C1 + R7C56 = 4{35} clashes with 27(5) cage = {34578}, combo blocker), no 1 -> R7C56 = 9,11 -> R6C1 = {57}
[The extra combo blocker elimination proves to be useful. Now the puzzle is cracked.]
34. R9C4 = 1 (hidden single in N8)
35. R7C1 = 1 (hidden single in N7) -> 19(4) cage at R6C1 = {1279/1378/1459}
35a. 5 of {1459} must be in R6C1 -> no 5 in R78C2
36. 14(3) cage at R8C1 = {347/356} (cannot be {239} because 2,3,9 only in R9C2, cannot be {257} which clashes with R6C1) -> R9C2 = 3, clean-up: no 7 in R56C2
37. 19(4) cage at R6C1 (step 35) = {1279/1459}, no 8, 9 locked for C2 and N7, clean-up: no 1 in R5C2
38. R56C2 = [64], R1C2 = 5, R1C3 = 6, R1C1 = 4, R2C2 = 8 (cage sum), clean-up: no 2 in R3C2 (step 4)
39. R3C3 = 2 (hidden single in N1)
40. 14(3) cage at R8C1 (step 36) = {356} (only remaining combination), 5,6 locked for C1 and N7 -> R6C1 = 7, R4C2 = 1
41. 21(3) cage at R7C3 = {678} (only remaining combination) -> R7C4 = 6
42. R9C3 = 4 (hidden single in N7), R8C4 = 5 (cage sum), R9C6 = 9, R9C5 = 2
43. 8 in N8 only in 12(3) cage at R8C5 = {138} (only remaining combination), 3,8 locked for R8 and N8
44. 27(5) cage at R5C7 (step 31c) = {34578} (only remaining combination) -> R6C6 = 5, R56C7 = {38}, locked for C7 and N6
45. R5C5 = 1 (hidden single in N5) -> 25(5) cage at R5C4 (step 32) = {1789} (only remaining combination) -> R6C5 = 9, R5C46 = {78}, locked for R5 and N5
and the rest is naked singles.
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