Prelims
a) R12C4 = {19/28/37/46}, no 5
b) R2C56 = {49/58/67}, no 1,2,3
c) R34C1 = {18/27/36/45}, no 9
d) R6C67 = {49/58/67}, no 1,2,3
e) R67C9 = {18/27/36/45}, no 9
f) R89C4 = {79}
g) R8C56 = {59/68}
h) 9(3) cage in N7 = {126/135/234}, no 7,8,9
i) 22(6) cage at R1C5 = {123457}, no 6,8,9
Steps resulting from Prelims
1a. Naked pair {79} in R89C4, locked for C4 and N8, clean-up: no 1,3 in R12C4, no 5 in R8C56
1b. Naked pair {68} in R8C56, locked for R8 and N8
1c. 22(6) cage at R1C5 = {123457}, CPE no 1,2,3,4,5,7 in R1C89
2. R1C89 = {68/69/89} -> R2C8 = {134}
3. 45 rule on R12 2 outies R3C78 = 1 innie R2C9 + 2, IOU no 2 in R3C78
3a. Min R3C78 = 4 -> min R2C9 = 2
4. 45 rule on R89 2 outies R7C23 = 1 innie R8C1 + 7, IOU no 7 in R7C23
4a. 45 rule on R89 3 innies R8C123 = 16 = {259/349/457}, no 1
4b. Killer pair 7,9 in R8C123 and R8C4, locked for R8
4c. 1 in R8 only in R8C789, locked for N9, clean-up: no 8 in R6C9
5. 8 in N7 only in R7C123, locked for R7, clean-up: no 1 in R6C9
5a. 45 rule on N7, R8C123 = 16, R9C123 = 9 -> R7C123 = 20 = {389/578} (cannot be {479/569} which don’t contain 8), no 1,2,4,6
5b. 7 of {578} must be in R7C1 -> no 5 in R7C1
5c. R8C123 (step 4a) = {349/457} (cannot be {259} which clashes with R7C123), no 2, 4 locked for R8 and N7
6. 9(3) cage in N7 = {126} (hidden triple in N7), locked for R9
7. 2 in R8 only in R8C789, locked for N9, clean-up: no 7 in R6C9
7a. 1,2 in N8 only in R7C456, locked for 32(7) cage at R6C3, no 1,2 in R6C34
8. 45 rule on N7 2 innies R78C1 = 13 = [85/94]
8a. R34C1 = {18/27/36} (cannot be {45} which clashes with R8C1), no 4,5
9. R7C123 (step 5a) = {389} (only remaining combination), locked for R7 and N7, clean-up: no 6 in R6C9
9a. Naked triple {457} in R8C123, locked for R8 and N7 -> R89C4 = [97]
9b. 3 in C4 only in R3456C4, CPE no 3 in R4C6
10. 3 in N8 only in R9C56, locked for R9 and 22(5) cage at R8C7, no 3 in R8C7
11. 3 in N9 only in 13(3) cage = {139/238} -> R9C9 = {89}
12. 32(7) cage at R6C3 = {1234589/1234679/1235678}, 3 only in R6C34, locked for R6, clean-up: no 6 in R7C9
12a. 6 in N9 only in R7C78, locked for 32(7) cage, no 6 in R6C34
12b. 32(7) cage at R6C3 = {1234679/1235678} (only combinations containing 6)
12c. 3,8,9 only in R6C34 = R6C34 = [38/83/93], no 4,5,7
12d. 6,7 only in R7C78 = {67}, locked for R7, clean-up: no 2 in R6C9
13. Naked pair {45} in R67C9, locked for C9
14. 16(3) cage at R2C9 = {178/367} (cannot be {169/268} which clash with R19C9, ALS block), no 2,9, 7 locked for C9
14a. 9 in N3 only in R1C89, locked for R1
14b. R1C89 = {69/89} -> R2C8 = {13}
15. 45 rule on C9 4 innies R1589C9 = 20 = {1289/2369}
15a. 1,2,3 only in R58C9 -> R5C9 = {123}
16. 2 in N3 only in R12C7, locked for C7 -> R8C7 = 1
16a. 2,4,5 in N3 only in R123C7 + R3C8, locked for 22(5) cage at R1C5, no 2,4,5 in R1C56
[The next CPE has been available since step 1 but I’ve only just spotted it. It is, however, more powerful now.]
17. 1 in N2 only in R1C56 + R3C456, CPE no 1 in R3C8
17a. 1 in 22(5) cage at R1C5 only in R1C56, locked for R1 and N2
[OOPS! I’ve realised, after looking at Ed’s alternative breakthrough, that this CPE should have been for 1,3 in N2.
]
18. R8C89 = {23} -> R9C9 = 8 (step 11)
19. 16(3) cage at R2C9 (step 14) = {367} (only remaining combination), locked for C9 -> R1C9 = 9, R8C89 = [32], R5C9 = 1, R2C8 = 1, R1C8 = 8 (cage sum) , clean-up: no 2 in R2C4
19a. 6 in N3 only in R23C9 -> no 6 in R4C9
20. 45 rule on N47 2 innies R4C1 + R6C3 = 10 = [19/28/73], no 3,6,8 in R4C1, clean-up: no 1,3,6 in R3C1
21. 1 in N1 only in R3C23, locked for 34(7) cage at R3C2, no 1 in R4C6
21a. 45 rule for N1 3 innies R3C123 = 16 = {178} (only remaining combination, cannot be {169} because 1,6,9 only in R3C23), locked for R3 and N1, clean-up: no 7 in R4C1, no 3 in R6C3 (step 20)
22. R6C4 = 3 (hidden single in R6)
22a. 45 rule on N5 2 remaining innies R46C6 = 13 = {49/58/67}, no 2
23. 1,2 in N6 only in 26(5) cage = {12689} -> R5C7 = 8, R456C8 = {269}, locked for C8 and N6 -> R7C78 = [67], clean-up: no 4,5,7 in R6C6, no 6,8,9 in R4C6 (step 22a)
23a. 3 in N6 only in R4C79, locked for R4
23b. R9C7 = 9 (hidden single in N9)
24. 12(3) cage in N1 = {246/345}, no 9, 4 locked for N1
24a. 3 of {345} must be in R12C1 (R12C1 cannot be {45} which clashes with R8C1), no 3 in R1C2
24b. 9 in N1 only in R2C23, locked for R2, clean-up: no 4 in R2C56
24c. Killer pair 6,8 in R12C4 and R2C56, locked for N2
25. 34(7) cage at R3C2 = {1234789} (only remaining combination), no 5
25a. Naked triple {347} in R4C679, locked for R4
25b. 4 in R4 only in R4C67, locked for 34(7) cage at R3C2 -> R3C4 = 2, clean-up: no 8 in R2C4
25c. Naked pair {46} in R12C4, locked for C4 and N2 -> R5C4 = 5, R7C4 = 1, R4C4 = 8, clean-up: no 7 in R2C56, no 5 in R6C7
25d. Naked pair {39} in R3C56, locked for R3, N2 and 34(7) cage at R3C2, no 3 in R4C7 -> R3C9 = 6
25e. Naked pair {47} in R4C67, locked for R4 and 34(7) cage at R3C2, no 7 in R3C23 -> R4C9 = 3, R2C9 = 7
26. Naked pair {47} in R46C7, locked for C7 and N6 -> R3C78 = [54], R67C9 = [54], R9C8 = 5
27. 32(7) cage at R6C3 (step 12b) = {1235678} (only remaining combination) -> R6C3 = 8, R3C23 = [81], R3C1 = 7, R4C1 = 2
28. R7C1 = 8 (hidden single in R7), R8C1 = 5 (step 8)
29. 45 rule on N4 3 remaining innies R5C1 + R6C12 = 14 = {149/167} (cannot be {347} = 3{47} which clashes with R6C7), no 3, 1 locked for R6 and N4
29a. 7 of {167} must be in R6C2 -> no 6 in R6C2
30. 3 in C1 only in R12C1 -> 12(3) cage in N1 (step 24) = {345} (only remaining combination) -> R1C2 = 5, R12C1 = {34}, locked for C1 and N1
31. Naked pair {69} in R4C2 and R5C1, locked for N4 -> R6C1 = 1, R9C1 = 6, R5C1 = 9, R6C2 = 4 (step 29), R6C7 = 7, R6C6 = 6
and the rest is naked singles.
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