Prelims
a) R1C12 = {14/23}
b) R12C8 = {15/24}
c) R12C9 = {16/25/34}
d) R2C12 = {39/48/57}, no 1,2,6
e) R5C23 = {14/23}
f) R5C78 = {19/28/37/46}, no 5
g) R89C1 = {19/28/37/46}, no 5
h) R89C2 = {17/26/35}, no 4,8,9
i) R8C67 = {69/78}
j) R9C78 = {18/27/36/45}, no 9
k) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
l) 20(3) cage at R4C1 = {389/479/569/578}, no 1,2
1. R12C9 = {16/34} (cannot be {25} which clashes with R12C8), no 2,5
1a. Killer pair 1,4 in R12C8 and R12C9, locked for N3
2. 45 rule on R12 1 outie R3C5 = 1
2a. 1 in N1 only in R1C12 = {14}, locked for R1 and N1, clean-up: no 8 in R2C12, no 2,5 in R2C8, no 3,6 in R2C9
2b. Naked pair {14} in R2C89, locked for R2
3. 45 rule on R89 1 outie R7C5 = 3
4. 21(3) cage at R1C3 = {579/678}, CPE no 7 in R2C12, clean-up: no 5 in R2C12
4a. Naked pair {39} in R2C12, locked for R2 and N1
4b. 21(3) cage at R1C3 = {678} (only remaining combination), no 5
4c. 20(3) cage at R4C1 = {479/569/578} (cannot be {389} which clashes with R2C1), no 3
5. 2,5 in N1 only in R3C123, locked for R3 and 21(4) cage at R3C1, no 2,5 in R4C2
5a. 21(4) cage contains 2,5 = {2568} (only remaining combination) -> R3C123 = {2568}, R4C2 = {68}
5b. 45 rule on N1 2(1+1) outies R2C4 + R4C2 = 14 = {68}, CPE no 6,8 in R4C2
5c. 7 in N1 only in R12C3, locked for C3
6. 45 rule on N3 2(1+1) outies R2C6 + R4C8 = 7 = {25}/[61]
6a. 24(4) cage at R3C7 = {1689/2679/3579} (cannot be {2589} because 2,5 only in R4C8, cannot be {3678} because R4C8 only contains 1,2,5), 9 locked for R3 and N3
6b. Killer pair 3,6 in R1C9 and 24(4) cage, locked for N3
7. 45 rule on N7 2(1+1) outies R6C2 + R8C4 = 10 = {19/28/46/55}/[37], no 7 in R6C2
7a. 7 in N4 only on R456C1, locked for C1, clean-up: no 3 in R89C1
7b. 20(3) cage at R4C1 (step 4c) contains 7 = {479/578}, no 6
8. R2C1 = 3 (hidden single in C1), R2C2 = 9, clean-up: no 1 in R8C4 (step 7)
9. 45 rule on N9 2(1+1) outies R6C8 + R8C6 = 17 = {89}, clean-up: no 8,9 in R8C7
9a. Naked pair {89} in R6C8 + R8C6, CPE no 8,9 in R6C6 + R8C8
10. 45 rule on C6789 3 innies R159C6 = 11 = {128/137/146/236/245}, no 9
[I should have spotted this when I did step 6a.]
11. 24(4) cage at R3C7 (step 6a) = {2679/3579} (cannot be {1689} which clashes with 21(4) cage at R3C1, ALS block), no 1,8, 7 locked for R3 and N3, clean-up: no 6 in R2C6 (step 6)
11a. Naked pair {25} in R2C6 + R4C8, CPE no 2,5 in R4C6
11b. Naked pair {25} in R14C8, locked for C8, clean-up: no 8 in R5C7, no 4,7 in R9C7
11c. 8 in N3 only in R12C7, locked for C7, clean-up: no 1 in R9C8
12. R159C6 = 11 = {128/137/146/236} (cannot be {245} which clashes with R2C6), no 5
13. 1,7,9 in N2 only in 25(5) cage at R1C4 = {12679/13579}, no 8
14. 45 rule on C12 3 outies R357C3 = 14 = {158/239/248/356} (cannot be {149} because no 1,4,9 in R3C3)
14a. 1 of {158} must be in R5C3 -> no 1 in R5C7
15. 45 rule on N6 4 innies R46C78 = 18
15a. Min R6C8 = 8 -> max R4C78 + R6C7 = 10, no 9 in R46C7
[At this stage I originally analysed 15(3) cage at R3C4 and 16(3) cage at R3C6 but this wasn’t very helpful so I’ve left these cages until later.]
16. 45 rule on C1 3 innies R137C1 = 12 = {129/156/246}, no 8
16a. R1C1 = {14} -> no 1,4 in R7C1
17. 45 rule on C9 2 innies R37C9 = 1 outie R8C8 + 6, IOU no 6 in R3C9
17a. R8C8 cannot be 2 more than R3C9 -> no 8 in R7C9
18. 45 rule on N9 1 outie R6C8 = 1 innie R8C7 + 2 -> R6C8 + R8C7 = [86/97]
18a. 23(4) cage at R6C8 = {1589/2489} (cannot be {1679/2678} which clash with R8C7, cannot be {2579/4568} which clash with R6C8 + R8C7, combo blocker), no 6,7
18b. 23(4) cage = {1589/2489}, CPE no 8 in R9C8, clean-up: no 1 in R9C7
19. Hidden killer pair 8,9 in 23(4) cage at R6C8 and 15(3) cage at R8C8 for N9, 23(4) cage must contain one of 8,9 in N9 -> 15(3) cage must contain one of 8,9 = {159/168/249/348} (cannot be {258} which clashes with 23(4) cage), no 7
19a. Killer pair 1,4 in 23(4) cage and 15(3) cage, locked for N9, clean-up: no 5 in R9C7
19b. Killer pair 6,7 in R8C7 and R9C78, locked for N9
19c. 15(3) cage = {159/249/348}
19d. 1 of {159} must be in R8C8 -> no 1 in R89C9
20. 45 rule on C9 4 innies R3789C9 = 21 = {1578/2478/3459} (cannot be {1389} which clashes with R12C9, cannot be {1479} which clashes with R2C9, cannot be {2379} which clashes with R9C78)
20a. R3789C9 = {2478/3459} (cannot be {1578} which clashes with 23(4) cage at R6C8, CCC), no 1, 4 locked for C9 and N9
21. R2C9 = 1, R1C9 = 6, R2C8 = 4, R1C8 = 2, R4C8 = 5, clean-up: no 6 in R5C7
21a. Naked pair {58}, locked for C7 and 15(3) cage at R1C7 -> R2C6 = 2
21b. Naked triple {379} in R3C789, locked for R3
21c. Naked triple {468} in R2C4 + R3C46, locked for N2
22. R46C78 (step 15) = 18 -> R4C7 + R6C78 = 13 = {139/148/238} (cannot be {247/346} because R6C8 only contains 8,9), no 6,7
23. R5C8 = 6 (hidden single in N6). R5C7 = 4, clean-up: no 1 in R5C23, no 3 in R9C7
23a. Naked pair {23} in R5C23, locked for R5 and N4, clean-up: no 7,8 in R8C4 (step 7)
24. 17(3) cage at R4C9 = {278} (only remaining combination), locked for C9 and N6 -> R6C8 = 9
24a. Naked pair {13} in R46C7, locked for C7
25. R159C6 (step 12) = {137} (only remaining combination, cannot be {146} because R1C6 only contains 3,7) -> R1C6 = 3, R59C6 = {17}, locked for C6
26. 5 in C6 only in 12(3) cage at R6C6 = {156/345}, no 8,9
27. 16(3) cage at R3C6 = {169/349} -> R4C6 = 9, R8C6 = 8, R8C7 = 7, R3C7 = 9, R3C89 = [73], R7C7 = 2, R9C78 = [63], R8C8 = 1, R7C8 = 8, R7C9 = 4 (cage sum), clean-up: no 4 in R8C1, no 2,5 in R8C2, no 2,9 in R9C1, no 1,7 in R9C2
28. 1 in R5 only in R5C46, locked for N5
28a. 8 in C5 only in R456C5, locked for N5
29. 3 in N5 only in R46C4
29a. 45 rule on N5 3 innies R4C4 + R6C46 = 13 = {346} (only remaining combination), locked for N5, 6 also locked for R6, clean-up: no 4 in R8C4 (step 7)
30. R7C6 = 5 (hidden single in C6), clean-up: no 5 in R6C2 (step 7)
31. 45 rule on N8 2 remaining innies R78C4 = 10 = [19], R6C2 = 1 (step 7), R6C7 = 3, R6C6 = 4 (cage sum)
and the rest is naked singles.
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