Prelims
a) R12C1 = {17/26/35}, no 4,8,9
b) R12C2 = {15/24}
c) R2C45 = {29/38/47/56}, no 1
d) R34C1 = {69/78}
e) R34C2 = {29/38/47/56}, no 1
f) R3C56 = {16/25/34}, no 7,8,9
g) R4C67 = {18/27/36/45}, no 9
h) R67C3 = {12}
i) R67C5 = {39/48/57}, no 1,2,3
j) R7C12 = {49/58/67}, no 1,2,3
k) R89C1 = {14/23}
l) R89C8 = {29/38/47/56}, no 1
m) R89C9 = {19/28/37/46}, no 5
n) 23(3) cage at R1C7 = {689}
o) 10(3) cage at R3C4 = {127/136/145/235}, no 8,9
p) 10(3) cage at R6C7 = {127/136/145/235}, no 8,9
q) 27(4) cage at R8C2 = {3789/4689/5679}, no 1,2
r) 39(8) cage at R1C6 = {12345789}, no 6
Steps resulting from Prelims
1a. Naked triple {689} in 23(3) cage at R1C7, locked for R1 and N3, clean-up: no 2 in R2C1
1b. Naked pair {12} in R67C3, locked for C3
1c. Killer pair 1,2 in R7C3 and R89C1, locked for N7
1d. 8,9 in R3 only in R3C123, locked for N1
1e. 27(4) cage at R8C2 = {3789/4689/5679}, 9 locked for R8, clean-up: no 2 in R9C8, no 1 in R9C8
2. 45 rule on R89 2 innies R8C67 = 6 = {15/24}
[I’ve moved the next step forward to simplify later steps …]
3. 1 in N1 only in R12C1 = {17} or R12C2 = {15} -> R12C1 = {17/26} (cannot be {35}, locking-out cages), no 3,5
3a. Killer pair 1,2 in R12C1 and R12C2, locked for N1, clean-up: no 9 in R4C2
3b. Killer pair 1,2 in R12C1 and R89C1, locked for C1
4. 45 rule on C1 3 innies R567C1 = 17 = {359/458} (cannot be {368/467} which clash with R34C1), no 6,7, clean-up: no 6,7 in R7C2
4a. 4 of {458} must be in R56C1 (R56C1 cannot be {58} because 18(3) cage at R5C1 cannot be {58}5), no 4 in R7C1, clean-up: no 9 in R7C2
4b. 45 rule on C1 2 outies R67C2 = 14 = [68/95], clean-up: no 9 in R7C1
4c. Naked pair {58} in R7C12, locked for R7 and N7, clean-up: no 4,7 in R6C5
4d. R34C2 = {38/47}/[92] (cannot be {56} which clashes with R67C2), no 5,6
5. 45 rule on R1234 3 outies R5C239 = 10 = {127/136/145/235}, no 8,9
6. Hidden killer pair 8,9 in R2C45 and R2C6 for N2, neither can contain both of 8,9 -> R2C45 = {29/38}, R2C6 = {89}
6a. 6 in N2 only in R3C456, locked for R3, clean-up: no 9 in R4C1
7. 45 rule on N14 2 innies R16C3 = 5 = [32/41], R1C3 = {34}
8. 5 in C3 only in R2345C3, locked for 27(5) cage at R2C3, no 5 in R5C2
8a. 5,8 in C3 only in 27(5) cage = {14589/15678/23589/34578}
9. 45 rule on C12 3 innies R589C2 = 14 = {167/239/347} (cannot be {149} which clashes with R12C2)
9a. 1 of {167} must be in R5C2 -> no 6 in R6C2
9b. 1 of {167} must be in R5C2, 7 of {347} must be in R89C2 (R89C2 cannot be {34} which clashes with R89C1), no 7 in R5C2
10. 45 rule on N2 3 innies R12C6 + R3C4 = 1 outie R1C3 + 15
10a. Min R1C3 = 3 -> min R12C6 + R3C4 = 18, cannot be {189} because 8,9 only in R2C6 -> no 1 in R1C6 + R3C4
11. 18(3) cage at R5C1 = {369/459/468}
11a. Hidden killer pair 8,9 in R4C123 and 18(3) cage for N4, 18(3) cage contains one of 8,9 -> R4C123 must contain one of 8,9
11b. 39(8) cage at R1C6 = {12345789}, R2C6 = {89} -> R4C89 must contain one of 8,9
11c. Killer pair 8,9 in R4C123 and R4C89, clean-up: no 1 in R4C67
12. 18(3) cage at R5C1 (step 11) = {369/459} (cannot be {468} = {48}6 which clashes with R34C1, combo blocker), no 8, 9 locked for N4
13. 8 in N4 only in R4C123, locked for R4
13a. 39(8) cage at R1C6 = {12345789} -> R2C6 = 8, 9 locked for N6, clean-up: no 3 in R2C45
13b. Naked pair {29} in R2C45, locked for R2 and N2, clean-up: no 4 in R1C2, no 5 in R3C56
14. 12(3) cage at R1C3 = {147/345}, 4 locked for R1
15. R12C6 + R3C4 = R1C3 + 15 (step 10), R2C6 = 8 -> R1C6 + R3C4 = R1C3 + 7
15a. R1C3 = {34} -> R1C6 + R3C4 = 10,11 = [37/56/73/74], no 5 in R3C6
16. 5 in N2 only in R1C456, locked for R1, clean-up: no 1 in R2C2
17. Hidden killer pair 3,5 in 12(3) cage at R1C3 and R1C6 for R1, 12(3) cage must contain both or neither of 3,5 but R1C6 cannot contain both of 3,5 -> 12(3) cage = {345} (only remaining combination), locked for R1 -> R1C6 = 7, clean-up: no 1 in R2C1, no 2 in R4C7
18. 1 in N2 only in R3C56 = {16}, locked for R3
19. 10(3) cage at R3C4 = {136/145/235} (cannot be {127} because R3C4 only contains 3,4), no 7
19a. R3C4 = {34} -> no 3,4 in R4C45
20. 45 rule on N3 3 innies R2C7 + R3C78 = 10 = {145/235}, 5 locked for N3 and 39(6) cage at R1C6, no 5 in R4C89 + R5C9
20a. 1 of {145} must be in R2C7 -> no 4 in R2C7
21. R5C239 (step 5) = {127/136/145/235}
21a. 5,6,7 only in R5C3 -> R5C3 = {567}
22. 1,3 in R2 only in R2C3789
22a. 45 rule on R12 4 remaining innies R2C3789 = 15 = {1347} (only remaining combination, cannot be {1356} = [65]{13} because 12(3) cage at R2C8 doesn’t contain 8), locked for R2 -> R12C1 = [26], R12C2 = [15], R7C12 = [58], R6C2 = 6 (step 4b), R56C1 = 12 = {39}, locked for C1 and N4, clean-up: no 3 in R3C2
[The rest is fairly straightforward.]
23. Naked pair {14} in R89C1, locked for N7 -> R7C3 = 2, R6C3 = 1, R1C3 = 4 (step 7), clean-up: no 7 in R4C2
23a. Killer pair 1,4 in R8C1 and R8C67, locked for R8, clean-up: no 7 in R9C8, no 6,9 in R9C9
23b. 3 in C2 only in R89C2, locked for N7
23. 27(4) cage at R8C2 = {3789/5679}, 7 locked for R8, clean-up: no 4 in R9C8, no 3 in R9C9
24. R3C4 = 4 (hidden single in N2), R4C45 = 6 = {15}, locked for R4 and N5, clean-up: no 4 in R4C67, no 7 in R7C5
25. 4,7 in N3 only in 12(3) cage at R2C8 = {147} -> R3C9 = 7, R2C89 = {14}, locked for N3 -> R2C7 = 3, R2C3 = 7, R3C123 = [893], R4C2 = 2, R4C13 = [78], R4C67 = [36], R5C23 = [45], clean-up: no 9 in R7C5, no 3 in R8C9
25a. 39(8) cage at R1C6 = {12345789} -> R5C9 = 1, R2C89 = [14], R4C89 = [49], clean-up: no 6 in R8C9
26. Naked pair {28} in R89C9, locked for C9 and N9 -> R1C9 = 6, clean-up: no 3 in R8C8, no 3,9 in R9C8
27. R67C9 = [53] = 8 -> R67C8 = 16 = [79], R1C67 = [98], R6C7 = 2, R3C78 = [52], R5C78 = [83], R56C1 = [93], R7C5 = 4, R6C5 = 8
28. Naked pair {16} in R37C6, locked for C6 -> R5C6 = 2, R8C6 = 5, R8C7 = 1 (step 2)
29. Naked pair {67} in R5C45, locked for 23(4) cage at R5C4 -> R7C4 = 1
and the rest is naked singles.
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