Prelims
a) R1C34 = {14/23}
b) R12C9 = {39/48/57}, no 1,2,6
c) R23C6 = {69/78}
d) R34C8 = {18/27/36/45}, no 9
e) R34C9 = {17/26/35}, no 4,8,9
f) R56C1 = {59/68}
g) R56C2 = {18/27/36/45}, no 9
h) R7C12 = {19/28/37/46}, no 5
i) R78C8 = {15/24}
j) R8C12 = {29/38/47/56}, no 1
k) R89C4 = {49/58/67}, no 1,2,3
l) 10(3) cage at R9C1 = {127/136/145/235}, no 8,9
m) 42(8) cage at R1C1 = {12456789}, no 3
n) 38(8) cage at R3C5 = {12345689}, no 7
1. R34C8 = {18/27/36} (cannot be {45} which clashes with R78C8), no 4,5
2. 45 rule on C9 1 innie R5C9 = 4, clean-up: no 8 in R12C9, no 5 in R6C2
2a. R34C9 = {17/26} (cannot be {35} which clashes with R12C9), no 3,5 in R34C9
3. 3 in N1 only in R13C3, locked for C3
3a. 3 in C1 only in R789C1, locked for N7, clean-up: no 7 in R7C1, no 8 in R8C1
4. 45 rule on N7 2 innies R78C3 = 14 = {59/68}
4a. R8C12 = {29/47}/[38] (cannot be {56} which clashes with R78C3), no 5,6
5. 45 rule on N4 2 innies R4C1 + R6C3 = 8 = {17/26}
5a. R56C2 = {18/36}/[54] (cannot be {27} which clashes with R4C1 + R6C3), no 2,7
6. 42(8) cage at R1C1 = {12456789}, 4,5,8,9 locked for N1, clean-up: no 1 in R1C4
7. 14(3) cage at R4C2 = {149/239/248/347} (cannot be {158/356} which clash with R56C1, cannot be {167/257} which clash with R4C1 + R6C3), no 5,6
7a. 5 in N4 only in R56C1 = {59} or in R56C2 = [54] -> 14(3) cage = {239/248/347} (cannot be {149}, locking-out cages), no 1
7b. 3 of {239/347} must be in R4C2 -> no 7,9 in R4C2
7c. 7 of {347} must be in R5C3 -> no 7 in R4C3
8. 45 rule on C1234 2 innies R25C4 = 5 = [23/32/41], R2C5 = {567}
8a. R1C34 and R25C4 must have different combinations because R25C4 “sees” R1C4, no 4 in R1C3 + R5C4 -> R12C4 must contain 4, locked for C4 and N2, clean-up: no 9 in R89C4
[Alternatively R1C4 = {234} -> R125C4 = 7,8,9 = {124/134/234}, 4 locked for C4 and N2 …]
9. 45 rule on R1 4 innies R1C1289 = 26 = {2789/3689/4589/4679/5678}, no 1
10. 45 rule on N7 3 outies R6C34 + R7C4 = 16 = {178/259/268/367} (cannot be {169/358} which clash with R78C3)
10a. 1 of {178} must be in R67C4 (R67C4 cannot be {78} which clashes with R89C4), no 1 in R6C3, clean-up: no 7 in R4C1 (step 5)
10b. 42(8) cage at R1C1 = {12456789}, 7 locked for N1
10c. 7 in N4 only in R56C3, locked by C3
11. 45 rule on C12 2 outies R29C3 = 1 innie R4C2 + 6
11a. Min R4C2 = 2 -> min R29C3 = 8, max R9C3 = 6 -> min R2C3 = 2
12. 45 rule on R9 2 innies R9C49 = 2 outies R8C56 + 8
12a. Min R8C56 = 3 -> min R9C49 = 11, max R9C4 = 8 -> min R9C9 = 3
12b. Max R9C49 = 17 -> max R8C56 = 9, no 9 in R8C56
13. 8 in C9 only in 21(4) cage at R6C9 = {1389/1578/2568}
13a. 6 of {2568} must be in R789C9 (R789C9 cannot contain both of 2,5 which would clash with R78C8) -> no 6 in R6C9
14. 17(3) cage at R3C3 = {179/269/359/368} (cannot be {278} which clashes with R89C4)
14a. 1 of {179} must be in R3C3 -> no 1 in R34C4
[The only way I can see to make progress from here is to use a forcing chain]
15. 30(5) cage at R6C3 = {15789/25689/35679}
15a. Consider combinations for 17(3) cage at R3C3 (step 14) = {179/269/359/368}
17(3) cage at R3C3 = {179/269/359}, 9 locked for C4 => 9 in 30(5) cage only in R78C3= {59} (step 4)
or 17(3) cage at R3C3 = {368} = 6{38} (cannot be 3{68} which clashes with R89C4) => R78C3 = {59} (step 4)
-> R78C3 = {59}, locked for C3, N7 and 30(5) cage at R6C3, no 5,9 in R67C4, clean-up: no 1 in R7C12, no 2 in R8C12
[Now the puzzle seems to be cracked, the remaining steps are fairly straightforward.]
16. 9 in N4 only in R56C1 = {59}, locked for C1 and N4, clean-up: no 4 in R6C2
17. 10(3) cage at R9C1 = {127/136}, no 4, 1 locked for R9
17a. 3 of {136} must be in R9C1 -> no 6 in R9C1
17b. R7C12 = {28/46} (cannot be [37] which clashes with 10(3) cage), no 3,7
18. R6C34 + R7C4 (step 10) = {178/268/367}
18a. 2 of {268} must be in R67C4 (R67C4 cannot be {68} which clashes with R89C4), no 2 in R6C3, clean-up: no 6 in R4C1 (step 5)
18b. 42(8) cage at R1C1 = {12456789}, 6 locked for N1
19. 9 in C4 only in 17(3) cage at R3C3 (step 14) = {179/269/359}, no 8
19a. R3C3 = {123} -> no 2,3 in R34C4
20. 4 in R6 only in R6C56, locked for N5 and 25(4) cage at R5C7, no 4 in R7C5
20a. 38(8) cage at R3C5 = {12345689} -> R3C7 = 4
20b. 4 in N8 only in R89C56, locked for 27(6) cage at R8C5, no 4 in R9C8
20c. 4 in R9 only in R9C56, locked for N8
21. 4 in N9 only in R78C8 = {24}, locked for C8 and N9, clean-up: no 7 in R34C8
21a. Killer pair 2,4 in R7C12 and R7C8, locked for R7
22. 32(7) cage at R5C8 contains 4 = {1234589/1234679} -> R6C7 = 2
23. R3C9 = 2 (hidden single in C9), R4C9 = 6, clean-up: no 3 in R3C8
24. 17(3) cage at R3C3 (step 19) = {179/359}, no 6
24a. Killer pair 5,7 in R34C4 and R89C4, locked for C4
25. R6C34 + R7C4 (step 18) = {178/367} -> R6C3 = 7, R4C1 = 1 (step 5), clean-up: no 8 in R3C8, no 8 in R56C2
25a. Naked pair {36} in R56C2, locked for C2, clean-up: no 4 in R7C1
25b. R13C3 = {13} (hidden pair in N1), locked for C3, clean-up: no 3 in R1C4
26. 45 rule on N3 2 remaining innies R1C7 + R3C8 = 13 = [76], R4C8 = 3, clean-up: no 5 in R12C9, no 9 in R2C6
26a. Naked pair {39} in R12C9, locked for C9 and N3
26b. R1C7 = 7 -> R1C56 = 7 = {16/25}, no 3,8,9
26c. Killer pair 1,2 in R1C34 and R1C56, locked for R1
27. 4 in R6 only in 25(4) cage at R5C7 = {4579/4678}, no 1,3 -> R7C5 = 7, clean-up: no 2 in R2C4, no 3 in R543 (step 8), no 6 in R89C4
27a. Naked pair {58} in R89C4, locked for C4 and N8
28. 17(3) cage at R3C3 (step 24) = {179} (only remaining combination) -> R3C3 = 1, R1C3 = 3, R1C4 = 2, R5C4 = 1, R2C4 = 4 (hidden single in C4), R2C5 = 5
28a. Naked pair {16} in R1C56, locked for R1 and N2, clean-up: no 9 in R3C6
28b. Naked pair {78} in R23C6, locked for C6 and N2 -> R34C4 = [97], R3C5 = 3
29. R5C2 = 3 (hidden single in R5), R6C2 = 6, R67C4 = [36], clean-up: no 4 in R7C2
29a. Naked pair {28} in R7C12, locked for R7 and N7 -> R78C8 = [42], R8C56 = [13], R7C6 = 9
29b. Naked pair {47} in R8C12, locked for R8 and N7
30. R89C56 = [13]{24} -> 27(6) cage at R8C5 = {123489} (only remaining combination) -> R9C78 = {89}, locked for R9 and N9 -> R9C4 = 5
31. Naked pair {18} in R2C78, locked for R2 and N3 -> R1C8 = 5, R2C6 = 7
and the rest is naked singles.
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