Prelims
a) R1C12 = {59/68}
b) R2C12 = {19/28/37/46}, no 5
c) R56C4 = {49/58/67}, no 1,2,3
d) R56C7 = {29/38/47/56}, no 1
e) R56C9 = {17/26/35}, no 4,8,9
f) R67C2 = {17/26/35}, no 4,8,9
g) R9C23 = {59/68}
h) R9C56 = {59/68}
i) R9C78 = {13}
j) 10(3) cage at R1C5 = {127/136/145/235}, no 8,9
k) 21(3) cage at R3C8 = {489/579/678}, no 1,2,3
l) 10(3) cage at R5C5 = {127/136/145/235}, no 8,9
m) 11(3) cage at R6C3 = {128/137/146/236/245}, no 9
Steps resulting from Prelims
1a. Naked pair {13} in R9C78, locked for R9 and N9
1b. Naked quad {5689} in R9C2356, locked for R9
2. 13(3) cage at R8C8 = {247/256}, no 8,9, 2 locked for N9
3. 45 rule on N9 2 outies R6C8 + R8C6 = 7 = {16/25/34}, no 7,8,9
4. 45 rule on C9 3 innies R789C9 = 15 = {249/258/267} (cannot be {456} because 4 of {456} must be in R9C9 and 13(3) cage at R8C8 cannot contain 4 and one of 5,6), 2 locked for C9 and N9, clean-up: no 6 in R56C9
4a. 45 rule on C9 1 innie R7C9 = 1 outie R8C8 + 2, no 4,5 in R7C9
5. 35(6) cage at R6C8 = {146789/245789/345689} (cannot be {236789} which clashes with 13(3) cage at R8C8)
5a. 1,2 of {146789/245789} must be in R8C6 (R8C6 cannot be 5,6 which clash with 13(3) cage), no 1,2 in R6C8, no 5,6 in R8C6
6. 2 in C8 only in R12C8, locked for N3, CPE no 2 in R2C6
7. 45 rule on N5 2 innies R4C4 + R6C5 = 1 outie R4C7 + 5, IOU no 5 in R6C5
8. 25(4) cage at R6C5 = {1789/2689/3589/3679/4579/4678}
8a. 1,2,3 of {1789/2689/3589/3679} must be in N8 (R7C56 + R8C5 cannot be {789/689/589/679} which clash with R9C56), no 1,2,3 in R6C5
9. 13(3) cage at R8C8 (step 2) = {247/256}, R6C8 + R8C6 (step 3) = {16/25/34}
9a. Consider combinations for 35(6) cage at R6C8 (step 5) = {146789/245789/345689}
35(6) cage = {146789} -> 13(3) cage = [562], R68C8 = [65] -> 21(3) cage at R3C8= {489}
or 35(6) cage = {245789} -> 13(3) cage = [652], R68C8 = [56] -> 21(3) cage = {489}
or 35(6) cage = {345689}, CPE no 4 in R8C8 -> 13(3) cage = 7{24} -> 21(3) cage = {489}
-> 21(3) cage = {489}, locked for C8, clean-up: no 6 in R7C9 (step 4a), no 3 in R8C6 (step 3)
9b. R789C9 (step 4) = {249/258/267}, R7C9 = {789} -> no 7 in R89C9
9c. 35(6) cage = {146789/245789/345689}, CPE no 4 in R8C9
10. 7 in R9 only in R9C14, CPE no 7 in R8C23
11. 16(4) cage at R8C2 = {1258/1267/1348/1357/1456/2347/2356} (cannot be {1249} which clashes with R8C6), no 9
12. 13(3) cage at R8C8 (step 2) = {247/256}, R7C8 = {567}
12a. Variable combined cage R7C8 + 13(3) cage = {56}[724]/7{56}2 -> 7 in R78C8, locked for C8 and N9, clean-up: no 5 in R8C8 (step 2a), no 6 in R8C9
12b. 35(6) cage at R6C8 (step 5) = {245789/345689} (cannot be {146789} which clashes with 13(3) cage), no 1, clean-up: no 6 in R6C8 (step 3)
13. R56C9 = {17} (cannot be {35} which clashes with R6C8), locked for C9 and N6, clean-up: no 4 in R56C7
14. R8C6 “sees” all cells in N9 containing 2,4 except for R9C9 -> R8C6 = R9C9
14a. 2 in N9 only in R89C9 -> R8C69 contains 2, locked for R8
15. 16(4) cage at R8C2 (step 11) = {1258/1348/1357/1456/2356} (cannot be {1267} which clashes with R8C8, cannot be {2347} which clashes with R8C6)
15a. 4,7 of {1348/1357/1456} must be in R9C4 -> no 4,7 in R8C234
16. 45 rule on R4 3 innies R4C189 = 14 = {149/158/239/248/347} (cannot be {167/257/356} because R4C8 only contains 4,8,9), no 6
16a. 1,2,7 only in R4C1 -> R4C1 = {127}
16b. 6 in C9 only in R123C9, locked for N3
17. R78C8 = {67} (hidden pair in C8), locked for N9
17a. 35(6) cage at R6C8 (step 12b) = {245789/345689}, CPE no 5 in R456C7, clean-up: no 6 in R56C7
[Cracked. The rest is fairly straightforward.]
18. 21(3) cage at R3C8 = {489}
18a. 4 of {489} must be in R45C8 (R45C8 cannot be {89} which clashes with R56C7), no 4 in R3C8, 4 locked for N6
19. R4C7 = 6 (hidden single in N6), R4C56 = 11 = {29/38/47}, no 1,5
20. 45 rule on N6 1 outie R3C8 = 2 remaining innies R4C9 + R6C5 + 1
20a. Min R4C9 + R6C5 = 8 -> R3C8 = 9, R4C9 + R6C5 = {35}, locked for N6, clean-up: no 8 in R56C7
21. R7C9 = 9 (hidden single in C9), R8C8 = 7 (step 4a), R89C9 = 6 = [24], R7C8 = 6, R8C6 = 4, R6C8 = 3 (step 3), R4C9 = 5, R9C78 = [31], clean-up: no 7 in R4C5 (step 19), no 2 in R6C2, no 5 in R7C2
21a. Naked pair {25} in R12C8, locked for N3, CPE no 5 in R2C6
22. 16(4) cage at R8C2 (step 15) = {1357/2356} (cannot be {1258} which clashes with R8C7), 3,5 locked for R8 -> R78C7 = [58]
23. R4C189 (step 16) = {158} (only remaining combination) -> R4C1 = 1, R4C8 = 8, R5C8 = 4, clean-up: no 9 in R2C2, no 3 in R4C56 (step 19), no 9 in R6C4, no 7 in R7C2
24. 1 in N5 only in 10(3) cage at R5C5 = {136} (only remaining combination, cannot be {127} which clashes with R4C56), locked for N5, 3 also locked for R5, clean-up: no 7 in R56C4
24a. R56C4 = {58} (only remaining combination, cannot be [94] which clashes with R4C56), locked for C4 and N5
[Alternatively 5 in N5 only in R56C4 = {58} …]
24b. 2 in N5 only in R4C456, locked for R4
25. 5 in N8 only in R9C56 = {59}, locked for R9 and N8
25a. Naked pair {68} in R9C23, locked for N7 -> R8C1 = 9, clean-up: no 5 in R1C2, no 1 in R2C2
26. 16(4) cage at R2C7 = {1267/1357/1456/2347} (cannot be {1258/2356} because 2,5,6,8 only in R2C8 + R3C6, cannot be {1348} because 3,8 only in R3C6)
26a. 3,6 only in R3C6 -> R3C6 = {36}
27. Naked triple {136} in R356C6, locked for C6, 1 also locked for N5
28. 8 in N2 only in R12C6, locked for C6
28a. 19(4) cage at R1C6 contains 8 = {2458} (only remaining combination) -> R1C7 = 4, R2C6 = 8, R1C68 = {25}, locked for R1, clean-up: no 9 in R1C2, no 2 in R2C12
28b. Naked pair {68} in R1C12, locked for R1 and N1 -> R1C9 = 3, R2C9 = 6, R3C9 = 8, clean-up: no 4 in R2C12
28c. Naked pair {37} in R2C12, locked for R2 and N1 -> R23C7 = [17]
29. 14(3) cage at R1C3 = {149} (only remaining combination) -> R2C4 = 4, R1C34 = {19}, locked for R1 -> R1C5 = 7
30. R1C5 = 7, R23C5 = 3 = [21], R2C8 = 5, R2C3 = 9, R1C34 = [19]
31. R8C5 = 6, R5C5 = 3, R7C5 = 8
32. 16(4) cage at R8C2 (step 22) = {1357} (only remaining combination) -> R9C4 = 7, R9C1 = 2, R4C4 = 2, clean-up: no 9 in R4C56 (step 19), no 6 in R6C2
32a. R4C56 = [47], R4C23 = [93]
33. R7C13 = {47} (hidden pair in N7)
33a. R89C1 = [92] = 11 -> R67C1 = 11 = {47}, locked for C1 -> R2C12 = [37], R3C12 = [54], R3C3 = 2, R3C4 = 6 (cage sum)
34. 11(3) cage at R6C3 = {146} (only remaining combination, cannot be {137} because 1,3 only in R7C4) -> R6C3 = 6, R7C3 = 4, R7C4 = 1
and the rest is naked singles.
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