Prelims
a) R12C3 = {49/58/67}, no 1,2,3
b) R1C45 = {29/38/47/56}, no 1
c) R1C78 = {16/25/34}, no 7,8,9
d) R12C9 = {14/23}
e) R4C67 = {59/68}
f) R6C67 = {29/38/47/56}, no 1
g) R67C9 = {29/38/47/56}, no 1
h) R78C4 = {19/28/37/46}, no 5
i) R78C5 = {59/68}
j) R9C89 = {18/27/36/45}, no 9
k) 28(4) cage at R3C9 = {4789/5689}, no 1,2,3
l) 17(5) cage at R7C6 = {12347/12356}, no 8,9
m) 33(5) cage at R8C3 = {36789/45789}, no 1,2
n) 23(6) cage at R6C3 = {123458/123467}, no 9
Steps resulting from Prelims
1a. R1C78 = {16/25} (cannot be {34} which clashes with R12C9), no 3,4
1b. Killer pair 1,2 in R1C78 and R12C9, locked for N3
1c. R1C45 = {29/38/47} (cannot be {56} which clashes with R1C78), no 5,6
1d. 28(4) cage at R3C9 = {4789/5689}, CPE no 8,9 in R6C9, clean-up: no 2,3
1e. 1 in N6 only in R5C78 + R6C8, locked for 33(7) cage at R4C5, no 1 in R4C5 + R5C56 + R7C8
1f. 17(5) cage at R7C6 = {12347/12356}, CPE no 2,3 in R7C8
2. 45 rule on N3 2 innies R2C7 + R3C9 = 17 = {89}, locked for N3
2a. Naked pair {89} in R2C7 + R3C9, CPE no 8,9 in R3C6
3. 45 rule on R9 2 outies R8C36 = 16 = {79}, locked for R8, clean-up: no 1,3 in R7C4, no 5 in R7C5
3a. Naked pair {79} in R8C36, CPE no 7,9 in R9C4
4. 33(5) cage at R8C3 = {36789/45789}, 8 locked for R9, 7,9 locked for N7, clean-up: no 1 in R9C89
5. 45 rule on R6789 2 innies R67C8 = 1 outie R5C1 + 11
5a. Min R67C8 = 12, no 1,2 in R6C8
5b. Max R67C8 = 17 -> max R5C1 = 6
5c. 1 in N6 only in R5C78, locked for R5
5d. 33(7) cage at R4C5 contains 2, CPE no 2 in R5C4
6. 45 rule on N89 3(2+1) innies R7C89 + R9C4 = 21
6a. Max R7C89 = 17 -> min R9C4 = 4
7. 45 rule on N8 2 innies R7C6 + R9C4 = 1 outie R9C7 + 2, IOU no 2 in R7C6
7a. Min R7C6 + R9C4 = 5 -> min R9C7 = 3
7b. 1 in R9 only in R9C56, locked for N8, clean-up: no 9 in R7C4
7c. Min R7C6 + R9C4 = 7 -> min R9C7 = 5
[Next I originally found
Min R8C6 + R9C7 = 14 (because 9 in R89 only in 33(5) cage at R8C3 and 19(4) cage at R8C6 which must both contain 9) -> max R9C56 = 5 = {12/13/14}
but this isn’t needed after I found the next step.]
8. 17(5) cage at R7C6 = {12347/12356}, 2 locked for N9, clean-up: no 7 in R9C89
8a. R9C56 = {12} (hidden pair in R9), locked for N8, clean-up: no 8 in R78C4
8b. R9C56 = {12} = 3 -> R8C6 + R9C7 = 16 = {79}
9. 8 in N9 only in R7C89, locked for R7, clean-up: no 6 in R8C5
10. 3 in N8 only in R7C6 + R8C4, CPE no 3 in R8C789
10a. 17(5) cage at R7C6 = {12347/12356}, 3 locked for R7
10b. 7 of {12347} must be in R7C7 (R7C7 + R8C789 cannot be {1234} which clashes with R9C89), no 7 in R7C6
10c. 6 of {12356} must be in R7C7 + R8C789 (R7C7 + R8C789 cannot be {1235} which clashes with R9C89), no 6 in R7C7
11. 36(6) cage at R1C6 = {156789/246789/345789}, 7 locked for N2, clean-up: no 4 in R1C45
11a. Killer pair 8,9 in R1C45 and 36(6) cage, locked for N2
12. R7C6 + R9C4 = R9C7 + 2 (step 7)
12a. R9C7 = {79} -> R7C6 + R9C4 = 9,11 = {45}/[38] (cannot be [36] which clashes with R78C4, cannot be [56] which clashes with R78C5), no 6 in R7C6
13. 45 rule on C1 2 innies R19C1 = 1 outie R6C2 + 11
13a. Max R19C1 = 17 -> max R6C2 = 6
13b. Min R19C1 = 12, no 1,2 in R1C1
14. 45 rule on C1 3 outies R126C2 = 1 innie R9C1 + 1
14a. Min R126C2 = 6 -> min R9C1 = 5
14b. Max R126C2 = 10, no 8,9 in R12C2
15. 45 rule on R1 3 outies R2C239 = 1 innie R1C6 + 3
15a. Min R2C239 = 7 -> min R1C6 = 4
16. 45 rule on R1 2 innies R1C36 = 2 outies R2C29 + 10
16a. Max R1C36 = 16 (cannot be {89} which clashes with R1C45) -> max R2C29 = 6, no 6,7 in R2C2
17. 45 rule on C9 2 innies R89C9 = 1 outie R4C8 + 1
17a. R89C9 cannot total 9 (which clashes with R9C89, CCC) -> no 8 in R4C8
17b. 28(4) cage at R3C9 = {4789/5689}, 8 locked for C9, clean-up: no 3 in R6C9
18. R7C8 = 8 (hidden single in N9)
19. R67C8 = R5C1 + 11 (step 5), R7C8 = 8 -> R6C8 = R5C1 + 3, no 5 in R5C1, no 3,4 in R6C8
20. R7C89 + R9C4 = 21 (step 6), R7C8 = 8 -> R7C9 + R9C4 = 13 = [58/94], R6C9 = {26}
20a. R7C6 + R9C4 = R9C7 + 2 (step 7)
20b. R9C4 is now only even, R9C7 is odd -> R7C6 must be odd -> R7C6 = {35}
20c. 4 in N8 only in R789C4, locked for C4
21. 5 in N8 only in R7C6 + R8C5, CPE no 5 in R8C789
21a. 17(5) cage at R7C6 = {12347/12356}
21b. 3,5,7 only in R7C67 -> R7C7 = {357}
21c. 5 of {12356} must be in R7C6 (R7C7 + R8C789 cannot be {1256} which clashes with R9C89), no 5 in R7C7
21d. 1,2 in N9 only in R8C789, locked for R8
22. 7 in N9 only in R79C7, locked for C7, clean-up: no 4 in R6C6
22a. 7 in N3 only in R23C8, locked for C8
23. 7 in N6 only in 28(4) cage at R3C9 = {4789} (only remaining combination), 4 locked for N6, clean-up: no 7 in R6C6
24. 33(7) cage at R4C5 contains 8 so must also contain 4, locked for N5
24a. 4 in R6 only in R6C123, locked for N4
25. R89C9 = R4C8 + 1 (step 17)
25a. R4C8 = {49} -> R89C9 = 5,10 cannot contain 5, no 5 in R9C9, clean-up: no 4 in R9C8
[Cracked, the rest is straightforward.]
26. R7C9 = 5 (hidden single in C9), R6C9 = 6, clean-up: no 3 in R9C8
26a. R9C8 = 6, R9C9 = 3, R7C67 = [37], R9C7 = 9, R8C36 = [97], R2C7 = 8, R3C9 = 9, R4C8 = 4, clean-up: no 2 in R12C9
26b. Naked pair {14} in R12C9, locked for N3 -> R8C9 = 2, clean-up: no 6 in R1C7
26c. Naked pair {25} in R1C78, locked for R1 and N3, clean-up: no 9 in R1C45
26d. R8C89 = [41], R78C4 = [46], R9C4 = 8, R1C45 = [38], R78C5 = [95]
26e. R4C7 = 5, R4C6 = 9, R6C8 = 9, R1C78 = [25], R6C7 = 3, R6C6 = 8, R3C7 = 6, R5C78 = [12]
26f. Clean-up: no 4,5 in R12C3
27. R1C1 = 9 (hidden single in R1) -> R12C2 = 3 = [12], R12C9 = [41], R1C6 = 6, R12C3 = [76], R7C2 = 6
28. R2C45 = [97] (hidden singles in N2), R23C8 = [37]
29. Naked pair {12} in R69C5, locked for C5 -> R3C5 = 4, R2C6 = 5, R2C1 = 4, R3C6 = 1 (cage sum), R3C4 = 2, R9C56 = [12], R6C5 = 2
30. R2C1 = 4 -> R34C1 = 8 = {35} (cannot be {17} because 1,2,6,7 only in R4C1) -> R3C1 = 5, R4C1 = 3, R9C1 = 7, R5678C1 = [6128], R6C2 = 5 (cage sum)
and the rest is naked singles.
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