Prelims
a) R45C4 = {17/26/35}, no 4,8,9
b) R45C6 = {14/23}
c) R56C1 = {16/26/34}, no 7,8,9
d) R56C5 = {59/68}
e) R56C9 = {29/38/47/56}, no 1
f) R89C1 = {49/58/67}, no 1,2,3
g) R8C34 = {15/24}
h) R89C5 = {18/27/36/45}, no 9
i) R8C67= {49/58/67}, no 1,2,3
j) R89C9 = {49/58/67}, no 1,2,3
k) 19(3) cage at R1C1 = {289/379/469/478/578}, no 1
l) 11(3) cage at R1C9 = {128/137/146/236/245}, no 9
m) 26(4) cage at R2C6 = {2789/3689/4589/4679/5678}, no 1
n) 33(5) cage at R3C8 = {36789/45789}, no 1,2
Step resulting from Prelims
1. 33(5) cage at R3C8 = {36789/45789}, CPE no 7,8,9 in R6C8
2. 45 rule on C1 2 innies R47C1 = 6 = {15/24}
2a. R56C1 = {16/34} (cannot be {25} which clashes with R478C1), no 2,5
2b. Killer pair 1,4 in R47C1 and R56C1, locked for C1, clean-up: no 9 in R89C1
3. 9 in C1 only in 19(3) cage at R1C1, locked for N1
3a. 19(3) cage = {289/379}, no 5,6
3b. 15(4) cage at R2C2 = {1248/1257/1347/1356/2346} (cannot be {1239} = {123}9 which clashes with 19(3) cage at R1C1), no 9
4. 45 rule on C9 2 innies R47C9 = 10 = {37/46}/[82/91], no 5, no 8,9 in R7C9
5. 45 rule on R789 2 innies R7C37 = 9 = {18/27/36/45}, no 9
5a. 9 in N7 only in R789C2 + R9C3, locked for 35(6) cage at R7C1, no 9 in R9C4
6. 45 rule on C1234 2 innies R37C4 = 12 = {39/48/57}, no 1,2,6
7. 45 rule on C6789 2 innies R37C6 = 7 = {16/25} (cannot be {34} which clashes with R45C6)
7a. Killer pair 1,2 in R37C6 and R45C6, locked for C6
8. 45 rule on N7 1 outie R9C4 = 2 innies R78C3 + 3
8a. Min R78C3 = 3 -> min R9C4 = 6
8b. Max R9C4 = 8 -> max R78C3 = 5, no 5,6,7,8 in R7C3, no 5 in R8C3, clean-up: no 1,2,3,4 in R7C7 (step 5), no 1 in R8C4
9. 35(6) cage at R7C1 must contain both of 8,9, CPE no 8 in R9C1, clean-up: no 5 in R8C1
10. 1,2,3 in N9 only in R7C89 + R8C8 + R9C78, locked for 24(6) cage at R7C8, no 3 in R9C6
11. 45 rule on N6 3 innies R5C7 + R6C78 = 1 outie R3C8 + 1
11a. Min R5C7 + R6C78 = 6 -> min R3C8 = 5
11b. Max R5C7 + R6C78 = 10, no 8,9 in R56C7
11c. Max R5C7 + R6C78 = 10 -> min R6C6 + R7C7 = 12, no 3 in R6C6
12. 45 rule on N5 3 innies R4C5 + R6C46 = 18 = {279/378/459/468} (cannot be {189/369/567} which clash with R56C5), no 1
13. R37C6 (step 7) = 7 = {16/25}, R45C6 = 5 = {14/23} -> combined cage R3457C6 = 12 = {1236/1245}
[Or for those who don’t like combined cages
45 rule on C6789 4 innies R3456C6 = 12 = {1236/1245}]
13a. 45 rule on N9 2 outies R89C6 = 1 innie R7C7 + 5
13b. Min R89C6 = 11 (R89C6 cannot be {46} = 10 which clashes with R3457C6) -> min R7C7 = 6, clean-up: no 4 in R7C3 (step 5)
14. 30(5) cage at R5C3 cannot contain more than one of 1,2,3
14a. R7C3 = {123} -> no 1,2,3 in R5C3 + R6C234
14b. 2 in N4 only in R4C123 + R5C2, locked for 21(5) cage at R3C2, no 2 in R3C2
15. 45 rule on N89 3(2+1) innies R7C7 + R89C4 = 18
15a. Max R7C7 = 8 -> min R89C4 = 10
15b. 45 rule on N8 4 innies R89C46 = 23 = {2489/2678/4568} (cannot be {2579} because min R89C4 = 10), 8 locked for N8, clean-up: no 4 in R3C4 (step 6), no 1 in R89C5
16. 1 in N8 only in 13(3) cage at R7C4, locked for R7, clean-up: no 5 in R4C1 (step 2), no 9 in R4C9 (step 4), no 8 in R7C7 (step 5)
16a. 13(3) cage = {139/157}, no 2,4,6, clean-up: no 8 in R3C4 (step 6), no 1,5 in R3C6 (step 7)
16b. R89C9 = {49/58} (cannot be {67} which clashes with R7C7), no 6,7
17. 1 in C9 only in 11(3) cage at R1C9, locked for N3
17a. 11(3) cage = {128/137/146}, no 5
18. 45 rule on N36 4(3+1) outies R126C6 + R7C7 = 28
18a. Max R7C7 = 7 -> min R126C6 = 21, no 3 in R12C6
19. 3 in C6 only in R45C6 = {23}, locked for C6 and N5 -> R3C6 = 6, R7C6 = 1 (step 7), clean-up: no 5,6 in R45C4, no 7 in R8C7
19a. Naked pair {17} in R45C4, locked for C4 and N5, clean-up: no 5 in R37C4 (step 6)
19b. Naked pair {39} in R37C4, locked for C4
20. 13(3) cage at R7C4 (step 16a) = {139} (only remaining combination, cannot be {157} because 5,7 only in R7C5), 3,9 locked for R7 and N8 -> R7C3 = 2, R7C7 = 7 (step 5), clean-up: no 4 in R4C1 (step 2), no 3,7,8 in R4C9 (step 4), no 4 in R8C7, no 6 in R89C5
21. Naked pair {46} in R47C9, locked for C9, clean-up: no 5,7 in R56C9, no 9 in R89C9
21a. Naked pair {58} in R89C9, locked for C9 and N9, clean-up: no 5,8 in R8C6
22. Naked pair {46} in R7C89, locked for R7 and N9 -> R7C1 = 5, R4C1 = 1 (step 2), R7C2 = 8, R45C4 = [71], R8C7 = 9, R8C6 = 4, R8C3 = 1, R8C4 = 5, R89C5 = [85], clean-up: no 6 in R56C1, no 3 in R56C9
22a. Naked pair {34} in R56C1, locked for C1 and N4
22b. Naked pair {29} in R56C9, locked for C9 and N6
23. R9C4 = 6 -> R89C1 = [67], R8C2 = 3, R8C8 = 2, R89C5 = [72], R9C6 = 8
[Step 23 simplified. When I first solved this puzzle I must have forgotten to manually eliminate 8 from R9C4 in step 22.]24. Naked triple {289} in 19(3) cage at R1C1, locked for N1
25. Naked triple {137} in 11(3) cage at R1C9, locked for N3
26. 33(5) cage at R3C8 must contain 9 -> R3C8 = 9, R4C4 = 3, R7C45 = [93]
27. 28(6) cage at R1C5 must contain 1,3,6 = {134569} (only remaining combination), no 8
28. R56C5 = {68} (hidden pair in C5), locked for N5 -> R6C4 = 4, R56C1 = [43]
29. 26(4) cage at R2C6 = {4589/5678} (cannot be {2789} because 7,9 only in R2C6), no 2, 8 locked for N3
29a. 7,9 only in R2C6 -> R2C6 = {79}
30. R1C7 = 2 (hidden single in C7), R12C4 = [82], R123C1 = [982]
31. 15(3) cage at R1C6 = {267} (only remaining combination) -> R1C68 = [76], R2C6 = 9
32. 14(3) cage at R1C2 = {158} (only remaining combination) -> R1C23 = [15]
33. R6C678 = [561], R5C7 = 3, R56C5 = [68]
34. Naked pair {79} in R6C23, locked for R6 and N4 -> R5C3 = 8
35. 15(4) cage at R2C2 (step 3b) = {2346} (only remaining combination) -> R3C3 = 4, R2C2 = 6, R2C3 = 3
and the rest is naked singles.
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