Prelims
a) R3C89 = {15/24}
b) R45C1 = {19/28/37/46}, no 5
c) R4C78 = {59/68}
d) R5C34 = {16/25/34}, no 7,8,9
e) R67C1 = {39/48/57}, no 1,2,6
f) 12(2) cage in N9 = {39/48/57}, no 1,2,6
g) 20(3) cage at R1C3 = {389/479/569/578}, no 1,2
h) 11(3) cage at R2C8 = {128/137/146/26/245}, no 9
i) 19(3) cage in N5 = {289/379/469/478/568}, no 1
j) 11(3) cage in N6 = {128/137/146/236/245}, no 9
k) 10(3) cage at R6C2 = {127/136/145/235}, no 8,9
l) 13(4) cage at R8C4 = {1237/1246/1345}, no 8,9
1. 13(4) cage at R8C4 = {1237/1246/1345}, 1 locked for R8
2. 45 rule on R89 2 innies R8C28 = 17 = {89}, locked for R8, CPE no 8,9 in R2C28 + R5C5 using diagonals
2a. R8C8 = {89} -> R7C7 = {34}
3. 45 rule on R123 2 outies R4C56 = 10 = {28/37/46}/[91], no 5, no 1 in R4C5
4. 45 rule on C1 2 outies R19C2 = 9 = {18/27/36/45}, no 9
5. 45 rule on D/ 2 innies R1C9 + R9C1 = 9 = {18/27/36/45}, no 9
6. 45 rule on N489 1 innie R9C4 = 9
6a. R9C4 = 9 -> R89C3 = 8 = {26/35}/[71], no 4,8, no 7 in R9C3
7. 9 on D/ only in R7C3 + R8C2, locked for N7, clean-up: no 3 in R6C1
8. 45 rule on N8 2 outies R89C7 = 6 = {15/24}
9. 45 rule on N6 2 outies R7C89 = 12 = {39/48/57}
10. 6 in N9 only in 15(3) cage = {168/267} (cannot be {456} which clashes with R89C7), no 3,4,5
10a. 3 in N9 only in R7C789, locked for R7, clean-up: no 9 in R6C1
10b. 9 in N9 only in R7C89 + R8C8, CPE no 9 in R5C8
11. 45 rule on R1234 2 outies R5C12 = 1 innie R4C9 + 10
11a. Min R5C12 = 11, no 1, clean-up: no 9 in R4C1
11b. Max R5C12 = 17 -> max R4C9 = 7
12. 45 rule on N1 3 outies R123C4 = 17 = {278/368/458/467}, no 1
13. 45 rule on N3 2 innies R12C7 = 1 outie R4C6 + 13
13a. Min R12C7 = 14, no 1,2,3,4
13b. Max R12C7 = 17 -> max R4C6 = 4 -> min R4C5 = 6 (step 3)
14. 45 rule on R12 3 innies R2C248 = 1 outie R3C1 + 3
14a. Min R2C248 = 6 -> min R3C1 = 3
15. 32(6) cage at R4C9 contains at least one of 1,2 which must be in N6
15a. 11(3) cage in N6 = {137/146/236/245} (cannot be {128} which clashes with 32(6) cage at R4C9), no 8
16. 14(3) cage in N8 = {158/167/248/257}
16a. 9 in N7 only in R7C3 + R8C2
16b. 45 rule on N7 4 remaining innies R7C123 + R8C2 = 24 = {1689/2589/2679} (cannot be {4569} = {456}9 which clashes with 14(3) cage in N8), no 4, clean-up: no 8 in R6C1
16c. 7 of {2679} must be in R7C1 -> no 7 in R7C23
17. R89C3 (step 6a) = {35}/[71] (cannot be {26} which clashes with R7C123 + R8C2), no 2,6
18. 45 rule on N47 2 innies R7C3 + R8C2 = 2 remaining outies R45C4 + 9
18a. Min R45C4 = 3 -> min R7C3 + R8C2 = 12, no 1,2 in R7C3
18b. Max R7C3 + R8C2 = 17 -> max R45C4 = 8, no 8 in R4C4
19. R7C123 + R8C2 (step 16b) = {1689/2589/2679} -> R7C2 = {12}
19a. 14(3) cage in N8 = {158/167/257} (cannot be {248} which clashes with R7C123 + R8C2, noting that {1689} can only be [816]9), no 4
20. 4 in R7 only in R7C789, locked for N7, clean-up: no 2 in R89C7 (step 8)
20a. Naked pair {15} in R89C7, locked for C7 and N9, clean-up: no 9 in R4C8, no 7 in R7C89 (step 9)
20b. Killer pair 8,9 in R7C89 and R8C8, locked for N9
20c. 8 in N9 only in R7C89 + R8C8, CPE no 8 in R5C8
21. 11(3) cage in N6 (step 15a) = {137/146/236/245}
21a. 1,5 of {137/245} must be in R6C8 -> no 4,7 in R6C8
22. Hidden killer pair 8,9 in R4C78 and R5C8 + R456C9 for N6, R4C78 contains one of 8,9 -> R5C8 + R456C9 must contain one of 8,9
22a. Hidden killer pair 3,4 in 11(3) cage and R5C8 + R456C9 for N6, 11(3) cage contains one of 3,4 -> R5C8 + R456C9 must contain one of 3,4
22b. R7C89 (step 9) = {39/48}
22c. Taking these steps together 32(6) cage at R4C9 must contain 3,4,8,9 = {134789/234689}, no 5
[This step is actually fairly easy, it just looks more complicated when written formally. Ed should be able to “see” this step in his head.
I probably wouldn’t have used this step if I’d seen step 24 earlier.]
23. 5 in N6 only in R46C8, locked for C8, clean-up: no 1 in R3C9
24. 45 rule on R9 3 outies R8C139 = 15 = {267/456} (cannot be {357} = {35}7 which clashes with R89C3, CCC), no 3, 6 locked for R8, clean-up: no 5 in R9C3 (step 6a)
24a. R8C3 = {57} -> no 5,7 in R8C19
24b. 3 in N7 only in R9C123, locked for R9
24c. 7 in N9 only in R9C89, locked for R9, clean-up: no 2 in R1C2 (step 4), no 2 in R1C9 (step 5)
[With hindsight this 45 could have been used a lot earlier; it was only while checking my walkthrough that I realised that after step 6a it could have been used to eliminate 7 from R8C9, because of a CCC between R8C13 and R89C3. However it’s much more powerful now, after the eliminations in step 17.]
25. 15(3) cage at R9C5 = {168/258/456}
25a. R9C7 = {15} -> no 1,5 in R9C56
25b. Killer pair 2,6 in 15(3) cage at R9C5 and 15(3) cage in N9, locked for R9, clean-up: no 3,7 in R1C2 (step 4), no 3,7 in R1C9 (step 5)
26. 4 in N7 only in 13(3) cage = {148/346}, no 2,5, clean-up: no 4 in R1C2 (step 4), no 4 in R1C9 (step 5)
27. R7C2 = 2 (hidden single in N7)
[Ed pointed out that my original step 27a was flawed so I’ve re-worked from here.]27a. 10(3) cage at R6C2 = {127/235}, no 4,6
[Maybe the next step is a bit heavier than necessary but it gets me back toward my original solving path quickly].
28. R7C2 = 2 -> R7C123 + R8C2 (step 16b) = {2589/2679}
28a. Consider permutations for R7C123 + R8C2
R7C123 + R8C2 = {2589}, locked for N7 => R89C3 = [71] (step 6a) => 10(3) cage at R6C2 = {235} (only combination, cannot be {127} because 1,7 only in R6C2)
or R7C123 + R8C2 = {2679} = [726]9 => R6C1 = 5
28b. -> 5 must be in R6C123, locked for R6 and N4, clean-up: no 2 in R5C4
29. R4C8 = 5 (hidden single in N6), R4C7 = 9, clean-up: no 1 in R4C6 (step 3)
30. R12C7 = R4C6 + 13 (step 13)
30a. Max R12C7 = 15 -> max R4C6 = 2 -> R4C6 = 2, R4C5 = 8 (step 3), clean-up: no 2,8 in R5C1
30b. R4C6 = 2 -> R12C7 = 15 = {78}, locked for C7 and N3, clean-up: no 1 in R9C1 (step 5)
31. R4C6 = 2 -> 11(3) cage at R2C8 = {236} (only remaining combination), 3,6 locked for N3 and D/
31a. R3C89 = {24} (only remaining combination, cannot be {15} which clashes with R1C9), locked for R3 and N3
[Rather than reverting to the last remaining combination for R7C123 + R8C2, this way looks more interesting.]
32. 7 on D/ only in R5C5 + R6C4, locked for N5
32a. 19(3) cage in N5 = {469} (only remaining combination), locked for N5, clean-up: no 1,3 in R5C3
33. R9C1 = 4 (hidden single on D/), R1C9 = 5 (step 5), placed for D/, R8C1 = 6, R9C2 = 3 (step 26), R1C2 = 6 (step 4), R8C9 = 2, R9C3 = 1, R8C3 = 7 (step 6a), R89C7 = [15]
34. 10(3) cage at R6C2 (step 27a) = {235} (only remaining combination), cannot be {127} because 1,7 only in R6C2) -> R6C23 = [53], R6C1 = 7, R7C1 = 5, R6C4 = 1, R5C5 = 7, placed for D\, R9C9 = 6, placed for D\, R4C4 = 3, placed for D\, R7C7 = 4, R8C8 = 8, both placed for D\, R6C6 = 9, placed for D\
and the rest is naked singles without using the diagonals.