Prelims
a) R12C1 = {29/38/47/56}, no 1
b) R12C5 = {29/38/47/56}, no 1
c) R12C9 = {39/48/57}, no 1,2,6
d) R23C4 = {16/25/34}, no 7,8,9
e) R23C6 = {17/26/35}, no 4,8,9
f) R3C12 = {16/25/34}, no 7,8,9
g) R34C5 = {18/27/36/45}, no 9
h) R3C89 = {19/28/37/46}, no 5
i) R5C34 = {13}
j) R5C67 = {19/28/37/46}, no 5
k) R5C89 = {69/78}
l) R67C5 = {39/48/57}, no 1,2,6
m) R78C1 = {16/25/34}, no 7,8,9
n) R78C9 = {18/27/36/45}, no 9
Steps resulting from Prelims
1a. Naked pair {13} in R5C34, locked for R5, clean-up: no 7,9 in R5C67
1b. Killer pair 6,8 in R5C67 and R5C89, locked for R5
2. 45 rule on N4 2(1+1) outies R5C4 + R7C2 = 5 = [14/32], R7C2 = {24}
3. 45 rule on R6789 1 outies R5C2 = 1 innie R6C4 + 1, R6C4 = {13468}
4. 45 rule on N2 3 innies R1C46 + R3C5 = 19, no 1, clean-up: no 8 in R4C5
5. 45 rule on N3 2 outies R14C6 = 15 = {69/78}
6. 45 rule on C1234 1 innie R9C4 = 1 outie R5C5 + 5, R5C5 = {24}, R9C4 = {79}
6a. Killer pair {24} in R5C5 and R5C67, locked for R5, clean-up: no 1,3 in R6C4 (step 3)
6b. 5 in R5 only in R5C12, locked for N4
7. 45 rule on N6 1 outie R7C8 = 1 innie R5C7 + 3, no 8 in R5C7, R7C8 = {579}, clean-up: no 2 in R5C6
8. 45 rule on R789 2 outies R6C56 = 2 innies R7C28 + 5
8a. Min R7C28 = 7 -> min R6C56 = 12, no 1,2 in R6C6
[Only just noticed …]
9. 45 rule on C6789 1 innie R9C6 = 2, clean-up: no 6 in R23C6
9a. 20(4) cage at R8C5 = {1289/2369/2378/2567} (cannot be {2468} because R9C4 only contains 7,9, cannot be {2459} which clashes with R5C5 + R9C4 = [49], combo blocker), no 4
[Maybe this block, and the one in step 14a, are better described as IOD blocker. Thanks Ed for pointing this out.]
9b. R9C4 = {79} -> no 7,9 in R89C5
9c. 1 in C5 only in R4C5 -> R34C5 = [81] or in R89C5 -> 20(4) cage = {1289}, 8 locked for C5 (locking cages), clean-up: no 3 in R12C5, no 4 in R67C5
9d. 1 in C5 only in R4C5 -> R34C5 = [81] or in R89C5 -> 20(4) cage = {1289} -> 20(4) cage = {1289/2369/2567} (cannot be {2378}, locking-out cages)
10. 8 in N2 only in R1C46 + R3C5 = 19 (step 4) = {289/478/568}, no 3, clean-up: no 6 in R4C5
11. Hidden killer pair 1,3 in R23C4 and R23C6, R23C6 contains one of 1,3 -> R23C4 must contain one of 1,3 -> R23C4 = {16/34}, no 2,5
11a. Killer pair 1,3 in R23C4 and R5C4, locked for C4
12. 2 in C4 in R14C4
12a. 45 rule on N1 4(3+1) outies R146C4 + R5C5 = 17
12b. R5C5 + R6C4 are both even -> R14C4 is odd and contains 2 -> R14C4 = {25/27/29}, no 4,6,8
13. R1C46 + R3C5 (step 10) = {289/478/568}
13a. 5,7 of {478/568} must be in R1C4 -> no 7 in R1C6, no 5,7 in R3C5, clean-up: no 2,4 in R4C5, no 8 in R4C6 (step 5)
14. R5C2 = R6C4 + 1 (step 3) = [54/76/98]
14a. 45 rule on R5 3 innies R5C124 = 16 must contain 5 = {259/457} = [574/592/952] (cannot be [754] which clashes with R5C2 + R6C4 = [54], combo blocker), no 7 in R5C1
15. 45 rule on R789 3 innies R7C258 = 1 outie R6C6 + 7
15a. R7C2 is even, R7C58 are both odd -> R7C258 must be even -> R6C6 must be odd -> R6C6 = {3579}
[I can see a contradiction move to eliminate 3 from R6C6 but I won’t use it, at least for now.]
16. 45 rule on R1234 3(2+1) outies R5C15 + R6C4 = 15
16a. R5C1 = {59} -> R5C5 + R6C4 = 6,10 = [24/28/46]
16b. 28(6) cage at R1C2 = {123589/124579/124678/134569/234568} (cannot be {123679/134578} which don’t contain any of the permutations for R5C5 + R6C4)
16c. 28(6) cage = {123589/124579/124678/234568} or 28(6) cage = {134569} => R1C4 = 2 (hidden single in C4), CPE no 2 in R1C1, clean-up: no 9 in R2C1
17. 4 in C6 only in R578C6
17a. 45 rule on N69 4 outies R5678C6 = 20 = {1469/1478/3458} (cannot be {3467} which clashes with R14C6)
17b. 7,9 of {1469/1478} must be in R6C6 -> no 7,9 in R78C6
17c. 6 of {1469} must be in R5C6 (R78C6 cannot be {16} which clashes with 20(4) cage at R8C5) -> no 6 in R78C6
[I’d forgotten that this is a Killer-X, since I don’t routinely draw the X on my Excel worksheet; I just have a note at the top of the page and I’m usually working further down so I can’t see the note, which is off the top of the screen.
I ought to have realised from the cage pattern, with four of the cages mainly on the diagonals, that it had been designed as a Killer-X.]
18. 45 rule on D/ 3 innies R1C9 + R5C5 + R6C4 = 1 outie R1C8 + 5
18a. R5C5 + R6C4 = 6,10 (step 16a) -> R1C8 = R1C9 + 1 or R1C9 + 5, no 1,2,3,7 in R1C8, no 9 in R1C9, clean-up: no 3 in R2C9
[I also spotted that R1C89 = [54] is blocked by [24] on D/, but I’ll leave it for now; it’s a bit more chainy than my other combo blockers.]
19. 20(4) cage at R8C5 (step 9d) = {1289/2369/2567}
19a. 45 rule on N7 2 outies R78C4 = 1 innie R7C2 + 11
19b. R7C2 = {24} -> R78C4 = 13,15 = {49/58/67/78} (cannot be {69} which clashes with 20(4) cage
19c. Variable combined cage R78C4 + 20(4) cage = {49}{2567}/{58}{2369}/{67}{1289}/{78}{2369} must contain 9, locked for C4 and N8, clean-up: no 3 in R6C5
19d. R789C4 must contain 9, CPE no 9 in R9C23
20. R146C4 + R5C5 = 17 (step 12a)
20a. R5C5 + R6C4 = 6,10 (step 16a) -> R14C4 = 7,11 but R14C4 (step 12b) = {25/27} -> R14C4 = {25}, locked for C4, R5C5 + R6C4 = 10 = [28/46], no 4 in R6C4, clean-up: no 5 in R5C2 (step 3)
21. R5C1 = 5 (hidden single in R5), clean-up: no 6 in R12C1, no 2 in R3C2, no 2 in R78C1
21a. 22(4) cage at R4C1 = {2569/2578/3568/4567} (cannot be {1579} which clashes with R5C2), no 1
22. 7 in C4 only in R789C4, locked for N8, clean-up: no 5 in R6C5
22a. 7 in C4 only in R789C4, CPE no 7 in R9C23
23. 45 rule on N7 4 innies R7C2 + R8C3 + R9C23 = 20 = {1289/2378/2459/2567/3458} (cannot be {1379/1568} because R7C2 only contains 2,4, cannot be {1469/1478/2369/2468/3467} which clash with R78C1)
23a. 7,9 of {1289/2378/2459/2567} must be in R8C3, 4 of {358} must be in R7C2 -> R8C3 = {35789}, no 1,2,4,6
24. R78C4 (step 19b) = {49/67/78}
24a. 31(5) cage at R7C4 = {16789/34789/45679} (cannot be {35689} which doesn’t contain any of the combinations for R78C4)
24b. 31(5) cage contains both of 7,9, one is in R78C4 -> R8C3 = {79}
25. 18(3) cage at R7C3 = {189/369/378/459/567} (cannot be {279} which clashes with R8C3, cannot be {468} which clashes with R6C4 on D/), no 2
26. R7C2 = 2 (hidden single in N7), R5C4 = 3 (step 2), R5C3 = 1, clean-up: no 4 in R23C4, no 6 in R3C5, no 7 in R8C9
[Cracked!]
27. Naked pair {16} in R23C4, locked for C4 and N2 -> R6C4 = 8, placed for D/, R5C5 = 2 (step 20a), placed for both diagonals, R4C4 = 5, placed for D\, R1C4 = 2, clean-up: no 5,9 in R12C5, no 7 in R23C6, no 4 in R3C5, no 7 in R4C5
28. R34C5 = [81], R1C6 = 9, R6C6 = 7, placed for D\, R4C6 = 6, placed for D/, R5C67 = [46], R7C8 = 9 (step 7), R6C5 = 9, R7C5 = 3, R9C4 = 7 (step 6), R78C4 = [49], R8C3 = 7, clean-up: no 2,4 in R2C1, no 4 in R2C9, no 2 in R3C8, no 1,2 in R3C9, no 9 in R5C9, no 3,4 in R8C1, no 5,6 in R8C9
28a. R5C2 = 9 (hidden single in R5)
29. Naked pair {16} in R78C1, locked for C1 and N7 -> R7C3 = 5, placed for D/, R8C2 + R9C1 = [49] (hidden singles in N7), both placed for D/
29a. Naked pair {38} in R9C23, locked for R9
29b. Naked pair {137} in R1C9 + R2C8 + R3C7, locked for N3
29c. Clean-up: no 8 in R2C9, no 3 in R3C1, no 1,6 in R3C2, no 9 in R3C9
30. Naked triple {346} in R6C123, locked for R6 and N4
30a. Naked triple {278} in R4C123, locked for R4
31. Naked pair {46} in R3C89, locked for R3 and N3 -> R3C1 = 2, R3C2 = 5, R23C4 = [61], R23C6 = [53], R3C3 = 9, R3C7 = 7, R12C9 = [39], R2C8 = 1, R1C8 = 8 (cage sum), R1C1 = 4, R1C3 = 6, R2C3 = 8 (cage sum)
and the rest is naked singles, without using the diagonals.
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The first one was essential to my solution. Andrew's style of walk-through lets you see how he actually found the puzzle rather than my highly optimised WTs that leave out all the searching.