No repeated candidates on the diagonals R1C1-R9C9, R1C3-R7C9, R1C5-R5C9, R3C1-R9C7, R5C1-R9C5, R1C5-R5C1, R1C7-R7C1, R1C9-R9C1, R3C9-R9C3 and R5C9-R9C5.
Prelims (not including any cage interactions)
a) R1C12 = {59/68}
b) 5(2) cage at R1C3 = {14/23}
c) R1C89 = {18/27/36/45}, no 9
d) 14(2) cage at R2C2 = {59/68}
e) 14(2) cage at R2C8 = {59/68}
f) R34C4 = {39/48/57}, no 1,2,6
g) R34C7 = {19/28/37/46}, no 5
h) 3(2) cage at R4C8 = {12}
i) 8(2) cage at R5C1 = {17/26/35}, no 4,8,9
j) R67C3 = {18/27/36/45}, no 9
m) R67C6 = {19/28/37/46}, no 5
n) 9(2) cage at R7C1 = {18/27/36/45}, no 9
o) 16(2) cage at R7C7 = {79}
p) 9(2) cage at R8C6 = {18/27/36/45}, no 9
q) R9C12 = {18/27/36/45}, no 9
r) R9C89 = {18/27/36/45}, no 9
Steps resulting from Prelims
1a. Naked pair {12} in 3(2) cage at R4C8, locked for N6 and R1C5-R5C9 diagonal, clean-up: no 8,9 in R34C7
1b. Naked pair {79} in 16(2) cage at R7C7, locked for N9 and R1C1–R9C9 diagonal, CPE no 7,9 in R6C8 + R8C6 using R5C9-R9C5 diagonal, clean-up: no 5 in R1C2, no 5 in 14(2) cage at R2C2, no 3,5 in R3C4, no 1,3 in R7C6, no 2 in 9(2) cage at R8C6, no 2 in R9C89
2. Naked pair {68} in 14(2) cage at R2C2, locked for N1 and R1C1–R9C9 diagonal, CPE no 6,8 in R4C2 using R1C5-R5C1 diagonal, clean-up: no 4 in R3C4, no 2,4 in R7C6, no 1,3 in R9C8
2a. R1C1 = 5, placed for R1C1–R9C9 diagonal, R1C2 = 9, clean-up: no 4 in R1C89, no 7 in R3C4, no 3 in R6C2, no 4 in R8C2, no 4 in R9C2, no 4 in R9C8
3. 2 on R1C1–R9C9 diagonal only in R5C5 + R6C6, locked for N5
3a. Generalised X-Wing for 2 in 3(2) cage at R4C8 and R5C5 + R6C6, no other 2 in R5 and R3C9-R9C3 diagonal, clean-up: no 6 in R6C2
3b. 2 in R5 only in R5C59, CPE no 2 in R1C9 using R1C9-R9C1 diagonal, no 2 in R9C5 using R5C9-R9C5 diagonal, clean-up: no 7 in R1C8
4. 9 in R9 only in R9C3456, CPE no 9 in R7C5 + R8C4 using R3C9-R9C3 diagonal
4a. 9 in N7 only in R8C13 + R9C3, CPE no 9 in R8C5
5. 17(3) cage at R5C7 = {269/278/359/368/458/467} (cannot be {179} because 7,9 only in R5C7), no 1
6. 18(3) cage at R3C1 = {279/378/459/567} (cannot be {189/369/468} because 6,8,9 only in R5C3), no 1
6a. 6,8,9 only in R5C3 -> R5C3 = {689}
6b. 5 of {459} must be in R4C2 -> no 4 in R4C2
7. R5C456 = {179/269/278/359/368/458/467}
7a. R5C5 = {1234} -> no 1,3,4 in R5C46
8. Consider placements for R5C9
R5C9 = 1 => R4C8 = 2, R6C6 = 1 (hidden single on R1C1-R9C9 diagonal)
R5C9 = 2 => R4C8 = 1, R6C6 = 2 (hidden single on R1C1-R9C9 diagonal)
-> R6C6 = {12}, clean-up: no 6,7 in R7C6
8a. Naked pair {12} in R4C8 + R6C6, locked for R3C9-R9C3 diagonal, CPE no 1 in R4C56
8b. Min R4C56 = {35} (cannot be {34} which clashes with R4C4) = 8 -> max R3C56 = 8, no 8,9 in R3C56
8c. Min R8C45 + R9C3 = 8 -> max R8C3 = 8
9. Consider placements for R5C9
R5C9 = 1 => no 1 in R1C9, no 1 in R9C5 using R5C1-R9C5 diagonal
R5C9 = 2 => R4C8 = 1, R6C6 = 2 => 1 on R1C1–R9C9 diagonal only in R5C5 + R9C9, CPE no 1 in R1C9 using R1C9-R5C5 diagonal, no 1 in R9C5
-> no 1 in R1C9 + R9C5, clean-up: no 8 in R1C8
10. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => no 9 in R5C7
R9C3 = 9, placed for R3C9-R9C3 diagonal => no 9 in R5C7
-> no 9 in R5C7
11. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => R6C6 = 2 => R5C9 = 2 (hidden single in R5) => no 2 in R7C9
R9C3 = 9 => R8C8 = 9 (hidden single in R8) => R7C7 = 7 => no 7 in R5C7
11a. 17(3) cage at R5C7 (step 5) = {368/458/467} (cannot be {278} because no 2 in R7C9 or no 7 in R5C7), no 2
11b. Killer pair 3,4 in 5(2) cage at R1C3 and 17(3) cage at R5C7, locked for R1C3-R7C9 diagonal
12. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => no 7 in R8C345
R9C3 = 9 => R8C345 = 7 = {124}
-> no 1,2,4 in R8C1, no 7 in R8C345
13. Consider placements for 1 in R9
1 in R9C12 = {18} => no 8 in R9C7 => no 1 in R8C6
1 in R9C467, CPE no 1 in R8C6
1 in R9C9 => R9C8 = 8 => no 8 in R9C7 => no 1 in R8C6
-> no 1 in R8C6, clean-up: no 8 in R9C7
14. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => no 7 in R8C2
R9C3 = 9, placed for R3C9-R9C3 diagonal => R8C8 = 9 (hidden single in R8) => 14(2) cage at R2C8 = {68}, locked for N3 => R1C9 = 7, placed for R1C9-R9C1 diagonal -> no 7 in R8C2
-> no 7 in R8C2, clean-up: no 2 in R7C1
14a. R8C18 = {79} (hidden pair in R8)
15. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => R6C6 = 2 => no 2 in R6C3 => no 7 in R7C3
R9C3 = 9 => R8C1 = 7 => no 7 in R7C3
-> no 7 in R7C3, clean-up: no 2 in R6C3
16. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => no 8 in R7C3
R9C3 = 9 => R35C3 = {68}, locked for C3
-> no 8 in R7C3, clean-up: no 1 in R6C3
17. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => no 9 in R7C4
R9C3 = 9, placed for R3C9-R9C3 diagonal => R8C8 = 9 (hidden single in R8) => 14(2) cage at R2C8 = {68} => naked pair {68} in R3C39 => R3C4 = 9 => no 9 in R7C4
-> no 9 in R7C4
18. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9, placed for R5C9-R9C5 diagonal => no 9 in R9C5
R9C3 = 9 => no 9 in R9C5
-> no 9 in R9C5
19. 9 in N7 only in R8C1 + R9C3
45 rule on N7 5 innies R7C23 + R8C13 + R9C3 = 27 = {12789/13689/14589/23679/24579/34569} (cannot be {14679/23589} which clash with the 9(2) cages in N7)
19a. Consider placements for 9 in N7
R8C1 = 9 => R7C23 + R89C3 = 18 must consist of R7C23 = 9, R89C3 = 9 (other combinations cause CCC clash with R67C3) => R8C45 = 7
R9C3 = 9 => R8C345 = 7 = {124}, locked for R8 => R7C8 = 2 (hidden single in R7), R8C1 = 7 => no 2 in 9(2) cages at R7C1 and R9C1 => R8C3 = 2 (hidden single in N7) => R89C3 = [29] => R8C45 = [41]
19b. -> R89C3 = 9 or [29], no 8 in R8C3
-> R8C45 = 7 or R8C45 = [41] -> R8C45 = [34/41/43/52/61], no 8 in R8C4, no 5,6,8 in R8C5
19c. R8C1 = 9 => R7C23 = 9
R9C3 = 9 => R8C13 = [72] = 9 => R7C23 = 9
-> R7C23 = 9 = {36/45}/[72/81], no 1,2 in R7C2
-> R7C23 = R67C3 -> R7C2 = R6C3
[With hindsight step 19 could have been done after step 14a.]
20. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => no 8 in R7C2
R9C3 = 9 => R8C3 = 2 (step 19a), R35C3 = {68}, locked for C3 => R67C3 = {45} => R7C23 = {45} (step 19c)
-> no 8 in R7C2, clean-up: no 1 in R7C3 (step 19c), no 8 in R6C3
21. R9C3 cannot be 9, here’s how
R9C3 = 9 => R8C345 = [241] (step 19a), R7C23 = {45} (step 20) => 4 in R9 only in R9C79
Now consider placements for 4 in R9C79
R9C7 = 4 => R8C6 = 5 => cannot place 5 in R9
R9C9 = 4 => R8C8 = 5, no 4 in R9C7 => no 5 in R8C6 => cannot place 5 in N8
-> no 9 in R9C3
22. R8C1 = 9 (hidden single in N7), R8C8 = 7, R7C7 = 9, R7C6 = 8, R6C6 = 2, placed for R3C9-R9C3 diagonal, R4C8 = 1, R5C9 = 2, clean-up: no 8 in R1C9, no 6 in R5C1, no 1 in R8C2, no 1 in R9C7
22a. 1 on R1C1-R9C9 diagonal only in R5C5 + R9C9, CPE no 1 in R9C1 using R1C9-R9C1 diagonal, clean-up: no 8 in R9C2
22b. 8 in R1 only in R1C457, CPE no 8 in R2C5
23. 18(3) cage at R3C1 (step 6) = {279/378/459} (cannot be {567} which clashes with 9(2) cage at R8C6 using R3C1-R9C7 diagonal), no 6
23a. {378} must be [738] (cannot be [378] which clashes with 8(3) cage at R5C1), no 3 in R3C1
24. R8C1 = 9 -> R89C3 (step 19a) = 9 = [27]/{36/45} (cannot be [18] because cannot place 1 in N9), no 1,8
24a. 8 in N7 only in R8C2 + R9C1, locked for R1C9-R9C1 diagonal, clean-up: no 6 in R3C9
24b. 8 in R9 only in R9C18 -> R9C12 = [81] or R9C89 = [81] (locking cages), 1 locked for R9
24c. 1 in N8 only in R7C4 + R8C5, CPE no 1 in R6C5
25. 1 in N7 only in R7C1 + R9C2, CPE no 1 in R6C2 using R1C7-R7C1 diagonal, clean-up: no 7 in R5C1
26. 1 in C3 only in R12C3, locked for N1, CPE no 1 in R2C4 using R1C3-R7C9 diagonal, clean-up: no 4 in R1C3
27. 2 on R1C9-R9C1 diagonal only in R7C3 + R8C2 + R9C1, locked for N7, clean-up: no 7 in R9C1, no 7 in R9C3 (step 24)
28. 16(4) cage at R8C3 = {1456/2356}, CPE no 5,6 in R7C3 + R8C2 using R5C1-R9C5 diagonal, clean-up: no 3,4 in R6C3, no 3,4 in R7C1, no 3,4 in R7C2 (step 19c)
28a. 1,2 only in R8C5 -> R8C5 = {12}
28b. R89C3 = 9 (step 24) -> R8C45 = 7 = [52/61], no 3,4
29. 2 on R1C9-R9C1 diagonal only in R7C3 + R8C2 + R9C1 -> R67C3 = [72] or 9(2) cage at R7C1 = [72] or 9(2) cage at R9C1 = [27] (locking cages), CPE no 7 in R6C2 using R1C7-R7C1 diagonal
29a. R6C2 = 5, placed for R1C7-R7C1 diagonal, R5C1 = 3, placed for R1C5-R5C1 diagonal, clean-up: no 2 in R1C3, no 4 in R7C3, no 4 in R9C1, no 6 in R9C2
[I overlooked that R5C1 was also placed for the R5C1-R9C5 diagonal. Since it’s so close to the finish, I haven’t re-worked the remaining steps.]
30. R89C3 = {45} (hidden pair), locked for C3 and 16(4) cage at R8C3 -> R8C4 = 6, placed for R3C9-R9C3 diagonal, R8C5 = 1 (step 28b), R5C5 = 4, placed for R1C1-R9C9 and R1C9-R9C1 diagonals, R4C4 = 3, placed for R1C1-R9C9 and R1C7-R7C1 diagonals, R3C4 = 9, R9C9 = 1, R9C8 = 8, clean-up: no 5 in R2C8, no 3 in R9C7
31. 9(2) cage at R7C1 = [18] (hidden pair in N7), 1 placed for R1C7-R7C1 diagonal, R2C2 = 6, R3C3 = 8, R2C8 = 9, placed for R1C9-R9C1 diagonal, R3C9 = 5, placed for R3C9-R9C3 diagonal, R89C3 = [54], R7C2 = 7, R9C2 = 3, R7C3 = 2, R6C3 = 7, R9C1 = 6, placed for R1C9-R9C1 diagonal, R45C3 = [69], R5C2 = 1, R4C2 = 2, placed for R1C5-R5C1 diagonal, R3C12 = [74], R2C1 = 2, R2C4 = 4, placed for R1C3-R7C9 diagonal, R1C3 = 1, R2C3 = 3, R3C7 = 3
32. R6C4 = 1, R7C45 = [53], R6C5 = 9 (cage sum)
and the rest is naked singles without using diagonals.