This is a Centre Dot Killer and a Killer-X, R258C258 must contain the numbers 1 to 9
Prelims
a) R1C34 = {19/28/37/46}, no 5
b) R1C67 = {18/27/36/45}, no 9
c) R56C1 = {14/23}
d) R56C9 = {39/48/57}, no 1,2,6
e) R89C1 = {15/24}
f) R89C5 = {29/38/47/56}, no 1
g) R89C9 = {16/25/34}, no 7,8,9
h) 21(3) cage at R1C8 = {489/579/678}, no 1,2,3
i) 19(3) cage at R9C6 = {289/379/469/478/568}, no 1
j) 28(4) cage at R2C3 = {4789/5689}, no 1,2,3
k) 13(4) cage at R2C7 = {1237/1246/1345}, no 8,9
l) 37(8) cage at R3C5 = {12345679}, no 8
m) 40(8) cage at R4C5 = {12346789}, no 5
1. R89C1 = {15} (only remaining combination, cannot be {24} which clashes with R56C1), locked for C1 and N7, clean-up: no 4 in R56C1
1a. Naked pair {23} in R56C1, locked for C1 and N4
2. 5 in N5 only in R4C456 + R5C46, locked for 37(8) cage at R3C5, no 5 in R3C5 + R4C37
2a. 8 in N5 only in R5C5 + R6C456, locked for 40(8) cage at R5C5, no 8 in R67C37
3. 45 rule on R9 3 innies R9C159 = 12 = {129/138/147/156/345} (cannot be {237/246} because R9C1 only contains 1,5)
3a. 9 of {129} must be in R9C5 -> no 2 in R9C5, clean-up: no 9 in R8C5
4. 35(6) cage at R2C1 = {146789} (only remaining combination), no 5, 1 locked for N4
5. R6C2 = 5 (hidden single in N4), clean-up: no 7 in R5C9
6. 8 in N4 only in R4C12 + R5C23, locked for 35(6) cage at R2C1, no 8 in R23C1
7. 5 in C3 only in R23C3, locked for 28(4) cage at R2C3, no 5 in R3C4
7a. 28(4) cage contains 5 = {5689}, no 4,7
8. 45 rule on N1 3 innies R1C3 + R23C1 = 1 outie R3C4 + 4
8a. Min R1C3 + R23C1 = 11 -> no 6 in R3C4
8b. 28(4) cage at R2C3 = {5689}, 6 locked for N1
8c. R3C4 = {89} -> R1C3 + R23C1 = 12,13 = {147/247} (only possible combinations) -> R1C3 = {12}, R23C1 = {47}, locked for C1, N1 and 35(6) cage at R2C1, no 4,7 in R4C12 + R5C23, clean-up: R1C4 = {89}
8d. R1C23 + R2C2 = {123} (hidden triple in N1), 3 locked for C2
8e.. R46C3 = {47} (hidden pair in N4), locked for C3
8f. Naked pair {89} in R1C14, locked for R1, clean-up: no 1 in R1C67
8g. Naked pair {89} in R13C4, locked for C4 and N2
9. 21(3) cage at R1C8 = {579/678} (cannot be {489} because 8,9 only in R2C8), no 4, 7 locked for R1 and N3, clean-up: no 2 in R1C67
9a. 8,9 only in R2C8 -> R2C8 = {89}
9b. Killer pair 5,6 in R1C67 and 21(3) cage, locked for R1
10. 8,9 in N3 only in R2C8 + R23C9, CPE no 8,9 in R45C8
11. Caged X-Wing for 4,7 in 37(8) cage at R2C5 and 40(8) cage at R5C5 in R46C3 + N5, no other 4,7 in these cages -> no 4,7 in R3C5 + R467C7
12. 45 rule on R6789 3 outies R5C159 = 14 = {239/248/257/347/356} (cannot be {149/158/167} because R5C1 only contains 2,3), no 1
12a. 3 of {239} must be in R5C19 (cannot be [239] because R6C19 cannot be [33]), 3 of {347/356} must be in R5C1 -> no 3 in R5C5
13. 18(4) cage at R1C5 = {1467/2367/2457} (cannot be {3456} which clashes with R1C6), 7 locked for R2 and N2 -> R23C1 = [47]
13a. 4 of {1467} only in R1C5 -> no 1 in R1C5
13b. 1 in R1 only in R1C23, locked for N1
14. 13(4) cage at R2C7 = {1246/1345}, CPE no 1,4 in R3C9
15. 8 on D/ only in R2C8 + R5C5 + R8C2, locked for Centre Dot group, no 8 in R5C2 + R8C58, clean-up: no 3 in R9C5
16. 45 rule on R9 3 outies R8C159 = 12 = {147/156/345} (cannot be {237/246} because R8C1 only contains 1,5), no 2, clean-up: no 9 in R9C5, no 5 in R9C9
16a. R9C159 (step 3) = {138/147/156/345}, no 2, clean-up: no 5 in R8C9
16b. 14(3) cage at R9C2 = {167/239/248/257/347/356} (cannot be {149/158} which clash with R9C159)
16c. 1,5 of {167/356} must be in R9C4 -> no 6 in R9C4
17. 27(6) cage at R2C9 = {123489/123579/124578} (cannot be {123678/124569/134568/234567} which clash with R56C9 because all cells of 27(6) cage “see” R56C9), no 6
[Alternatively, with hindsight, combined cage 27(6) + R56C9 = 39(8) cage, no 6. Similarly there’s another 8-cell combined cage on the other side of the cage pattern, but the early naked pair in R56C1 makes it’s use unnecessary.]
17a. Combined cage 27(6) + R56C9 contains both of 4,7 in N6, locked for N6
17b. 5 in N6 only in R4C89 + R5C789, CPE no 5 in R23C9
18. 45 rule on N5 7 outies R3C5 + R467C37 = 32 form split 7-cell cage {1234679} (because only cages in N5 are 37(8) cage at R2C5 and 40(8) cage at R5C5), R46C3 = {47} -> R3C5 + R46C7 + R7C37 form split 5-cell cage {12369}, CPE no 1,2,3,6 in R3C7 using D/
19. 33(6) cage at R6C2 contains 5 = {145689/235689/245679/345678} (cannot be {135789} which clashes with R8C159)
19a. 4,7 in N7 only in R789C2 -> 33(6) cage must contain at least one of 4,7 = {145689/245679/345678} (cannot be {235689} which doesn’t contain 4 or 7)
20. 45 rule on N7 2 remaining outies R89C4 = 1 innie R7C3 + 3
20a. R7C3 = {2369} -> R89C4 = 5,6,9,12
20b. 14(3) cage at R9C2 = {239/248/257/347/356} (cannot be {167} = [761] which clashes with 33(6) cage at R6C2 because R89C4 cannot be [71] = 8), no 1 in R9C4
20c. 1 in R9 only in R9C19 -> R9C159 (step 16a) = {138/147/156}
20d. 7 of {147} must be in R9C5 -> no 4 in R9C5, clean-up: no 7 in R8C5
20e. R8C159 (step 16) = {156/345}, 5 locked for R8
21. Consider combinations for 21(3) cage at R1C8 (step 9) = {579/678}
21(3) cage = {579} => R3C7 = 4
or 21(3) cage = {678}, 6 locked for R1 => R13C7 = {45}
-> 4 must be in R13C7, locked for C7 and N3, 5 must be in R13C7 + R1C89, locked for N3
21a. 13(4) cage at R2C7 (step 14) = {1246/1345}
21b. 1,3 of {1345} must be in R2C7 + R3C8 -> no 3 in R3C6
22. R89C4 = 5,6,9,12 (step 20a) = [14]/{23}/[15]/{24}/{27}/[63/75]
22a. 33(6) cage at R6C2 (step 19a) = {145689/245679/345678}
22b. Consider combinations for R89C5 = [38/47]/{56}
R89C5 = [38] => 33(6) cage = {145689/245679}, 9 locked for N7 => R7C3 = {236}
or R89C5 = [47]/{56} => R89C4 cannot be [75] = 12 => no 9 in R7C3
-> R7C3 = {236}
[Looking at this forcing chain a different way …]
22c. All combinations for 33(6) cage contain 6
22d. Consider placements for R7C3 = {236}
R7C3 = {23} => 6 in 33(6) cage must be in R7C12 + R8C23, locked for N7
or R7C3 = 6 => R8C4 = 6 => R89C4 = 9 = [63] (cannot be {27}
-> 6 must be in R7C12 + R8C23 or in R7C3, CPE no 6 in R9C3
and R89C4 = [14]/{23}/[15]/{24}/[63], no 7 in R89C4
23. R3C5 + R46C7 + R7C37 form split 5-cell cage (step 18) = {12369}, 9 locked for C7
[I can take the forcing chains in step 22 a bit further …]
24. 33(6) cage at R6C2 (step 19a) = {145689/245679/345678}, R89C4 (step 22d) = [14]/{23}/[15]/{24}/[63], 14(3) cage at R9C2 (step 20b) = {239/248/257/347/356}
24a. Consider combinations for R89C5 = [38/47]/{56}
R89C5 = [38]/{56} => R89C4 cannot be [63]
or R89C5 = [47] => R9C159 (step 16a) = {147} = [174] => R8C9 = 3 => 33(6) cage = {145689/245679}, 9 locked for N7 => 14(3) cage at R9C2 = {356} => R9C4 = 5 = R89C4 = [15]
-> R89C4 = [14]/{23}/[15]/{24}, no 6
24b. 33(6) cage = {145689/245679/345678}, 6 locked for N7
25. 7 in 40(8) cage at R5C5 only in R5C5 + R6C4567
25a. R5C159 (step 12) = {239/248/257/347} (cannot be {365} = [365], blocking-out cages), no 6
25b. 8 in 40(8) cage only in R5C5 + R6C67
25b. R5C159 (step 12) = {239/248/257} (cannot be {347} = [374], blocking-out cages), 2 locked for R5
25c. Consider combinations for R5C159
R5C159 = {239} = [293/329] => R6C19 = [39/23]
or R5C159 = {248/257} => R6C1 = 3
-> 3 must be in R6C19, locked for R6
26. 3 in N5 only in R4C456 + R5C46, locked for 37(8) cage at R3C5, no 3 in R3C5 + R4C7
26a. 3 in R3 only in R3C89, locked for N3, clean-up: no 6 in R1C6
26b. 6 in R1 only in R1C789, locked for N3
27. 13(4) cage at R2C7 (step 14) = {1246/1345}
27a. 6 of {1246} must be in R3C6, 4,5 of {1345} must be in R3C67 -> R3C6 = {456}
27b. 13(4) cage = {1246/1345}, 1 locked for N3
28. 45 rule on N12 3 remaining innies R1C6 + R3C56 = 10 = {136/145/235}
28a. 1,2 only in R3C5 -> R3C5 = {12}
29. R3C5 + R46C7 + R7C37 (step 18) form split 5-cell cage {12369}, 6 locked for C7, clean-up: no 3 in R1C6
30. R1C6 + R3C56 (step 28) = {145} (only remaining combination) -> R3C5 = 1, R13C6 = {45}, locked for C6 and N2
30a. Naked pair {45} in R13C7, locked for C7 and N3
30b. 13(4) cage at R2C7 (step 14) = {1345} (only remaining combination) -> R3C8 = 3, R2C7 = 1
30c. R2C3 = 5 (hidden single in R2)
31. R1C89 = {67} = 13 -> R2C8 = 8, placed for D/
31a. R2C9 = 9 (hidden single in R2), R3C9 = 2, clean-up: no 3 in R56C9
32. 1,8 in N5 only in R6C456, locked for R6, clean-up: no 4 in R5C9
32a. Naked pair {47} in R6C39, locked for R6
33. 3 in 40(8) cage at R5C5 only in R7C37, locked for R7
33a. 4,7 in 40(8) cage only in R5C5 + R7C3 -> R5C5 = {47}
34. R5C1 = 2 (hidden single in R5), R6C1 = 3
35. R9C159 (step 20c) = {138/156} (cannot be {147} which clashes with R5C5 using both diagonals), no 4,7, clean-up: no 4 in R8C5, no 3 in R8C9
36. 19(3) cage at R9C6 = {289/379/469/478} (cannot be {568} which clashes with R9C159), no 5
36a. 5 in R9 only in R9C45, locked for N8, clean-up: no 6 in R9C5
37. 18(3) cage at R7C4 = {189/279} (cannot be {468} which clashes with R89C5), no 4,6, 9 locked for R7 and N8
38. 19(3) cage at R9C6 (step 36) = {289/379/478} (cannot be {469} because 4,9 only in R9C8), no 6
38a. 4,9 of {379/478} must be in R9C8 -> no 7 in R9C8
39. R9C159 (step 35) = {156} (only remaining combination, cannot be {138} which clashes with 19(3) cage at R9C6) -> R9C5 = 5, R9C1 = 1, placed for D/, R9C9 = 6, placed for D\, R8C1 = 5, R8C5 = 6, R8C9 = 1, R1C9 = 7, placed for D/, R1C8 = 6, R5C5 = 4, placed for both diagonals, R46C3 = [47], R6C9 = 4, R5C9 = 8, R7C9 = 5, R4C9 = 3, R5C7 = 7
40. Naked pair {89} in R1C1 + R3C3, locked for N1 and D\ -> R3C2 = 6
41. Naked pair {23} in R2C2 + R7C7, locked for D\ -> R6C6 = 1, R8C8 = 7, placed for Centre Dot group, R4C4 = 5
42. Naked pair {23} in R2C25, locked for R2 and Centre Dot group -> R8C2 = 9, placed for D/, R5C2 = 1, R4C2 = 8, R45C8 = [15]
43. Naked pair {23} in R23C5, locked for C5
44. R7C1 = 6 (hidden single in N7), R4C1 = 9, R1C1 = 8, R1C4 = 9, R1C3 = 1
45. Naked pair {23} in R7C37, locked for R7 and 40(8) cage at R5C5 -> R6C4 = 6, R4C6 = 2, placed for D/ -> R7C3 = 3, R7C7 = 2, placed for D\ -> R2C2 = 3
and the rest is naked singles.
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