Prelims
a) R12C2 = {29/38/47/56}, no 1
b) R1C89 = {49/58/67}, no 1,2,3
c) R23C9 = {18/27/36/45}, no 9
d) R4C12 = {15/24}
e) R4C89 = {17/26/35}, no 4,8,9
f) R5C12 = {69/78}
g) R5C89 = {69/78}
h) 19(3) cage at R3C4 = {289/379/469/478/568}, no 1
i) 14(4) cage at R3C6 = {1238/1247/1256/1346/2345}, no 9
j) 14(4) cage at R6C8 = {1238/1247/1256/1346/2345}, no 9
k) 32(5) cage at R4C5 = {26789/35789/45689}, no 1
1. Naked quad {6789} in R5C1289, locked for R5
2. 32(5) cage at R4C5 = {35789/45689} (cannot be {26789} because R5C56 only contain 2,3,4,5), no 2
2a. R5C56 = {35/45} -> no 3,4,5 in R4C5 + R6C67
2b. R5C56 = {35/45}, 5 locked for R5 and N5
2c. 32(5) cage = {35789/45689}, CPE no 8,9 in R6C45
3. 45 rule on R1234 1 innie R4C5 = 1 outie R5C7 + 7, R4C5 = {89}, R5C7 = {12}
3a. 3,4 in R5 only in R5C3456, CPE no 3,4 in R6C45
4. 45 rule on N1 3 innies R123C3 = 9 = {126/135/234}, no 7,8,9
4a. Max R123C3 + R5C3 = 13(4), must contain 1, locked for C3
4b. 45 rule on N1 2 outies R12C4 = 12 = {39/48/57}, no 1,2,6
5. 45 rule on N3 2 outies R12C6 = 9 = {18/27/36/45}, no 9
5a. 45 rule on N3 2 innies R12C7 = 8 = {17/26/35}, no 4,8,9
6. 45 rule on C12 3 outies R8C3 + R9C34 = 1 innie R6C2 + 21
6a. Max R8C3 + R9C34 = 24 -> max R6C2 = 3
6b. Min R8C3 + R9C34 = 22, no 1,2,3,4
6c. Min R8C3 + R9C34 = 22 -> max R8C1 + R9C12 = 10, no 8,9 in R8C1 + R9C12
[Alternatively 45 rule on C12 4(1+3) innies R6C2 + R8C1 + R9C12 = 11 -> max R8C1 + R9C12 = 10, no 8,9 in R8C1 + R9C12]
7. R4C12 = {15/24}, R4C89 = {17/26/35} -> combined cage R4C1289 = {15}{26}/{24}{17}/{24}{35}, 2 locked for R4
7a. 19(3) cage at R3C4 = {379/469/478/568} (cannot be {289} which clashes with R4C5), no 2
8. 45 rule on N7 2(1+1) outies R6C1 + R9C4 = 1 innie R7C3 + 9
8a. Min R7C3 = 2 -> min R6C1 + R9C4 = 11, no 1 in R6C1
9. 45 rule on N9 3(2+1) outies R6C89 + R9C6 = 11
9a. Min R6C89 = 3 -> max R9C6 = 8
10. 45 rule on R123 2 innies R3C46 = 9, no 9 in R3C4
10a. Min R3C3 = 3 -> max R3C6 = 6
11. 30(7) cage at R5C3 = {1234569/1234578}, 5 locked for N8
11a. 30(7) cage at R6C2 = {1234569/1234578}, 5 locked for C3
11b. Caged X-Wing for 5 in 32(5) cage at R4C5 and 30(7) cage at R5C3, no other 5 in C56, clean-up: no 4 in R12C6 (step 5), no 4 in R3C4 (step 10)
[With hindsight 5 in C4 only in R123C4, locked for N2 … is simpler.]
12. R123C3 (step 4) = {126/234}, 2 locked for C3 and N1, clean-up: no 9 in R12C2
12a. R12C2 = {38/47/56}, R123C3 = {126/234} -> combined cage R12C2 + R123C3 = {38}{126}/{47}{126}/{56}{234}, 6 locked for N1
12b. Min R7C3 = 3 -> min R6C1 + R9C4 = 12 (step 8), no 2 in R6C1
13. 2 in N5 only in R5C4 + R6C45, locked for 30(7) cage at R5C3, no 2 in R7C56 + R8C6
14. 32(6) cage at R8C1 cannot contain all four of 6,7,8,9 (because 1,2,6,7,8,9 total 33)
14a. R8C3 + R9C34 = {6789} -> no 6,7 in R8C1 + R9C12
15. 4 in N3 only in R1C89 = {49} or 15(3) cage at R2C8 or R23C9 = {45} -> 15(3) cage cannot be {159} (blocking cages)
15a. 9 in N3 only in R1C89 = {49} or 15(3) cage = {249} (locking cages), 4 locked for N3, clean-up: no 5 in R23C9
15b. 15(3) cage = {168/249/258/267/357} (cannot be {348} because of step 15a, 15(3) cage must contain both or neither of 4,9)
16. 9 in N2 only in R12C4 (step 4b) = {39} or in 15(3) cage at R1C5 = {249} -> 15(3) cage = {168/249/267} (cannot be {348} (blocking cages), no 3
16a. 9 in N2 only in R12C4 = {39} or in 15(3) cage = {249} -> R12C4 = {39/57} (cannot be {48}, locking-out cages), no 4,8 in R12C4
16b. 15(3) cage = {168/249/267}, R12C6 (step 5) = {18/27/36} -> combined cage 15(3) + R12C6 = {168}{27}/{249/267}with rest of R12C6, 2 locked for N2, clean-up: no 7 in R3C4 (step 10)
17. 19(3) cage at R3C4 (step 7a) = {379/469/478/568}
17a. 3 of {379} must be in R3C4 -> no 3 in R4C34
18. 45 rule on C6789 2 innies R78C6 = 2 outies R45C5 + 1
18a. Min R45C5 = 11 -> min R78C6 = 12, no 1 in R78C6
19. 1,2 in 30(7) cage at R5C3 must be in R5C34 + R6C45 (R5C34 + R6C45 cannot contain both of 3,4 which would clash with R5C56, R5C34 + R6C45 cannot contain both of 6,7 because 30(7) cage only contains one of 6,7) -> no 1 in R7C5
19a. One of 1,2 in 30(7) cage must be in R5C34 (R5C34 cannot be {34} which clashes with R5C56) and one in 1,2 must be in R6C45 (because R6C45 cannot contain both of 6,7) -> R6C45 must contain one of 6,7 -> no 6,7 in R7C56 + R8C6 (because 30(7) cage only contains one of 6,7)
19b. Killer pair 6,7 in R6C45 and 32(5) cage at R4C5, locked for R6
19c. Min R78C6 = 12 (step 18a) -> R78C6 must contain one of 8,9 (cannot be both because 30(7) cage only contains one of 8,9) -> no 8,9 in R7C5
20. Variable hidden killer triple 1,2,3 in R4C89, R5C7 and R6C89 for N6, R4C89 contains one of 1,2,3, R5C7 = {12} -> R6C89 cannot contain more than one of 1,2,3 (may contain none of them)
20a. R6C89 + R9C6 = 11 (step 9)
20b. Min R6C89 = 5 (because cannot contain more than one of 1,2,3) -> max R9C6 = 6
21. Hidden killer pair 6,7 in R4C3 and R5C12 for N4, R5C12 must contain one of 6,7 -> R4C3 = {67}
21a. 19(3) cage at R3C4 (step 7a) = {379/478/568} (cannot be {469} because 4,9 only in R4C4)
21b. R4C3 = {67} -> no 6,7 in R34C4, clean-up: no 3 in R3C6 (step 10)
21c. 9 in R4 only in R4C45, locked for N5
22. Hidden killer pair 8,9 in R5C12 and R6C13 for N4, R5C12 must contain one of 8,9 -> R6C13 must contain one of 8,9
22a. Killer pair 8,9 in R6C13 and R6C67, locked for R6
23. 9 in N6 only in R5C89 = {69} or in R6C7 = 9 -> no 6 in R6C7 (locking-out cages)
23a. 6 in R6 only in R6C456, locked for N5
24. 3,4 in R5 only in R5C3456
24a. Whichever of 3,4 is in R5C56 must also be in R7C56 + R8C6 and in R123C4
24b. R123C4 doesn’t contain 4 -> no 4 in R5C56
24c. Naked pair {35} in R5C56, locked for R5 and N5
24d. 4 in R5 only in R5C34, locked for 30(7) cage at R5C3, no 4 in R7C56 and R8C6
24e. 3,5 in 30(7) cage at R5C3 only in R7C56 + R8C6, locked for N8
24d. Killer pair 3,5 in R5C6 and R78C6, locked for C6, clean-up: no 6 in R12C6 (step 5)
24e. R12C7 (step 5a) = {26/35} (cannot be {17} which clashes with R12C6)
24f. Killer pair 1,4 in R4C12 and R5C3, locked for N4
25. R5C56 = {35} -> 32(5) cage at R5C5 (step 2c) = {35789}, 7 locked for R6
26. 6 in R6 only in 30(7) cage at R5C3 = {1234569} (only remaining combination), 9 locked for N8
27. Min R8C3 + R9C34 = 22 (step 6c), must contain 9 in R89C3, locked for C3 and N7
28. 30(7) cage at R6C2 = {1234578} (only remaining combination), no 6
28a. 6 in N8 only in R9C4, locked for R9
[I’d seen this 45 a long time ago, but it’s much more powerful now …]
29. 45 rule on C6789 5(4+1) innies R5678C6 + R6C7 = 33
29a. R578C6 = {359} = 17 -> R6C67 = 16 = [79], clean-up: no 2 in R12C6 (step 5), no 6 in R5C89
30. Naked pair {18} in R12C6, locked for C6 and N2 -> R34C6 = [64], R9C6 = 2, R9C4 = 6 (hidden single in N8), R3C4 = 3 (step 10), clean-up: no 9 in R12C4 (step 4b), no 3,6 in R2C9, no 2 in R4C12
30a. Naked pair {15} in R4C12, locked for R4 and N4 -> R5C3 = 4
30b. R3C4 = 3 -> R4C34 = 16 = [79], R4C5 = 8, R5C7 = 1 (step 3), R4C7 = 3 (cage sum), clean-up: no 5 in R12C7 (step 5a)
30c. R6C89 = {45} (hidden pair in N6), locked for R6 and 14(4) cage at R6C8, no 4,5 in R78C9
31. 30(7) cage at R6C2 (step 28) = {1234578} -> R6C2 = 2, R67C3 = [35]
32. R6C1 = 8, 6,7 in N7 only in R7C12 + R8C2 -> R7C12 + R8C2 = 14 = {167}, locked for N7
32a. Naked pair {34} in R9C12, locked for R9 and N7 -> R8C1 = 2
33. Naked pair {57} in R12C4, locked for C4 and N2
33a. Naked pair {26} in R12C7, locked for C7 and N3, clean-up: no 7 in R1C89, no 7 in R23C9
33b. Naked pair {18} in R12C9, locked for C9 and N3 -> R5C89 = [87], clean-up: no 5 in R1C89
33c. Naked pair {49} in R1C89, locked for R1 and N3 -> R1C5 = 2, clean-up: no 7 in R2C2
33d. Naked pair {57} in R3C78, locked for R3 and N3 -> R2C8 = 3, clean-up: no 8 in R1C2
34. R1C6 = 8 (hidden single in R1), R2C6 = 1, R23C9 = [81], R123C3 = [162], clean-up: no 3,5 in R1C2, no 5 in R2C2
34a. R12C2 = [74]
35. R6C89 = {45} = 9 -> R78C9 = 5 = [23]
36. R7C56 = [39], R8C6 = 5
37. 17(3) cage at R7C7 = {467} (only remaining combination) -> R7C8 = 6, R78C7 = {47}, locked for C7 and N9 -> R3C7 = 5
and the rest is naked singles.