Prelims
a) R2C3 = 1
b) R8C7 = 1
c) R12C1 = {49/58/67}, no 1,2,3
d) R12C2 = {29/38/47/56}, no 1
e) R45C4 = {89}
f) R4C67 = {49/58/67}, no 1,2,3
g) R56C6 = {15/24}
h) R6C34 = {29/38/47/56}, no 1
i) R89C8 = {19/28/37/46}, no 5
j) R89C9 = {16/25/34}, no 7,8,9
k) 11(3) cage at R1C5 = {128/137/146/236/245}, no 9
l) 6(3) cage at R4C1 = {123}
m) 21(3) cage at R8C6 = {489/579/678}, no 1,2,3
n) 18(5) cage at R4C8 = {12348/12357/12456}, no 9
1. R2C3 = 1
2. R8C7 = 1, clean-up: no 9 in R89C8, no 6 in R89C9
3. 45 rule on N4 1 innie R6C3 = 9, R6C4 = 2
4. 45 rule on N6 1 innie R4C7 = 9, R4C6 = 4, R45C4 = [89]
4a. 9 in N9 only in R7C89, locked for R7
5. Naked pair {15} in R56C6, locked for C6 and N5
6. Naked triple {367} in 16(3) cage at R4C5, locked for C5
7. Naked triple {123} in 6(3) cage at R4C1, locked for R4 and N4
8. 5 in R4 only in R4C89, locked for N6
8a. 18(5) cage at R4C8 = {12357/12456} (cannot be {12348} which doesn’t contain 5), no 8, 1 locked for R5 and N6
8b. 1,2,3,4 of 18(5) cage must be in R5C789, no 6,7 in R5C789
8c. 8 in N6 only in R6C789, locked for R6
9. R56C6 = [51]
10. 45 rule on N3 2 innies R12C7 = 7 = {25/34}, no 6,7,8
11. 45 rule on N7 2 innies R89C3 = 9 = {27/36/45}, no 8
12. 45 rule on R6 3 innies R6C125 = 15 = {357/456}
12a. 3 of {357} must be in R6C5 -> no 7 in R6C5
12b. R6C5 = {36} -> no 6 in R6C12
12c. 6 in N4 only in R5C123, locked for R5
13. 11(3) cage at R1C5 = {128/137/146/236/245}
13a. 1 of {128} must be in R1C5 -> no 8 in R1C5
13b. 6,7 of {137/236} must be in R1C6 -> no 3 in R1C6
14. Hidden killer triple 1,2,3 in R3C1, R4C1 and 16(3) cage at R7C1 for C1, R4C1 = {123}, 16(3) cage cannot contain more than one of 1,2,3 -> R3C1 = {23}, 16(3) cage must contain one of 1,2,3
15. 45 rule on N1 2 outies R3C45 = 1 innie R1C3 + 8, min R1C3 = 2 -> min R3C45 = 10, no 1,2 in R3C5
16. 45 rule on C6789 3 outies R127C5 = 11 = {128/245}, no 9, 2 locked for C5
17. 45 rule on C1234 3 outies R389C5 = 18 = {189/459}
17a. 1 of {189} must be in R9C5 -> no 8 in R9C5
18. 2 in N8 only in R7C56, locked for R7
18a. 45 rule on N9 4 outies R7C56 + R89C6 = 23 contains 2 = {2489/2579/2678}, no 1,3
19. 3 in C6 only in R23C6, locked for N2 and 19(3) cage at R2C5, no 3 in R2C7, clean-up: no 4 in R1C7 (step 10)
19a. 19(4) cage at R2C5 = {2359/2368/3457}
19b. 3,6,7,9 only in R23C6 -> R23C6 = {36/37/39}, no 2,8
20. 14(3) cage at R1C3 = {158/167/257/347/356} (cannot be {248} because 2,8 only in R1C3)
20a. 3 of {347/356} must be in R1C3 -> no 4,5 in R1C3
21. 11(3) cage at R1C5 = {128/236/245} (cannot be {146} because 1,4 only in R1C5, cannot be {137} which clashes with 14(3) cage at R1C3), no 7, 2 locked for R1, clean-up: no 9 in R2C2
21a. 4 of {245} must be in R1C5 -> no 5 in R1C5
22. 45 rule on N1 4 outies R12C4 + R3C45 = 22 = {1579/1678/4567} (cannot be {1489} because 8,9 only in R3C5), 7 locked for C4 and N2
23. 19(4) cage at R2C5 (step 19a) = {2359/2368}, no 4, 2 locked for R2, clean-up: no 9 in R1C2, no 3 in R1C7 (step 10)
24. Naked pair {25} in R12C7, locked for C7 and N3
24a. R5C89 = {12} (hidden pair in N6)
25. 11(3) cage at R1C5 (step 21) = {128/245}, no 6
25a. 1,4 only in R1C5 -> R1C5 = {14}
26. R127C5 (step 16) = {128/245}
26a. R1C5 = {14} -> no 4 in R7C5
27. 45 rule on N9 2 outies R7C56 (which must contain 2) = 1 innie R9C7 + 2, no 5 in R7C5, no 4 in R9C7
28. 21(3) cage at R8C6 = {678} (only remaining combination), no 9
28a. Naked quad {2678} in R7C56 + R89C6, locked for N8
29. 9 in C6 only in R23C6, locked for N2
29a. 19(4) cage at R2C5 (step 23) = {2359} (only remaining combination), no 6,8
29b. R2C57 = {25}, locked for R2, clean-up: no 8 in R1C1, no 6 in R1C2
30. 14(3) cage at R1C3 (step 20) = {167/347/356} (cannot be {158} because 1,5 only in R1C4), no 8
30a. 5 of {356} must be in R1C4 -> no 6 in R1C4
31. R3C45 = R1C3 + 8 (step 15)
31a. R1C3 = {367} -> R3C45 = 11,14,15 = [65/74/68/78], R3C4 = {67}
32. 1 in N2 only in R1C45, locked for R1
33. 2,5 in R3 only in 28(5) cage at R3C1 = {25678} (only remaining combination, cannot be {24589} because R3C4 only contains 6,7), no 3,4,9 -> R3C1 = 2
33a. R3C6789 = {1349}, no 6,7,8, 4 locked for N3
33b. Naked pair {34} in R35C7, locked for C7
33c. Killer triple 6,7,8 in R79C7 and R89C8, locked for N9
34. 9 in N1 only in R12C1 = {49}, locked for C1 and N1, clean-up: no 7 in R12C2
35. R389C5 (step 17) = {189/459}
35a. R3C5 = {58} -> no 5 in R89C5
36. 5 in N2 only in R23C5, locked for N2
36a. 14(3) cage at R1C3 (step 30) = {167/347}
37. 19(4) cage at R7C4 = {1369/1459/2359/3457} (cannot be {1279/1567/2467} because 2,6,7 only in R8C3)
37a. 2,6,7 of {1369/2359/3457} -> no 3 in R8C3, clean-up: no 6 in R9C3 (step 11)
38. 20(4) cage at R1C8 = {1379/1469/1478/3467}
38a. 1 of {1379/1469} must be in R3C8 -> no 9 in R3C8
38b. 6,7,8,9 only in R12C8 -> R12C8 = {6789}, no 3,4
39. 16(3) cage at R7C1 = {178/358/367}
39a. Killer pair 5,7 in R6C1 and 16(3) cage, locked for C1
40. 9 in N7 only in 20(4) cage at R7C2 = {1289/1469/2369/2459} (cannot be {1379} which clashes with 16(3) cage at R7C1), no 7
40a. 8 of {1289} must be in R7C3 -> no 8 in R789C2
40b. 2,9 of {2369/2459} must be in R89C2 -> no 3,5 in R89C2
41. 29(5) cage at R7C5 = {23789/24689/25679}
41a. 20(4) cage at R7C2 (step 40) = {1289/1469/2459} (cannot be {2369} = {36}{29} which clashes with 29(5) cage), no 3
42. 45 rule on R7 4 innies R7C1234 = 16 must contain 1 = {1348/1456} (cannot be {1357} because 20(4) cage at R7C2 only contains one of 1,5), no 7, 4 locked for R7
43. 29(5) cage at R7C5 (step 41) = {23789/25679}
43a. Killer pair 3,5 in 29(5) cage and R89C9, locked for N9, clean-up: no 7 in R89C8
43b. 7 in N9 only in R79C7, locked for C7
44. 20(4) cage at R1C8 (step 38) = {1379/1478/3467} (cannot be {1469} which clashes with R5C8 + R89C8, killer ALS block), 7 locked for C8 and N3
[With hindsight this step can be seen
44. Hidden killer triple 1,2,6 in 20(4) cage at R1C8, R5C8 and R89C8 for C8, R5C8 = {12}, R89C8 contains one of 2,6 -> 20(4) cage contains one of 1,6 in C8
44a. 20(4) cage at R1C8 (step 38) = {1379/1478/3467} (cannot be {1469} which contains both of 1,6 in C8), 7 locked for C8 and N3]
45. 12(3) cage at R9C3 = {129/147/345} (cannot be {237} because no 2,7 in R9C3)
45a. 7 of {147} must be in R9C3, 4 of {345} must be in R9C5 -> no 4 in R9C3, clean-up: no 5 in R8C3 (step 11)
46. 19(4) cage at R7C4 (step 37) = {1369/1459/2359/3457}
46a. 4 of {1459} must be in R8C3, 4 of {3457} must be in R8C5 -> no 4 in R78C4
47. 4 in R7 only in R7C23, locked for N7, clean-up: no 5 in R9C3 (step 11)
47a. 20(4) cage at R7C2 (step 41a) contains 4 = {1469/2459}, no 8
48. Killer pair 2,3 in R4C3 and R89C3, locked for C3
49. 14(3) cage at R1C3 (step 36a) = {167} (only remaining combination) -> R1C4 = 1, R1C5 = 4, R8C5 = 9, R9C5 = 1, R9C4 = 4 (hidden single in C4), R9C3 = 7 (step 45), R8C3 = 2 (step 11)
50. Naked pair {68} in R9C67, locked for R9 and 21(3) cage at R8C6 -> R8C6 = 7, R9C8 = 2, R8C8 = 8
51. R8C9 = 4 (hidden single in N9), R9C9 = 3
and the rest is naked singles.