Prelims
a) R1C78 = {49/58/67}, no 1,2,3
b) R2C12 = {14/23}
c) R3C12 = {18/27/36/45}, no 9
d) R67C2 = {29/38/47/56}, no 1
e) R67C4 = {16/25/34}, no 7,8,9
f) R6C78 = {18/27/36/45}, no 9
g) R89C8 = {18/27/36/45}, no 9
h) R89C9 = {79}
i) 19(3) cage in N1 = {289/379/469/478/568}, no 1
j) 9(3) cage in N2 = {126/135/234}, no 7,8,9
k) 8(3) cage in N3 = {125/134}
l) 9(3) cage at R7C6 = {126/135/234}, no 7,8,9
m) 10(3) cage in N7 = {127/136/145/235}, no 8,9
n) 31(5) cage at R8C5 must contain 9
o) And of course 45(9) cage at R4C1 = {123456789}
Steps resulting from Prelims
1a. Naked pair {79} in R89C9, locked for C9 and N9, clean-up: no 2 in R89C8
1b. 8(3) cage in N3 = {125/134}, 1 locked for C9 and N3
1c. 9 in 31(5) cage at R8C5 only in R8C5 + R9C456, locked for N8
1d. 9 in R7 only in R7C123, locked for N7
1e. 9 in 45(9) cage at R4C1 only in R4567C1, locked for C1
2. 45 rule on R1 4 innies R1C4569 = 13 = {1237/1246/1345}, no 8,9
3. 45 rule on R1 2 innies R1C49 = 1 outie R2C5 + 4, IOU no 4 in R1C9
4. 45 rule on N1 2 innies R23C3 = 12 = {39/48/57}, no 1,2,6
5. 45 rule on N2 2 outies R2C37 = 12 = {39/48/57}, no 1,2,6
6. 45 rule on N9 2 innies R89C7 = 4 = {13}, locked for C7 and N9, clean-up: no 9 in R2C3 (step 5), no 3 in R3C3 (step 4), no 6,8 in R6C6, no 6,8 in R89C8
6a. Naked pair {45} in R89C8, locked for C8 and N9, clean-up: no 8,9 in R1C7
6b. Naked triple {268} in 16(3) cage in N9, locked for R7, clean-up: no 3,5,9 in R6C2, no 1,5 in R6C4
6c. 3 in N3 only in 8(3) cage + R23C8, CPE no 3 in R4C9
7. 9(3) cage at R7C6 = {135/234} (cannot be {126} because 2,6 only in R8C6), no 6, CPE no 3 in R8C45
7a. 2 of {234} must be in R8C6 -> no 4 in R8C6
8. 45 rule on R9 3 outies R8C589 = 21 = {489/579} (cannot be {678} because R8C8 only contains 4,5), no 1,2,6
8a. R8C8 = {45} -> no 4,5 in R8C5
9. 31(5) cage at R8C5 = {16789/34789/35689} (other combinations don’t contain 1 or 3), no 2, 8 locked for N8
9a. R9C7 = {13} -> no 1,3 in R9C456
9b. 2 in N8 only in R8C46, locked for R8
10. 2 in R9 only in 10(3) cage = {127/235}, no 4,6
11. 6 in R9 only in R9C456, locked for N8
11a. 31(5) cage at R8C5 (step 9) = {16789/35689}, no 4
11b. 6 in R8 only in R8C123, locked for 45(9) cage at R4C1, no 6 in R456C1
12. R9C8 = 4 (hidden single in R9), R8C8 = 5
12a. R8C589 (step 8) = {579} (only remaining combination) -> R8C59 = {79}, locked for R8
12b. 4 in R8 only in R8C1234, locked for 45(9) cage at R4C1, no 4 in R4567C1 + R7C5
12c. 8 in R8 only in R8C123, locked for 45(9) cage at R4C1, no 8 in R456C1
13. 9(3) cage at R7C6 (step 7) = {135/234}
13a. 4,5 only in R7C6 ->R7C6 = {45}
13b. R8C67 = {13/23}, 3 locked for R8
[I was slow spotting the next step but since it simplified the lower part of the grid I’ve re-worked this area to include it.]
14. 45 rule on N89 3 innies R7C45 + R8C4 = 9 = {135/234}, no 7, 3 locked for N8
14a. 1,2 of {135/234} must be in R8C4 -> R8C4 = {12}
14b. Naked pair {12} in R8C46, locked for R8 and N8 -> R8C7 = 3, R9C7 = 1, clean-up: no 6 in R6C4, no 7 in 10(3) cage in N7 (step 10)
14c. Naked triple {345} in R7C456, locked for R7 and N8, clean-up: no 6,7,8 in R6C2
14d. 7 in 45(9) cage at R4C1 only in R4567C1, locked for C1, clean-up: no 2 in R3C2
15. 1 in N1 only in R2C12 = {14} or R3C12 = {18} -> R3C12 = {18/36}/[27] (cannot be {45}, locking-out cages), no 4,5
15a. R2C12 = {14} or R3C12 = {18} -> R23C3 = [39/57/75] (cannot be {48}, locking-out cages), no 4,8, clean-up: no 4,8 in R2C7 (step 5)
[I hope I’ve now described these steps correctly.]
15b. Min R3C3 = 5 -> max R4C23 = 9, no 9 in R4C23
16. 19(3) cage in N1 = {289/469/568} (cannot be {379} which clashes with R23C3, cannot be {478} which clashes with R1C78), no 3,7
16a. 7 in N1 only in R2C3 + R3C23, CPE no 7 in R3C45
16b. 3 in R1 only in R1C4569 (step 2) = {1237/1345}, no 6
17. 45 rule on N36 2 innies R26C7 = 1 outie R5C6 + 4
17a. Min R26C7 = 7 -> min R5C6 = 3
18. 14(3) cage in N6 = {158/248/257/347/356} (cannot be {149/167} because 1,7,9 only in R6C8, cannot be {239} which clashes with 8(3) cage in N3), no 9
18a. 8 of {248} must be in R56C9 (R56C9 cannot be {24} which clashes with 8(3) cage in N3) -> no 8 in R6C8
18b. 6 of {356} must be in R56C9 (R56C9 cannot be {35} which clashes with 8(3) cage in N3) -> no 6 in R6C8
19. 45 rule on C89 2 innies R15C8 = 1 outie R7C7 + 5
19a. Max R7C7 = 8 -> max R15C8 = 13, min R1C8 = 6 -> max R5C8 = 7
20. Killer pair 4,5 in 8(3) cage in N3 and 14(3) cage in N6, locked for C9
[I ought to have spotted this killer pair while I was analysing step 18.]
20a. 22(4) cage at R2C8 = {1678/2389} -> R4C8 = {123} (R23C8 + R4C9 cannot contain both of 2,3 which would clash with 8(3) cage in N3 because R23C8 + R4C9 are common peers of the 8(3) cage)
21. 9 in N6 only in R45C7, locked for C7 and 29(5) cage at R3C7, no 9 in R5C6, clean-up: no 3 in R2C3 (step 5), no 9 in R3C3 (step 4)
22. Naked pair {57} in R2C37, locked for R2
23. Naked pair {57} in R23C3, locked for C3 and N1, clean-up: no 2 in R3C1
24. 19(3) cage in N1 (step 16) = {289/469}, 9 locked for R1, clean-up: no 4 in R1C7
25. Killer pair 5,7 in R1C78 and R2C7, locked for N3, clean-up: no 2 in 8(3) cage in N3
26. Naked triple {134} in 8(3) cage in N3, locked for C9 and N3
27. 22(4) cage at R2C8 (step 20a) = {2389} (only remaining combination) -> R4C8 = 3, R23C8 = {289}, R4C9 = {28}
28. 5 in C9 only in R56C9, locked for N6, clean-up: no 4 in R6C6
28a. 14(3) cage in N6 (step 18) = {158/257}, no 6
28b. Naked triple {258} in R456C9, locked for C9 and N6 -> R7C9 = 6, clean-up: no 1,7 in R6C6
29. Naked triple {289} in R237C8, locked for C8, clean-up: no 5 in R1C7
29a. Naked pair {67} in R1C78, locked for R1 and N3 -> R2C8 = 5, R23C3 = [75]
30. 19(3) cage in N1 (step 24) = {289} (only remaining combination), locked for R1 and N1, clean-up: no 3 in R2C12, no 1 in R3C12
30a. Naked pair {14} in R2C12, locked for R2 -> R2C9 = 3, R13C9 = [14]
30b. Naked pair {36} in R3C12, locked for R3
31. 9(3) cage in N2 = {234} (only remaining combination) -> R2C5 = 2, R1C56 = {34}, locked for R1 -> R1C4 = 5, clean-up: no 2 in R6C4
32. Naked pair {34} in R67C4, locked for C4
33. R2C7 = 5 -> R23C6 = 13 = [67]
34. 45 rule on N36 2 outies (no remaining innies) R56C6 = 10 = [82], R6C7 = 7, R1C78 = [67], R3C7 = 2, R56C8 = [61], R6C2 = 4, R7C2 = 7, R67C4 = [34], R78C6 = [51], R7C5 = 3, R7C78 = [82], R8C4 = 2, R9C6 = 9, R8C5 = 7, R89C9 = [97], R4C6 = 4, R45C7 = [94], R2C12 = [41]
34a. R8C3 = 4 (hidden single in R8)
35. 14(3) cage in N6 (step 28a) = {158} (only remaining combination) -> R56C9 = [58]
36. R5C23 = {23} (hidden pair in R5) = 5 -> R5C4 + R67C3 = 22 = [769]
and the rest is naked singles.