Prelims
a) R12C1 = {49/58/67}, no 1,2,3
b) R12C5 = {29/38/47/56}, no 1
c) R12C9 = {89}
d) R23C4 = {16/25/34}, no 7,8,9
e) R23C6 = {19/28/37/46}, no 5
f) R56C6 = {18/27/36/45}, no 9
g) R67C1 = {16/25/34}, no 7,8,9
h) R78C4 = {19/28/37/46}, no 5
i) R78C5 = {29/38/47/56}, no 1
j) R78C8 = {14/23}
k) R89C9 = {19/28/37/46}, no 5
l) 9(3) cage at R3C3 = {126/135/234}, no 7,8,9
m) 21(3) cage at R3C7 = {489/579/678}, no 1,2,3
n) 19(3) cage at R5C4 = {289/379/469/478/568}, no 1
o) 19(3) cage at R6C2 = {289/379/469/478/568}, no 1
p) 24(3) cage at R7C2 = {789}
q) 9(3) cage at R8C1 = {126/135/234}, no 7,8,9
Steps resulting from Prelims
1a. Naked pair {89} in R12C9, locked for C9 and N3, clean-up: no 1,2 in R89C9
1b. Naked triple {789} in 24(3) cage at R7C2, locked for N7
2. Killer pair 3,4 in R78C8 and R89C9, locked for N9
3. 45 rule on R9 2 innies R9C19 = 8 = [17/26/53], clean-up: no 6 in R8C9
4. 45 rule on N5 1 outie R3C5 = 1 innie R6C4 + 3, R3C5 = {56789}, R6C4 = {23456}
5. 19(3) cage at R6C2 = {289/379/469/478/568}
5a. 2,3 of {289/379} must be in R6C4 -> no 2,3 in R6C23
6. 45 rule on R1 3 outies R2C159 = 22 = {589/679}, 9 locked for R2, clean-up: no 9 in R1C1, no 7,8,9 in R1C5, no 1 in R3C6
7. Hidden killer triple 7,8,9 in R12C1 and R345C1 for C1, R12C1 contains one of 7,8,9 -> R345C1 must contain two of 7,8,9
7a. 24(5) cage at R3C1 cannot contain more than two of 7,8,9 -> no 7,8,9 in R5C23
8. 45 rule on R6789 2 innies R6C56 = 12 = {48/57}/[93], no 1,2,6, no 3 in R6C5, clean-up: no 3,7,8 in R5C6
9. 21(3) cage at R3C7 = {489/579/678}
9a. 4 of {489} must be in R3C7 -> no 4 in R4C78
10. 45 rule on R9 3 outies R8C129 = 11 = {137/146/236} (cannot be {245} because 9(3) cage at R8C1 cannot be {25}2), no 5
10a. 4 of {146} must be in R8C9 -> no 4 in R8C12
11. 9(3) cage at R8C1 = {126/135}, 1 locked for N7, clean-up: no 6 in R6C1
12. Max R9C23 = 11 -> min R9C4 = 4
13. 45 rule on C123 3 outies R169C4 = 13
13a. Min R69C4 = 6 -> max R1C4 = 7
14. 45 rule on C1234 2 innies R45C4 = 15 = {69/78}
15. R169C4 = 13 (step 13) = {139/148/157/238/247/256/346}
15a. Killer quad 6,7,8,9 in R169C4, R45C4, R78C4, locked for C4, clean-up: no 1 in R23C4
16. R5C45 cannot be 15 (it would clash with R45C4, CCC) -> no 4 in R6C5, clean-up: no 8 in R6C6 (step 7), no 1 in R5C6
16a. Similarly, and possibly harder to spot, R5C4 + R6C5 cannot be 15 (because of the same CCC) -> no 4 in R5C5
17. 1 in N5 only in R4C56, locked for R4
17a. Min R4C23 = 5 -> max R3C3 = 4
18. 45 rule on C1 3(2+1) outies R58C2 + R5C3 = 8
18a. Min R58C2 = 3 -> max R5C3 = 5
19. 45 rule on N1245 2(1+1) innies R1C6 + R6C1 = 11, no 1 in R1C6 + R6C1, clean-up: no 6 in R7C1
19a. Max R6C1 = 5 -> min R1C6 = 6
20. 1 in N4 only in R5C123, locked for R5 and 24(5) cage at R3C1, no 1 in R3C1
20a. 1 in N6 only in R6C789, locked for 21(5) cage at R6C7, no 1 in R7C79
21. 45 rule on N7 1 outie R9C4 = 1 innie R7C1 + 3, no 4,9 in R9C4
21a. R169C4 = 13 (step 13), min R69C4 = 7 -> max R1C4 = 6
22. 45 rule on N2 3 innies R1C46 + R3C5 = 17 = {179/269/278/368/467} (cannot be {359/458} which clash with R23C4), no 5, clean-up: no 2 in R6C4 (step 4)
22a. 1,2,3,4 only in R1C4 -> R1C4 = {1234}
23. 20(4) cage at R3C5 must contain 1 = {1289/1379/1469/1478/1568} -> R4C56 = {12345}
24. 19(3) cage at R5C4 = {289/379/568}
24a. 2,3 of {289/379} must be in R5C5 -> no 7,9 in R5C5
25. Killer quad 6,7,8,9 in R12C5, R3C5, R56C5 and R78C5, locked for C5
26. 8,9 in N5 only in R45C4 (step 14) = {69/78} and 19(3) cage at R5C4 (step 24) = {289/379/568} -> 19(3) cage can only be [829/928/739/658/685] (using interactions between these overlapping cages which place both of 8,9), no 6 in R5C5, no 7 in R6C5, clean-up: no 5 in R6C6 (step 8), no 4 in R5C6
27. 1,2 in R6 only in R6C1789
27a. 45 rule on R789 4 outies R6C1789 = 14 = {1238/1247/1256}, no 9
28. 45 rule on N5 4 innies R4C456 + R6C4 = 17 and must contain 1 = {1268/1349/1367/1457} (cannot be {1259/1358} which clash with 19(3) cage at R5C4)
28a. 7,8 of {1268/1367} must be in R4C4 -> no 6 in R4C4, clean-up: no 9 in R5C4 (step 14)
29. 45 rule on R6789 3 outies R5C456 = 16 = [682/736/826] (cannot be [835] because 19(3) cage at R5C4 cannot be [838]), no 5, clean-up: no 4 in R6C6, no 8 in R6C5 (step 8)
30. 19(3) cage at R5C4 (step 24) = {289/568} (cannot be {379} which clashes with R6C6), no 3, 7
31. Naked triple {268} in R5C456, locked for R5 and N5, clean-up: no 9 in R3C5 (step 4)
32. R169C4 (step 15) = {148/157/256/346} (cannot be {238/247} which clash with R23C4)
32a. 6 of {256} must be in R9C4 -> no 5 in R9C4, clean-up: no 2 in R7C1 (step 21), no 5 in R6C1, no 6 in R1C6 (step 19)
33. 19(3) cage at R6C2 = {469/478/568} (cannot be {379} which clashes with R6C6), no 3, clean-up: no 6 in R3C5 (step 4)
33a. R6C4 = {45} -> no 4,5 in R6C23
34. Killer pair 4,5 in R23C4 and R6C4, locked for C4, clean-up: no 6 in R78C4
35. 20(4) cage at R3C5 (step 23) = {1379/1478}, no 5
36. 19(3) cage at R6C2 (step 33) = {469/478/568}, R6C56 (step 8) = 12 = [57/93]
36a. 5 in N5 only in R6C45, locked for R6
36b. 5 in N5 only in R6C45 -> 19(3) cage at R6C2 = {568} or R6C56 = [57] -> 19(3) cage at R6C2 = {469/568} (cannot be {478}, locking-out cages), no 7, 6 locked for R6 and N4
37. 7 in N4 only in R45C1, locked for C1, clean-up: no 6 in R12C1
38. R2C159 (step 6) = {589} (only remaining combination, cannot be {679} because 6,7 only in R2C5), locked for R2, clean-up: no 4,5 in R1C5, no 2 in R3C4, no 2 in R3C6
39. 45 rule on R1 3 innies R1C159 = 19 = {289/469/568}, no 3, clean-up: no 8 in R2C5
40. R1C46 + R3C5 (step 22) = {179/278}, no 3, 7 locked for N2, clean-up: no 3 in R23C6
[Ed pointed out that an alternative way to eliminate 3 from R1C4 is
45 rule on N14 2 outies R16C4 = 1 innie R6C1 + 3, no 3 in R1C4 (IOU)]
[Cracked at last!]
41. 3 in N2 only in R23C4 = {34}, locked for C4 and N2 -> R6C4 = 5, R6C5 = 9, R6C6 = 3 (step 8), R5C6 = 6, R5C45 = [82], R4C4 = 7, R123C5 = [658], R9C4 = 6, R3C6 = 9, R2C6 = 1, R1C46 = [27], R4C56 = [14], R7C1 = 3 (step 21), R6C1 = 4, clean-up: no 8 in R1C1, no 9 in R2C1, no 3 in R8C5, no 2 in R8C8, no 4 in R8C9, no 2 in R9C1 (step 3)
42. R12C1 = [58], R12C9 = [89]
43. R9C5 = 3 (hidden single in C5), R9C1 = 1. R89C9 = [37], clean-up: no 2 in R7C8
43a. Naked pair {68} in R6C23, locked for R6 and N4
43b. Naked pair {14} in R78C8, locked for C8 and N9 -> R1C8 = 3, R1C7 = 4 (cage sum)
44. Naked pair {26} in R28C1, locked for C1 -> R45C1 = [97]
45. R3C9 = 1 (hidden single in N3), R6C9 = 2, R6C8 = 7, R6C7 = 1
46. R6C789 = 172] = 10 -> R7C79 = 11 = {56}, locked for R7 and N9
47. 13(3) cage at R2C7 = {256} (only remaining combination) -> R3C8 = 5, R2C78 = {26}, locked for R2 and N3, R3C7 = 7, R5C8 = 9
48. 21(3) cage at R3C7 = {678} (only remaining combination) -> R4C78 = {68}, locked for R4 -> R4C9 = 5
49. 9(3) cage at R3C3 = {234} (only remaining combination) -> R3C3 = 4, R4C23 = {23}, locked for N4
50. 16(3) cage at R2C2 = {367} (only remaining combination) -> R3C2 = 6
and the rest is naked singles.
I was fairly slow to spot some important 45s, particularly the ones in steps 19 and 29; if I’d spotted them earlier my solving path might have been a bit shorter. Also, after working through wellbeback's neat breakthrough and Ed's walkthrough, I realise that I missed the 45 on C6789 which would have given an early placement.
An interesting thing I noticed, after going through my walkthroughs for A218 and A219, is that they both used killer quad 6,7,8,9 to eliminate the {16} combination from a 7(2) cage, in C3 for A218 (step 9) and in C4 for A219 (step 15a).
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Fortunately, there is a neat (though quite long) way through that doesn't use any sneaky contradictions (chains). Took some good digging to find! Satisfying.
Took me a couple of days of thinking to see how you did that.