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Another challenging puzzle from my Unfinished folder.

That was an ingenious step! I'll agree with "extra-tricky" so I don't go along with Afmob's suggestion about the rating using that step; I'd put it in the same rating range as Afmob's walkthrough and my one.

My solving path was fairly similar to Afmob's but the way I saw some of the key steps was different so I'm belatedly posting my walkthrough.


Here is my walkthrough for A124 V2.

Prelims

a) R1C23 = {49/58/67}, no 1,2,3
b) R1C78 = {18/27/36/45}, no 9
c) R5C12 = {18/27/36/45}, no 9
d) R5C89 = {19/28/37/46}, no 5
e) R9C23 = {13}
f) R9C78 = {29/38/47/56}
g) 10(3) cage at R2C6 = {127/136/145/235}, no 8,9
h) 19(3) cage at R3C4 = {289/379/469/478/568}, no 1
i) 20(3) cage at R3C6 = {389/479/569/578}, no 1,2
j) 14(4) cage in N2 = {1238/1247/1256/1346/2345}, no 9

1. Naked pair {13} in R9C23, locked for R9 and N7, clean-up: no 8 in R9C78

2. 45 rule on R1234 2 innies R34C5 = 6 = {15/24}

3. 45 rule on R6789 2 innies R67C5 = 13 = {49/58/67}, no 1,2,3

4. 3 in 45(9) cage at R3C5 only in R5C34567, locked for R5, clean-up: no 6 in R5C12, no 7 in R5C89

5. 45 rule on C12 2 innies R19C2 = 10 = [73/91], R1C23 = [76/94]

6. 45 rule on C89 2 outies R19C7 = 11 = [29/47/56/65/74], no 1,3,8 in R1C7, no 2 in R9C7, clean-up: no 1,6,8 in R1C8, no 9 in R9C8

7. 45 rule on C9 3 innies R456C9 = 10 = {127/136/145/235}, no 8,9, clean-up: no 1,2 in R5C8

8. 45 rule on C1 2 innies R46C1 = 1 outie R5C2 + 8, IOU no 8 in R46C1

9. 45 rule on R9 2 innies R9C19 = 1 outie R8C5 + 10, max R9C19 = 17 -> max R8C5 = 7

10. 45 rule on N1 3 outies R2C4 + R4C12 = 20
10a. Min R4C12 = 11, no 1
10b. Max R4C12 = 17 -> min R2C4 = 3

11. 45 rule on N7 3 outies R6C12 + R8C4 = 8
11a. Min R6C12 = 4 (cannot be 3 because R78C2 cannot be 16 = {79} which clashes with R1C2) -> max R8C4 = 4
11b. Max R6C12 = 7, no 7,8,9
11c. Max R6C12 = 7 -> min R78C2 = 12, no 2

12. R46C1 = R5C2 + 8 (step 8)
12a. Min R46C1 = 9, no 2 in R4C1

13. 45 rule on C1 3 innies R456C1 = 17
13a. Max R46C1 = 15 -> min R5C1 = 2, clean-up: no 8 in R5C2

14. 45 rule on R5 4 outies R3467C5 = 4 innies R5C1289 must contain the same 4 numbers because of the 45(9) cage (Law of Leftovers)
14a. 8 in R67C5 must be in {58} (step 3) -> 8 in R5C1289 must be in R5C8 because 5 can only be in R5C12, otherwise 8 in 45(9) cage at R3C5 must be in R5 -> no 8 in R5C1, clean-up: no 1 in R5C2
14b. R67C5 = {49/58} would lock 4,5 in R3467C5 and therefore in R5C1289
14c. R67C5 = {67} => R5C1289 = {27}+{46}
14d. -> 4 locked in R5C1289, locked for R5
14e. 4 in 45(9) cage at R3C5 only in R3467C5, locked for C5

15. R1C23 = [76/94], R1C78 = {27/45}/[63] -> combined cage R1C2378 = [76]{45}/[94]{27}/[94][63], 4 locked for R1

16. 14(4) cage in N2 = {1238/1256}, no 7, 1,2 locked for N2, clean-up: no 4,5 in R4C5

17. R456C1 (step 13) = 17 = {179/269/359/467}
17a. 9 of {359} must be in R4C1 -> no 3,5 in R4C1
17b. 2,5 of {269/359} must be in R5C1 -> no 2,5 in R6C1

18. R789C1 = {259/268/457}
18a. R123C1 = {129/138/345} (cannot be {147} which clashes with R1C23, cannot be {156/237/246} which clash with R789C1), no 6,7
18b. 8 in C1 must be in R123C1 = {138} or R789C1 = {268} -> R123C1 cannot be {129} (locking-out cages)
18c. R123C1 = {138/345}, no 2,9, 3 locked for C1 and N1, clean-up: no 5 in R5C1 (step 17), no 4 in R5C2

19. Max R6C12 = 7 (step 11c)
19a. R6C12 cannot be [12] (step 11a), cannot be {23/25} because 2,3,5 only in R6C2, cannot be [42] which clashes with R5C12 -> no 2 in R6C2

20. 10(3) cage at R2C6 = {127/136/145/235}
20a. 7 of {127} must be in R2C6 -> no 7 in R23C7

21. 45 rule on N3 3 outies R2C6 + R4C89 = 11, min R2C6 = 3 -> max R4C89 = 8, no 8,9
21a. Max R4C89 = 8 -> min R23C8 = 11, no 1

[I originally got as far as here when I first tried this puzzle. I’ve done some minor editing of earlier steps, for example in step 18b I didn’t know the term locking-out cages when I did that step.]

[Just spotted some IOUs and a CPE which I ought to have seen earlier.]

22. 45 rule on N1 2 innies R23C2 = 1 outie R2C4 + 4, IOU no 4 in R3C2

23. 45 rule on N3 2 innies R23C8 = 1 outie R2C6 + 8, IOU no 8 in R3C8

24. 45 rule on N9 2 innies R78C8 = 1 outie R8C6 + 2, IOU no 2 in R7C8

25. 9 in N2 only in R2C4 + R3C46, CPE no 9 in R3C3

26. R456C1 (step 17) = {179/269/467}, R789C1 (step 18) = {259/268/457}
26a. 19(4) cage at R6C1 = {1378/1459/1468/1567/3457} (cannot be {1369} which clashes with R19C2)
Cannot be {1378} = [13]{78} which clashes with R789C1 which must be {268} when R6C1 = 1
Cannot be {3457} = [43]{57} which clashes with R789C1 which must be {259} when R6C1 = 4
26b. -> 19(4) cage at R6C1 = {1459/1468/1567}, no 3, 1 locked for R6 and N4
26c. 1 in N5 locked in R4C5 + R5C456, locked for 45(9) cage, no 1 in R5C7
26d. Min R6C12 = 5 -> max R8C4 = 3 (step 11)

[At this stage I made the mistake of thinking that [45] can be eliminated from R5C12 because of clashes with the 19(4) cage at R6C1; I hadn’t looked carefully enough at what happens when {1468} has 4 in R78C2. Therefore I’ve had to re-work from here.]

27. 24(4) cage at R2C2 = {1689/2679/3489/3579/3678/4569} (cannot be {2589/4578} which clash with 16(4) cage at R6C1)
27a. Hidden killer triple 2,5,8 in 24(4) cage at R2C2, R5C2 and 19(4) cage at R6C1 for C2, 24(4) cage contains one of 2,5,8, 19(4) cage contains one of 5,8 -> R5C2 must contain one of 2,5 -> R5C2 = {25}, no 7, clean-up: no 2 in R5C1
27b. 2 in C1 only in R789C1, locked for N7
27c. R789C1 (step 18) = {259/268}, no 4,7
27d. 7 in C1 only in R45C1, locked for N4

28. Hidden killer triple 7,8,9 in R1C2, R234C2 and R78C2 for C2, R1C2 = {79}, R78C2 contains one of 7,8,9 -> R234C2 must contain one of 7,8,9
28a. 24(4) cage at R2C2 (step 27) = {1689/2679/3489/3579/3678/4569}
28b. {1689/2679/3489/3579/3678} must have one of 7,9 in R4C1 because R234C2 only contains one of 7,8,9
28c. 9 of {1689} must be in R4C1 with 6 in R4C2 (otherwise {1689} clashes with R1C23 = [76]), 3 of {3489/3678} must be in R4C2, other combinations don’t contain 8 -> no 8 in R4C2
28d. 8 in N4 only in R456C3, locked for C3
28e. Max R4C12 = 16 -> min R2C4 = 4 (step 10)

29. 24(4) cage at R2C2 (step 27) cannot be {3678}, here’s how
{3678} => R5C2 = 2 (hidden single in C2), R5C1 = 7 clashes with {3678} which must have 7 in R4C1 (step 28b)
-> 24(4) cage at R2C2 (step 27) = {1689/2679/3489/3579/4569}
29a. 24(4) cage at R2C2 = {1689/2679/3579/4569} (cannot be {3489} because 9 must be in R4C1 (step 28b), 3 in R4C2 and {48} clashes with R123C1)

30. 24(4) cage at R2C2 (step 29a) = {1689/2679/3579/4569}
30a. 2 of {2679} cannot be in R23C2, here’s how
R23C2 = {26} => R4C12 = [79] => R1C2 = 7, R1C3 = 6 clashes with R23C2
R23C2 = {27} => 9 must be in R4C1 (step 28b), R4C2 = 6, R5C2 = 5, R45C2 = [65] clash with all combinations for 19(4) cage at R6C1
R23C2 = {29} => 7 must be in R4C1 (step 28b), R4C2 = 6, R5C2 = 5, R45C2 = [65] clash with all combinations for 19(4) cage at R6C1
-> no 2 in R23C2

31. 2 in N1 only in R23C3, locked for C3
31a. 16(3) cage at R2C3 must contain 2 = {259/268}, no 1,4,7
31b. 8 of {268} must be in R2C4 -> no 6 in R2C4

32. R9C3 = 1 (hidden single in C3), R9C2 = 3, R1C2 = 7 (step 5), R1C3 = 6, clean-up: no 2,3 in R1C78, no 4,5,9 in R9C7 (step 6), no 2,6,7 in R9C8
32a. Naked pair {45} in R1C78, locked for R1 and N3
32b. Naked pair {45} in R19C8, locked for C8, clean-up: no 6 in R5C9

33. R1C9 = 9 (hidden single in R1)

34. 2 in R1 only in R1C456, locked for N2
34a. 14(4) cage in N2 (step 16) = {1238} (only remaining combination, cannot be {1256} because 5,6 only in R2C5), locked for N2

35. 7 in C3 only in R78C3 -> 14(3) cage at R7C3 = {257/347}, no 1,9

36. Naked triple {259} in 16(3) cage at R2C3, CPE no 5,9 in R2C12

37. 24(4) cage at R2C2 (step 29a) = {1689/4569} (cannot be {2679} because 2,6,7 only in R4C12), no 2,7, 6 locked for R4 and N4
37a. 4 of {4569} must be in R2C2 -> no 4 in R4C12

38. R5C1 = 7, R5C2 = 2 (hidden singles in C1 and C2), R5C9 = 4 (hidden single in R5), R5C8 = 6

39. R67C5 = {67} (hidden pair in 45(9) cage at R3C5), locked for C5, R34C5 = [42] (hidden pair in 45(9) cage at R3C5)
[Alternatively R3467C5 = R5C1289 (step 14) = {2467} -> R4C5 = 2, R67C5 = {67}, locked for C5, R3C5 = 4
Or naked quint {13589} in R5C34567, locked for 45(9) cage at R3C5 ...]

40. R456C9 (step 7) = 10
40a. R5C9 = 4 -> R46C9 = 6 = [15]

41. Naked pair {14} in R6C12, locked for R6, N4 and 19(4) cage at R6C1

42. R78C3 = {47} (hidden pair in C3), R8C4 = 3 (step 35)

43. 19(4) cage at R2C8 = {1378} (only remaining combination) -> R2C8 = 8, R34C8 = {37}, locked for C8

44. R123C9 = {279/369}
44a. Killer pair 3,7 in R23C9 and R3C8, locked for N3

45. 10(3) cage at R2C6 = {127} (only remaining combination) -> R2C6 = 7, R23C7 = {12}, locked for C7 and N3, clean-up: no 7 in R3C9 (step 44)
45a. Naked pair {36} in R23C9, locked for C9 and N3 -> R34C8 = [73]

46. Naked triple {278} in R789C9, locked for N9 -> R9C7 = 6, R9C8 = 5, R1C78 = [54]

47. R7C7 = 3, R8C7 = 4 (hidden singles in C7), R8C6 = 8 (cage sum), R9C5 = 9, R78C3 = [47], R8C9 = 2

48. 20(4) cage in N8 = {2459} (only remaining combination) -> R8C5 = 5, R9C46 = {24}, locked for R9 and N8 -> R9C1 = 8, R9C9 = 7, R7C9 = 8
48a. Naked triple {167} in R7C456, locked for R7 -> R7C8 = 9, R7C2 = 5, R7C1 = 2, R8C1 = 6 (cage sum)

49. R123C1 (step 18c) = {345} (only remaining combination) -> R1C1 = 3, R23C1 = [45], R4C12 = [96], R23C2 = [18]

50. 20(3) cage at R3C6 = {479} (only remaining combination, cannot be {569} because 6,9 only in R3C6, cannot be {578} because 7,8 only in R4C7) -> R4C6 = 4, R3C6 = 9, R4C7 = 7

and the rest is naked singles.