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Congratulations Afmob for finding that powerful IOU for A142 V2! It's a bit like the ones I found for A123.


Here is my walkthrough for A142 V2.

Prelims

a) R12C1 = {15/24}
b) R1C23 = {18/27/36/45}, no 9
c) R1C78 = {79}
d) R12C9 = {16/25/34}, no 7,8,9
e) R56C2 = {59/68}
f) R56C3 = {12}
g) R56C7 = {15/24}
h) R56C8 = {49/58/67}, no 1,2,3
i) 12(4) cage at R3C7 = {1236/1245}, no 7,8,9
j) 14(4) cage in N8 = {1238/1247/1256/1346/2345}, no 9

Steps resulting from Prelims
1a. R1C23 = {18/27/36} (cannot be {45} which clashes with R12C1), no 4,5
1b. Naked pair {79} in R1C78, locked for R1 and N3, clean-up: no 2 in R1C23
1c. Naked pair {12} in R56C3, locked for C3 and N4, clean-up: no 8 in R1C2
1d. 12(4) cage at R3C7 = {1236/1245}, CPE no 1,2 in R2C8

2. 8 in N3 locked in R2C78, locked for R2 and 33(6) cage at R2C6, no 8 in R234C6 + R4C7

3. 45 rule on N7 2 innies R7C13 = 13 = {49/58/67}, no 1,2,3

4. 45 rule on N9 2 innies R7C79 = 5 = {14/23}
4a. Max R7C9 = 4 -> min R456C9 = 20, no 1,2 in R456C9

5. 45 rule on R789 3 innies R7C159 = 18
5a. Max R7C19 = 13 -> min R7C5 = 5
5b. Max R7C59 = 13 -> min R7C1 = 5, clean-up: no 9 in R7C3 (step 3)

6. 45 rule on R1234 4(1+3) innies R3C5 + R4C159 = 27
6a. Max R4C159 = 24 -> min R3C5 = 3

7. 45 rule on R1 3 outies R2C159 = 9 = {126/135/234}, no 7,9

8. 24(4) cage at R4C1 = {3489/3579/3678} (cannot be {4569/4578} which clash with R12C1), 3 locked for C1 and N4

9. 45 rule on N3 2 innies R2C78 = 1 outie R4C8 + 10
9a. Min R2C78 = 11, no 1,2 in R2C7
9b. Max R2C78 = 14 -> max R4C8 = 4

10. 45 rule on C1 1 outie R9C2 = 1 innie R3C1 + 2, no 8,9 in R3C1, no 1,2,5 in R9C2

11. 45 rule on C9 1 outie R9C8 = 1 innie R3C9 + 4, no 6 in R3C9, no 1,2,3,4 in R9C8

12. 45 rule on N6 2 innies R4C78 = 1 outie R7C9 + 2
12a. Max R7C9 = 4 -> max R4C78 = 6, no 6,7,9 in R4C7
12b. R4C78 = {13/15/23/24} (cannot be {12/14} which clash with R56C7) = 4,5,6 -> R7C9 = {234}, clean-up: no 4 in R7C7 (step 4)

13. Hidden killer pair 7,9 in R1C7 and R89C7 for C7 -> R89C7 must contain one of 7,9
13a. 18(3) cage in N9 cannot be {279} which clashes with R89C7, no 2 in R89C9

14. 45 rule on N47 3(2+1) innies R4C23 + R7C3 = 17, min R47C3 = 9 -> max R4C2 = 8

15. 45 rule on C789 4(3+1) innies R2C78 + R47C7 = 17 and must contain 8 in R2C78 = 8{135/234} (cannot be {68}{12} which clashes with R56C7), no 6
[At the time I overlooked that there’s also [8441]; this permutation is eliminated by step 16a.]
15a. R2C78 = {38/48/58} -> R4C8 = {123} (step 9)

16. 45 rule on N69 3(2+1) innies R4C78 + R7C3 = 7 -> no 2 in R4C7 (because no 4 in R4C8 or R7C3 and R4C8 + R7C3 = {23} are common peers of R4C7)
16a. R4C78 + R7C7 cannot be [412/421] (which clash with R56C7) -> no 4 in R4C7

17. 45 rule on C1 3 innies R389C1 = 15 = {159/168/249/267} (cannot be {258/456} which clash with R12C1)
17a. 4 of {249} must be in R3C1 (cannot be 2{49} because 17(3) cage in N7 cannot be {49}4), no 4 in R89C1

18. 45 rule on N1 2 innies R2C23 = 1 outie R4C2 + 6, IOU no 6 in R2C3

19. 16(4) cage in N2 = {1258/1348/1456/2356}
19a. 4 of {1348/1456} must be in R1C456 (R1C456 cannot be {138/156} which clash with R1C23), no 4 in R2C5

20. 24(4) cage at R4C9 = {2589/2679/3489/3678/4578} (cannot be {3579/4569} which clash with R56C8)
20a. 45 rule on C9 3 innies R389C9 = 14 = {149/158/167/239/257/347} (cannot be {248/356} which clash with 24(4) cage at R4C9)
20b. 2,4 of {239/347} must be in R3C9 (cannot be 3{47} because 18(3) cage in N9 cannot be {47}7), no 3 in R3C9, clean-up: no 7 in R9C8 (step 11)

21. Min R89C3 = 7 -> max R78C2 = 8, no 8,9 in R78C2

22. 45 rule on N9 2 outies R78C6 = 1 innie R7C9 + 8, IOU no 8 in R8C6
22a. 13(3) cage at R7C6 = {139/148/157/238/247/256/346}
22b. 8 of {238} must be in R7C6, 2 of {247/256} must be in R7C7 -> no 2 in R7C6

23. 45 rule on N7 2 outies R78C4 = 1 innie R7C1 + 5, IOU no 5 in R8C4

24. 24(4) cage at R3C1 = {1689/2589/2679/3489/3579/3678/4569/4578}
24a. 6 of {1689} must be in R4C2 (R3C123 cannot be {169} which clashes with R1C23) -> 1 of {1689} must be in R3C1 -> no 1 in R3C2

25. 45 rule on C12 4 innies R1278C2 = 1 outie R3C3 + 5
25a. Min R1278C2 = 10 -> min R3C3 = 5
25b. Max R1278C2 = 14, no 9

[This is how far I got when I first tried this puzzle. A few earlier steps have been edited.]

26. 7 in N1 only in R2C23 + R3C123, CPE no 7 in R3C4
26a. 9 in N1 only in R2C3 + R3C23, CPE no 9 in R3C4

27. 45 rule on N8 2 outies R7C37 = 1 innie R7C5
27a. Min R7C37 = 6 (cannot be 5 which clashes with R7C79, CCC) -> min R7C5 = 6

28. R7C79 (step 4) = [14/23/32] cannot be [14], here’s how
[14] => R56C7 = {24}, R4C78 = 6 (step 12) = [51] => 1 in N3 only in R12C9 = {16}, locked for N3 => 12(4) cage at R3C7 = {1245} has no candidate in R3C7
28a. -> R7C79 = {23}, locked for R7 and N9

29. R7C9 = {23} -> R4C78 = 4,5 (step 12) = {13/23}, 3 locked for R4 and N6, CPE no 3 in R2C8 + R3C7

30. Killer triple 1,2,3 in R4C7, R56C7 and R7C7, locked for C7

31. R2C78 = {48/58} = 12,13 -> R4C8 = {23} (step 9)

32. 12(4) cage at R3C8 = {1236/1245}, 1 locked for R3 and N3, clean-up: no 6 in R12C9, no 3 in R9C2 (step 10)
32a. 6 in N3 only in R3C78 -> 12(4) cage = {1236} (only remaining combination) -> R3C7 = 6, clean-up: no 8 in R9C2 (step 10)
32b. Killer pair 2,3 in R12C9 and R7C9, locked for C9 -> R3C9 = 1, R9C8 = 5 (step 11), clean-up: no 8 in R56C8
32c. Killer pair 7,9 in R1C8 and R56C8, locked for C8

33. R9C8 = 5 -> R89C9 = 13 = {49/67}, no 8

34. R389C1 (step 17) = {159/249/267} (cannot be {168} because no 1,6,8 in R3C1), no 8
34a. 5 of {159} must be in R3C1 -> no 5 in R8C1

35. 8 in C1 only in 24(4) cage at R4C1 = {3489/3678} (step 8), no 5, clean-up: no 8 in R7C3 (step 3)
35a. 5 in C1 only in R123C1, locked for N1

36. 18(3) cage at R7C3 cannot be {189} because no 1,8,9 in R7C3 -> no 1 in R78C4

37. 17(3) cage in N7 = {179/269/467}
37a. 15(4) cage in N7 = {1239/1347/2346} (cannot be {1248/1257/1356} which clash with 17(3) cage), no 5,8

38. R7C13 = [85] (hidden pair in N7)
38a. R7C3 = 5 -> R78C4 = 13 = {49/67}, no 2,3,8

39. R7C37 = R7C5 (step 27), R7C3 = 5 -> R7C5 = R7C7 + 5 -> R7C5 = 7, R7C7 = 2, R7C9 = 3, clean-up: no 4 in R12C9, no 4 in R56C7, no 6 in R78C4 (step 38a)
[45 rule on N89 2 remaining innies R7C59 = 10 = [73] ..., as in Afmob’s walkthrough, is a bit more direct.]

40. Naked pair {15} in R56C7, locked for C7 and N6 -> R4C7 = 3, R34C8 = [32]

41. Naked pair {49} in R78C4, locked for C4 and N8
41a. R7C4 = 9 (hidden single in R7), R8C4 = 4, clean-up: no 9 in R9C9 (step 33)

42. R7C7 = 2 -> R78C6 = 11 = [65]

43. Naked pair {48} in R2C78, locked for R2 and 33(6) cage at R2C6, no 4 in R34C6, clean-up: no 2 in R1C1
43a. R2C78 = {48} = 12, R4C7 = 3 -> R234C6 = 18 = {279} (only remaining combination), locked for C6, 2 also locked for N2

44. 16(4) cage in N2 (step 19) = {1348/1456}, 1,4 locked for N2, 4 also locked for R1, clean-up: no 2 in R2C1

45. Naked pair {15} in R12C1, locked for C1 and N1, clean-up: no 8 in R1C3

46. Naked pair {36} in R1C23, locked for R1 and N1
46a. 16(4) cage in N2 (step 44) = {1348/1456}
46b. 3,6 only in R2C5 -> R2C5 = {36}

47. Naked triple {279} in R2C236, locked for R2 -> R2C9 = 5, R1C9 = 2, R1C12 = [51]

48. Naked triple {148} in R1C456, locked for N2, R2C5 = 3 (step 44), R2C4 = 6, R3C4 = 5, R3C5 = 9

49. R2C3 = 9 (hidden single in R2)
49a. R4C6 = 9 (hidden single in C6)

50. 9 in N7 only in 17(3) cage (step 37) = {269} (only remaining combination), locked for N7, 2 also locked for C1

51. 8 in N1 only in 24(4) cage at R3C1 (step 24) = {4578} (only remaining combination) -> R4C2 = 5, R3C123 = {478}, locked for R3 and N1 -> R2C2 = 2, R23C6 = [72], clean-up: no 9 in R56C2

52. Naked pair {68} in R56C2, locked for C2 and N4 -> R1C23 = [36], R9C2 = 9

53. R2C234 + R3C4 = [2965] = 22 -> R4C34 = 8 = [71], R4C1 = 4, R89C3 = [34], R78C2 = [17], R7C8 = 4, R2C78 = [48], clean-up: no 9 in R56C8, no 9 in R8C9, no 6 in R9C9 (both step 38a)

54. R89C9 = [67], R4C9 = 8, R4C5 = 6

55. R5C4 = 7 (hidden single in C4)

56. R34C5 + R5C4 = [967] = 22 -> R5C56 = 6 = [24]

and the rest is naked singles.

Having come back to A142 V2 a few months ago, I had a go at the harder variants earlier this month. I finished the V3 just over a week ago but decided to try the others before going through the posted WTs.

Afmob’s step 2e was a neat and very powerful hidden killer, I took several steps to get the same result. I also liked his step 9c, another neat hidden killer which looked difficult to spot.


Here is my walkthrough for A142 V3.

Prelims

a) R12C1 = {16/25/34}, no 7,8,9
b) R1C23 = {29/38/47/56}, no 1
c) R1C78 = {17/26/35}, no 4,8,9
d) R12C9 = {17/26/35}, no 4,8,9
e) R56C2 = {19/28/37/46}, no 5
f) R56C3 = {18/27/36/45}, no 9
g) R56C7 = {18/27/36/45}, no 9
h) R56C8 = {18/27/36/45}, no 9
i) 6(3) cage in N9 = {123}
j) 10(4) cage in R6C4 = {1234}
k) 11(3) cage at R7C6 = {128/137/146/236/245}, no 9
l) 26(4) cage in N8 = {2789/3689/4589/4679/5678}, no 1
m) 28(4) cage in N9 = {4789/5689}, no 1,2,3

Steps resulting from Prelims
1a. 6(3) cage in N9 = {123}, locked for N9
1b. 10(4) cage in R6C4 = {1234}, CPE no 1,2,3,4 in R45C5
1c. 28(4) cage in N9 = {4789/5689}, 8,9 locked for N9

2. Killer triple 1,2,3 in R12C9 and R89C9, locked for C9

3. 45 rule on N7 2 innies R7C13 = 11 = {29/38/47/56}, no 1
3a. 45 rule on N9 2 innies R7C79 = 11 = {47/56}

4. 45 rule on N8 2 outies R7C37 = 1 innie R7C5 + 10
4a. R7C37 cannot be 11 (which clashes with R7C13, CCC) -> min R7C37 = 12, no 2,3,4 in R7C3, clean-up: no 7,8,9 in R7C1 (step 3)
4b. Min R7C37 = 12 -> min R7C5 = 2
4c. 1 in 10(4) cage in R6C4 only in R6C456, locked for R6 and N5, clean-up: no 9 in R5C2, no 8 in R5C3, no 8 in R5C7, no 8 in R5C8
[I noticed later that 45 rule on N78 2 innies R7C15 = 1 outie R7C7, IOU no 1 in R7C5 makes the same elimination, similarly for 45 rule on N89. Maybe that’s technically slightly simpler than my CCC.]

5. 45 rule on C9 1 innie R3C9 = 1 outie R9C8 + 6 -> R3C9 = {789}

6. 4 in C9 only in 25(4) cage at R4C9 = {4579/4678}, 7 locked for C9, clean-up: no 1 in R12C9, no 1 in R9C8 (step 5)

7. 11(3) cage at R7C6 = {137/146/236/245} (cannot be {128} because no 1,2,8 in R7C7), no 8
7a. 7 of {137} must be in R7C7 -> no 7 in R78C6

8. 45 rule on N6 2 innies R4C78 = 1 outie R7C9 + 2
8a. Max R7C9 = 7 -> max R4C78 = 9, no 9 in R4C78
8b. 9 in N6 only in R456C9, locked for C9 -> R3C9 = 8, R9C8 = 2 (step 5), clean-up: no 6 in R1C7, no 7 in R56C8
8c. 2 in C9 only in R12C9 = {26}, locked for C9 and N3, no 5 in R7C7 (step 3a)
8d. R56C7 = [18]/{27/36} (cannot be {45} which clashes with R456C9, ALS block), no 4,5
8e. R56C8 = [18]/{36} (cannot be {45} which clashes with R456C9, ALS block), no 4,5

9. 20(4) cage at R3C7 = {1478/3458} (cannot be {1568} which clashes with R1C78 because 6 only in R4C8), no 6,9, CPE no 4 in R2C8
9a. 4 in N3 only in R2C7 + R3C78, CPE no 4 in R3C6

10. 45 rule on N3 2 innies R2C78 = 1 outie R4C8 + 9, IOU no 9 in R2C7
10a. R2C8 = 9 (hidden single in N3)
[Make sure all eliminations are made for 31(6) cage at R2C6.]
10b. 45 rule on N3 1 remaining innie R2C7 = 1 outie R4C8 [Used for clean-ups]

11. Hidden killer pair 1,8 in R4C78, R56C7 and R56C8 for N6, R56C7 and R56C8 must contain both or neither of 1,8 -> R4C78 must contain both of 1,8 as [81] or neither of them -> no 1 in R4C7
11a. Similarly hidden killer pair 3,6 for the same cells -> R4C78 must contain both of 3,6 as [63] or neither of them -> no 3 in R4C7

12. R4C78 = R7C9 + 2 (step 8)
12a. R7C9 = {457} -> R4C78 = 6,7,9 -> no 7 in R4C7 (because no 2 in R4C8)

13. 45 rule on R1 3 outies R2C159 = 12 = {156/237/246} (cannot be {138/147/345} because R2C9 only contains 2,6), no 8
13a. 7 of {237} must be in R2C5 -> no 3 in R2C5
13b. 7(2) cage at R1C1 and 8(2) cage at R1C9 -> R2C9 cannot be 1 more than R2C1 because this would make R1C1 equal to R1C9
-> {156} can only be [156], no 5 in R2C1, no 1 in R2C5, clean-up: no 2 in R1C1

14. 9 in C7 only in R89C7
14a. 45 rule on C89 4 outies R1389C7 = 23 = {1589/3479} (cannot be {1679/3569} which clash with R1C78, CCC in N3 because 6,9 only in R89C7), no 6
14b. 8,9 of {1589} must be in R89C7 -> no 5 in R89C7

15. 45 rule on N1 2 innies R2C23 = 1 outie R4C2 + 9
15a. Min R2C23 = 10 -> no 1 in R2C23
15b. Max R2C23 = 15 -> max R4C2 = 6

16. 45 rule on C7 3 remaining innies R247C7 = 13 = {247/256/346} (cannot be {148/157/238} which clash with R1389C7), no 1,8, clean-up: no 1 in R4C8 (step 10b)
16a. 2 of {256} must be in R4C7 -> no 5 in R4C7
16b. 1 in N6 only in R5C78, locked for R5, clean-up: no 9 in R6C2, no 8 in R6C3
16c. 8 in N6 only in R6C78, locked for R6, clean-up: no 2 in R5C2
16d. 5 in C7 only in R123C7, locked for N3, clean-up: no 3 in R1C7

17. R4C78 = 6,7,9 (step 12a)
17a. R4C78 must contain both of 3,6 as [63] or neither of them (step 11a) -> R4C78 cannot be [43], R4C78 cannot be [45] which clashes with R456C9 (ALS block) -> no 4 in R4C7

18. 45 rule on C1 1 innie R3C1 = 1 outie R9C2, no 2 in R3C1, no 8 in R9C2

19. 45 rule on R789 3 innies R7C159 = 12 = {237/246/345}
19a. {237} = {23}7 => R7C79 = [47] (step 3a)
or {246/345}
-> 4 must be in R7C1579, locked for R7

20. 11(3) cage at R7C6 (step 7) = {137/146/236/245}
20a. R7C37 = R7C5 + 10 (step 4)
20b. R7C5 = {234} -> R7C37 = 12,13,14
20c. R7C357 = [527/736/824/846/934] (cannot be [637] which clashes with 11(3) cage = {13}7) -> no 6 in R7C3, clean-up: no 5 in R7C1 (step 3)
20d. If 11(3) cage = {245} => R7C357 = [934] (cannot be [824] which clashes with 11(3) cage) => R7C1 = 2 (step 3) => 5 of {245} must be in R7C6 -> no 5 in R8C6

[Consider the combinations in C7 in more detail.]
21. R1389C7 (step 14a) = {1589/3479}
21a. R247C7 (step 16) = {247/256/346}
{247}, locked for C7
or {256} => R1389C7 = {3479} must have 7 in R1C7, locked for C7
or {346} = [364] => R7C9 = 7 (step 3a), locked for N9
-> no 7 in R89C7
21b. 1,5 of {1589} must be in R13C7, 3 of {3479} must be in R3C7 -> R3C7 = {135}

22. 20(4) cage at R3C7 (step 9) = {1478/3458}, 4 locked for C8
22a. 1 of {1478} must be in R3C7 -> no 1 in R3C8

23. 18(3) cage at R7C3 = {189/279/378/459/567} (cannot be {369} = 9{36} which clashes with R7C357 = [934], cannot be {468} = [864] which clashes with R7C357 = [846] and with R7C357 = [824] => 11(3) cage at R7C6 = {16}4)
23a. {378} = [873] (cannot be 7{38} which clashes with R7C357 = [736], cannot be [837] which clashes with R7C13 = [38], step 3) -> no 3 in R7C4
23b. {459} = [594/954], {567} = [756] (cannot be [567] which clashes with R7C13 = [65], step 3, cannot be [576] which clashes with R7C357 = [527], cannot be [765] which clashes with R7C357 = [736]) -> no 6 in R7C4, no 5 in R8C4
23c. {378} = [873] (step 23a), {567} = [756] (step 23b) -> no 7 in R8C4

24. R56C2 = {37/46}/[82], R56C3 = {27/36/45} -> combined cage R56C23 = {37/45}/{46}{27}/[82]{36}/[82]{45}

25. 22(4) cage at R4C1 = {1489/2389/2479/2569/3469} (cannot be {1678/2578/3478} which clash with combined cage R56C23 because no 7,8 in R7C1, cannot be {1579} because R7C1 only contains 2,3,4,6, cannot be {3568/4567} which clash with R12C1), 9 locked for C1 and N4, clean-up: no 9 in R9C2 (step 18)

26. 45 rule on C1 3 innies R389C1 = 16 = {178/358/367/457} (cannot be {268} which clashes with 22(4) cage at R4C1), no 2
26a. 16(3) cage in N7 contains the same combinations as R389C1 because R3C1 = R9C2 (step 18)

27. 45 rule on N4 2 innies R4C23 = 1 outie R7C1 + 4
27a. R7C1 = {2346} -> R4C23 = 6,7,8,10 = {15/16/35}/[17/28/37] (cannot be {26} which clashes with R4C7, cannot be {46} which clashes with combined cage R56C23), no 4 in R4C2, no 2,4 in R4C3
[Note that for R7C1 = {23}, 1 in N4 must be in R4C23 because 22(4) cage at R4C1 cannot contain 1 and one of 2,3.]

28. Hidden killer quad 5,6,7,8 in R456C1, R4C23, R56C2 and R56C3 for N4, R4C23 contains one of 5,6,7,8, R56C2 contains one of 6,7,8, R56C3 contains one of 5,6,7 -> R456C1 must contain one of 5,6,7,8
28a. 22(4) cage at R4C1 (step 25) = {1489/2389/2479/2569/3469} must have {234} of {1489/2389/2479/3469} in R7C1 and 6 of {2569} must be in R7C1 (because no 5 in R7C1 and cannot have both of 5,6 in R456C1)
28b. 6 of {2569} in R7C1 -> R4C23 = 10 (step 27a) cannot be [28] which clashes with {2569} -> no 2 in R4C2, no 8 in R4C3
28c. 8 in N4 only in R45C1 or R56C2 = [82] -> 22(4) cage = {1489/2389/2479/3469} (cannot be {2569} = {259}6 which clashes with R56C2 = [82]) -> no 6 in R7C1 (because 6 of {3469} must be in R456C1), clean-up: no 5 in R7C3 (step 3)

29. R7C357 (step 20c) = [736/824/846/934] cannot be [824], here’s how
[824] => 11(3) cage at R7C6 = {16}4 and R7C9 = 7 (step 3a) block 18(3) cage at R7C3 = 8[19/73/91]
-> no 2 in R7C5
29a. 2 in 10(4) cage at R6C4 only in R6C456, locked for R6 and N5, clean-up: no 8 in R5C2, no 7 in R5C3, no 7 in R5C7
29b. R56C3 = [27/45/54] (cannot be {36} which clashes with R56C2), no 3,6
29c. Killer pair 4,7 in R56C2 and R56C3, locked for N4
29d. 8 in N4 only in R45C1, locked for C1

30. R389C1 (step 26) = {367/457}, no 1
30a. 16(3) cage in N7 (step 26a) = {367/457}, no 1, 7 locked for N7, clean-up: no 4 in R7C1 (step 3)

31. 22(4) cage at R4C1 (step 28c) = {2389} (only remaining combination), locked for C1, clean-up: no 4,5 in R1C1, no 4 in R2C1

32. Naked pair {16} in R12C1, locked for C1 and N1, clean-up: no 5 in R1C23
32a. 6 in R3 only in R3C456, locked for N2

33. R389C1 = {457} -> 16(3) cage in N7 = {457} (step 26a), 4,5 locked for N7

34. R7C1 = {23} => R4C23 (step 27a) = 6,7 = {15/16}, no 3
34a. Consider placements for R7C1
R7C1 = 2 -> no 2 in R4C1
R7C1 = 3 => R4C23 = 7 = {16} => R4C7 = 2 => no 2 in R4C1
-> no 2 in R4C1

35. R4C7 = 2 (hidden single in R4), clean-up: no 7 in R6C7
[Make sure all eliminations are made for 31(6) cage at R2C6.]
35a. Naked quad {1368} in R56C78, locked for N6, clean-up: no 3 in R2C7 (step 10b)

36. R2C159 (step 13) = {156/246}, no 7
36a. 4,5 only in R2C5 -> R2C5 = {45}

37. R2C23 = R4C2 + 9 (step 15)
37a. R2C23 cannot total 14 because no 6,9 in R2C23 -> no 5 in R4C2
37b. R4C2 = {16} -> R2C23 = 10,15 = {28/37/78}, no 4,5
37c. R1C23 = {29/47} (cannot be {38} which clashes with R2C23), no 3,8
37d. Killer pair 2,7 in R1C23 and R2C23, locked for N1
37e. 8 in N1 only in R2C23, locked for R2 and 31(6) cage at R2C2, no 8 in R4C4

38. 5 in N1 only in R3C123, locked for R3
38a. 18(4) cage at R3C1 = {1359/3456}, 3 locked for R3 and N1 -> R3C7 = 1, clean-up: no 7 in R1C78, no 8 in R6C7

39. R1C78 = [53], clean-up: no 5 in R4C8 (step 10b)
39a. Naked pair {36} in R56C7, locked for C7 and N6 -> R56C8 = [18], clean-up: no 5 in R7C9 (step 3a)

40. Naked pair {47} in R7C79, locked for R7 and N9 -> R7C5 = 3, R7C1 = 2, R7C3 = 9 (step 3), clean-up: no 2 in R1C2

41. R7C357 (step 29) = [934] -> R7C7 = 4, R7C9 = 7, R2C7 = 7, R34C8 = [47], R3C1 = 5, R3C3 = 3, R3C2 = 9, R4C2 = 1 (step 38a)
[Make sure all eliminations are made for 31(6) cage at R2C6.]

42. Naked pair {28} in R2C23, locked for R2, N1 and 31(6) cage at R2C2, no 2 in R3C4
42a. R12C9 = [26], R12C1 = [61], R2C5 = 5 (step 36)

43. R3C6 = 6, R3C4 = 7, R3C5 = 2

44. 31(5) cage at R3C5 = {25789} (only remaining combination), no 3,4,6

45. R4C4 = 6, R4C6 = 3 (hidden singles in N5), R4C3 = 5, R2C6 = 4, R2C4 = 3, clean-up: no 4 in R56C3

46. R56C3 = [27], R1C23 = [74], R2C23 = [28]
46a. Naked pair {16} in R89C3, locked for N7 -> R7C2 = 8, R8C2 = 3, R89C9 = [13], R89C3 = [61], R78C8 = [65], R8C6 = 2, R6C6 = 1, R7C6 = 5, R7C4 = 1, R8C4 = 8 (step 23)

and the rest is naked singles.

Then I had a go at the V4. I enjoyed this because I made even more use of hidden killers than in the earlier variants but also made several silly mistakes; I've had to re-work some of the steps in the middle of my walkthrough and hope they are right now.

manu's walkthrough had a nice forcing chain, with a built-in contradiction, in his step 8.


Here is my walkthrough for A142 V4.

Prelims

a) R12C1 = {69/78}
b) R1C23 = {17/26/35}, no 4,8,9
c) R1C78 = {17/26/35}, no 4,8,9
d) R12C9 = {29/38/47/56}, no 1
e) R56C2 = {19/28/37/46}, no 5
f) R56C3 = {19/28/37/46}, no 5
g) R56C7 = {16/25/34}, no 7,8,9
h) R56C8 = {79}
i) 20(3) cage at R7C3 = {389/479/569/578}, no 1,2
j) 37(6) at R2C6 = {256789/346789}, no 1

1. Naked pair {79} in R56C8, locked for C8 and N6, clean-up: no 1 in R1C7
1a. 1 in N3 only in R1C8 + R3C789, CPE no 1 in R4C8

2. 45 rule on N7 2 innies R7C13 = 10 = [19/28]/{37/46}, no 5, no 8,9 in R7C1

3. 45 rule on N9 2 innies R7C79 = 12 = {39/48/57}, no 1,2,6

4. 45 rule on R1 3 outies R2C159 = 12 = {129/138/147/156/237/246} (cannot be {345} because R2C1 only contains 6,7,8,9)
4a. R2C1 = {6789} -> no 6,7,8,9 in R2C59, clean-up: no 2,3,4,5 in R1C9
4b. 1 of {156} must be in R2C5 -> no 5 in R2C5

5. 45 rule on N1 2 innies R2C23 = 1 outie R4C2 + 4, IOU no 4 in R2C3

6. 45 rule on N3 2 innies R2C78 = 1 outie R4C8 + 6, IOU no 6 in R2C7

7. 45 rule on C9 1 outie R9C8 = 1 innie R3C9 + 1, no 6,8,9 in R3C9, no 1 in R9C8

8. 45 rule on N78 2 innies R7C15 = 1 outie R7C7 + 2, IOU no 2 in R7C15, clean-up: no 8 in R7C3 (step 2)

9. 45 rule on N89 2 innies R7C59 = 1 outie R7C3 + 4, IOU no 4 in R7C59, clean-up: no 8 in R7C7 (step 3)

10. 45 rule on N4 2 innies R4C23 = 1 outie R7C1 + 10, no 1 in R4C23

11. 45 rule on N47 3(2+1) innies R4C23 + R7C3 = 20
11a. Max R47C3 = 17 -> min R4C2 = 3

12. 45 rule on N8 2 outies R7C37 = 1 innie R7C5 + 8
12a. Max R7C37 = 16 -> max R7C5 = 8
12b. R7C357 = [415/437/459/613/635/679/954/965/987] (cannot be [657] which clashes with R7C79 (step 3) = [75], cannot be [734] which clashes with R7C13 (step 2) = [37], cannot be [789] because R7C13 (step 2) = [37] clashes with R7C79 (step 3) = [93]) -> no 3,7 in R7C3, clean-up: no 3,7 in R7C1 (step 2)

13. 20(3) cage at R7C3 = {389/479/569} (cannot be {578} because R7C3 only contains 4,6,9), CPE no 9 in R7C6

14. Hidden killer pair 1,9 in R456C1, R56C2 and R56C3 for N4, R56C2 and R56C3 can only contain both or neither of 1,9 -> R456C1 can only contain 9 if it also contains 1
[Note. The possibility of 9 in R4C23 isn’t relevant to this step.]
14a. 15(4) cage at R4C1 = {1248/1257/1347/1356/2346} (cannot be {1239} = {239}1, which doesn’t contain 1 in R456C1), no 9

15. R4C23 = R7C1 + 10 (step 10)
15a. R7C1 = {146} -> R4C23 = 11,14,16 cannot be [92], here’s how
R7C1 = 1 => 1 in N4 only in R56C2 or R56C3 = {19} which clashes with [92]
-> no 2 in R4C3

16. 15(4) cage at R4C1 (step 14a) = {1248/1257/1347/1356/2346} cannot be {2346}, here’s how
{2346} => 1 in N4 only in R56C2 or R56C3 = {19}
{2346} = {236}4 => R4C23 (step 10) = 14 but cannot be {59} which clashes with R56C2 or R56C3 = {19} and cannot be {68} which clashes with R456C1 = {236}
or {2346} = {234}6 => R4C23 (step 10) = 16 but cannot be {79} which clashes with R56C2 or R56C3 = {19}
-> 15(4) cage = {1248/1257/1347/1356}, 1 locked for C1

[There’s a forcing chain possible here but I didn’t spot it until step 26.]

17. 45 rule on C1 3 innies R389C1 = 15 = {249/258/348/357} (cannot be {267} which clashes with R12C1, cannot be {456} which clashes with 15(4) cage at R4C1)
17a. 45 rule on C1 1 innie R3C1 = 1 outie R9C2, clean-up: no 1 in R9C2
17b. R3C1 = R9C2 -> combinations for 15(3) cage in N7 are the same as for R389C1
17c. 15(3) cage = {258/348/357} (cannot be {249} which clashes with R7C13), no 9, clean-up: no 9 in R3C1 (step 17a)

18. 9 in C1 only in R12C1 = {69}, locked for C1 and N1, clean-up: no 2 in R1C23, no 4 in R7C3 (step 2), no 6 in R9C2 (step 17a)

19. R2C159 (step 4) = {129/156/246}, no 3, clean-up: no 8 in R1C9

20. 4,8 in R1 only in R1C456, locked for N2
20a. 15(4) cage at R1C4 contains both of 4,8 = {1248} (only remaining combination), 1,2 locked for N2

21. Hidden killer pair 1,6 in R56C7 and 20(4) cage at R4C9 for N6, R56C7 contains both or neither of 1,6 -> 20(4) cage at R4C9 can only contain 6 if it also contains 1 (because no other 1 in N6 and no 6 in R7C9)
[Note. The possibility of 6 in R4C78 isn’t relevant to this step.]
21a. 20(4) cage at R4C9 = {1289/1478/1568/2378/3458} (cannot be {1379} because 7,9 only in R7C9, cannot be {2369/2468} which clash with R56C7, cannot be {2459} which clashes with R2C9, cannot be {1469/2567} which clash with R12C9, cannot be {3467} which contains 6 but no 1), 8 locked for C9

22. 20(4) cage at R4C9 (step 21a) = {1289/1478/1568/2378/3458} cannot be {1289/1478}, here’s how
{1289} = {128}9 => R56C7 = {34}, R7C79 (step 3) = [39] clashes with R56C7
{1478} = {148}7 => R56C7 = {25}, R7C79 (step 3) = [57] clashes with R56C7
-> 20(4) cage = {1568/2378/3458}, no 9, clean-up: no 3 in R7C7 (step 3)

23. R7C37 = R7C5 + 8 (step 12), min R7C37 = 11 (cannot be 10 because of clash with R7C13, CCC) -> no 1 in R7C5

24. 45 rule on C9 3 innies R389C9 = 14 = {149/167/239/347} (cannot be {257/356} which clash with 20(4) cage at R4C9), no 5, clean-up: no 6 in R9C8 (step 7)

25. 15(3) cage in N9 = {159/168/249/267/348/357/456} (cannot be {258} because 5,8 only in R9C8)
25a. 5,8 of {348/357} only in R9C8 -> no 3 in R9C8, clean-up: no 2 in R3C9 (step 7)

26. Consider placements for R7C1
R7C1 = 1 => R7C3 = 9 => no 9 in R56C3, clean-up: no 1 in R56C3
R7C1 = 4 => 1 in C1 only in R456C1, locked for N4, clean-up: no 9 in R56C3
-> R56C3 = {28/37/46}, no 1,9

27. 15(4) cage at R4C1 (step 16) = {1248/1257/1347}
27a. Consider the effect of these permutations on R4C23
{1248/1347} => R7C1 = {14}, 5 in N4 only in R4C23 (step 10) = 11,14 = {56/59}
{1257} => R456C1 = {257}, locked for N4, 1 only in R56C2 = {19}, R56C3 = {46} => R4C23 = {38}
-> R4C23 = {38/56/59}, no 4,7

28. 1 in N6 only in 20(4) cage at R4C9 (step 22) = {1568} or in R56C7 = {16} -> killer pair 1,6 in 20(4) cage at R4C9 + R56C7, locked for N6

29. R7C79 = 12 (step 3)
29a. 2 in 14(3) cage at R7C6 cannot be in R8C6 (R7C67 cannot total 12 which would clash with R7C79, CCC) -> no 2 in R7C6
[Alternatively 45 rule on N9 2 outies R78C6 = 1 innie R7C9 + 2, IOU no 2 in R8C6]

30. 14(3) cage at R7C6 = {149/158/167/239/248/257/347/356}
30a. 5 of {158/356} must be in R7C7, 2 of {257} must be in R7C6 -> no 5 in R7C6
30b. Hidden killer pair 1,2 in 14(3) cage at R7C6 and 19(4) cage for N8, 14(3) cage at R7C6 contains one of 1,2 except for {347/356} -> 19(4) cage must contain at least one of 1,2 and both = {1279} when 14(3) cage = {347/356}
30c. 7 of {167} must be in R7C7, 2 of {257} must be in R7C6, 7 of {347} must be in R7C7 (R78C6 + R7C7 cannot be {37}4 which clashes with 19(4) cage = {1279} -> no 7 in R7C6

31. R2C78 = R4C8 + 6 (step 6)
31a. R2C78 cannot be {26} which clashes with R2C159, cannot be {35} because 37(6) cage cannot contain both of 3,5 -> R2C78 cannot total 8 -> no 2 in R4C8

32. 45 rule on N6 2 innies R4C78 = 1 outie R7C9 + 2
32a. Max R7C9 = 8 -> max R4C78 = 10, no 8 in R4C7

33. 45 rule on N69 3(2+1) innies R4C78 + R7C7 = 14 = [239/257/284/347/437/455] -> no 5 in R4C7

34. 37(6) at R2C6 = {256789/346789}
34a. 2 of {256789} must be in R4C7 -> no 2 in R2C78 + R4C6

35. R2C78 = R4C8 + 6 (step 6)
R4C8 = {345} => R2C78 = 9,10,11 => no 9 in R2C7
R4C8 = 8 => 8 in N3 only in R2C7
-> no 9 in R2C7
35a. 9 in 37(6) cage at R2C6 only in R234C6, locked for C6

36. Hidden killer pair 2,9 in R12C9 and 20(4) cage at R3C7 for N3, R12C9 can only contain both or neither of 2,9 -> 20(4) cage can only contain 2 if it also contains 9
[Note. The possibility of 2 in R1C78 isn’t relevant to this step.]
36a. Hidden killer pair 1,7 in R1C78 and 20(4) cage at R3C7 for N3, R1C78 can only contain both or neither of 1,7 -> 20(4) cage can only contain 7 if it also contains 1
[Note. The possibility of 7 in other cells of N3 isn’t relevant to this step.]
36b. Hidden killer pair/triple 2,6,9 in R1C78, R12C9 and 20(4) cage at R3C7 for N3, R1C78 can contain both or neither of 2,6, R12C9 can contain both or neither of 2,9 -> 20(4) cage can only contain both of 6,9 if it also contains 2
[Note the possibility of 6 in R2C8 isn’t relevant to this step.]
36c. 20(4) cage at R3C7 = {1379/1478/1568/2459/3458} (cannot be {1469} because cage can only contain both of 6,9 if it also contains 2, cannot be {2369} because 2,6,9 only in R3C78, cannot be {2378/2468/2567} because cage can only contain 2 if it also contains 9, cannot be {3467} because cage can only contain 7 if it also contains 1, cannot be {1289} = [9218] which clashes with R3C9 + R9C8 = [12], step 7)

[At this stage I made a serious error in my original solving path. I’ve tried to get back to it as much as possible by bringing a couple of steps forward, as step 37, and then adding a contradiction move.]

37. Hidden killer quad 1,2,4,8 in 18(4) cage at R3C1 and 20(4) cage at R3C7 for R3, 20(4) cage cannot contain more than two of 1,2,4,8 in R3 -> 18(4) cage at R3C1 must contain at least two of 1,2,4,8 in R3
37a. 18(4) cage at R3C1 = {1278/1458/1467/2349/2358/2457} (cannot be {1269} because 6,9 only in R4C2, cannot be {1359/1368/2367} which clash with R1C23, cannot be {3456} which only contains one of 1,2,4,8)
37b. All combinations for 18(4) cage contain two of 1,2,4,8 in R3 except for {1458} = {148}5 (cannot be {145}8 which clashes with R1C23) -> 20(4) cage at R3C7 must contain two of 1,2,4,8 in R3 or one of 1,2,4,8 in R3 and not clash with {145}8
37c. 20(4) cage at R3C7 (step 36c) = {1478/1568/2459/3458} (cannot be {1379} which only contains one of 1,2,4,8 and clashes with {145}8)
37d. All combinations for 20(4) cage at R3C7 must contain two of 1,2,4,8 in R3 (because {1478/3458} clash with {145}8, an ALS block in the case of {3458})

38. R1C78 = {26/35}/[71] cannot be {26}, here’s how
R1C78 = {26} => R12C9 = [74]
20(4) cage at R3C7 (step 37c) = {1478/1568/2459/3458}
{1478/3458} clash with R12C9 (because 4,8 of {3458} must be in R3, step 37d) and {1568/2459} clash with R1C78
-> R1C78 = [35/53/71], no 2,6

39. Naked quad {1357} in R1C2378, locked for R1, clean-up: no 4 in R2C9
39a. R2C5 = 1 (hidden single in N2)

40. Killer pair 2,5 in R2C9 and 20(4) cage at R4C9, locked for C9

41. 15(3) cage in N9 (step 25) = {159/168/249/267/348/357/456}
41a. 2,5,8 of {249/348/456} must be in R9C8 -> no 4 in R9C8, clean-up: no 3 in R3C9 (step 7)

42. R2C78 = R4C8 + 6 (step 6)
42a. R4C8 = {3458} -> R2C78 = 9,10,11,14 = [36/46/38/56/74/83/86] (cannot be {45} because R1C78 = [71] clashes with R3C9 = {17}, cannot be [73] which clashes with R1C78) -> no 5 in R2C8

43. Hidden killer pair 1,7 in R1C23 and 18(4) cage at R3C1 for N1, R1C23 can only contain both or neither of 1,7 -> 18(4) cage can only contain 7 if it also contains 1
[Note. The possibility of 7 in R2C78 isn’t relevant to this step.]
43a. 18(4) cage at R3C1 (step 37a) = {1278/1458/2349/2358} (cannot be {1467} which clashes with R3C9, cannot be {2457} which contains 7 but not 1), no 6

44. R2C23 = R4C2 + 4 (step 5)
44a. R4C2 = {3589} -> R2C23 = 7,9,12,13 = [43/27/45/72/48/58/85] (cannot be {25} which clashes with R2C9, cannot be {57} which clashes with R1C23) -> no 3 in R2C2

45. 18(4) cage at R3C1 (step 43a) = {1278/1458/2349/2358}
45a. 2 of {1278} must be in R3C1 (18(4) cage cannot be 7{12}8 which clashes with 15(4) cage at R4C1) -> no 7 in R3C1, clean-up: no 7 in R9C2 (step 17a)

46. R4C23 = {38/56/59} (step 27a)
46a. R2C23 = R4C2 + 4 (step 5)
46b. R4C2 = {3589} cannot be 9, here’s how
R4C2 = 9 => R2C23 = 13 = {58} clashes with R4C23 = [95] (because R2C23 and R4C3 are all in the 28(6) cage at R2C2)
46c. R4C2 = {358} -> no 5 in R4C3
46d. R4C2 = {358} -> R2C23 = 7,9,12 (step 44a) = [43/27/45/72/48], no 5,8 in R2C2

47. R2C78 = R4C8 + 6 (step 6)
47a. 8 in R2 only in R2C378 -> R2C23 = [48] (step 46d) and R2C78 (step 42a) = [36/56] or R2C78 = [38/83/86] (cannot be [48/58] = 12,13 because no 6,7 in R4C8) -> no 4 in R2C7
47b. 4 in N3 only in R2C8 + R3C789, CPE no 4 in R4C8

48. R4C8 = {358} -> R2C78 (step 42a) = 9,11,14 = [36/38/56/74/83/86]
48a. R4C78 = R7C9 + 2 (step 32)
48b. R7C9 = {3578} -> R4C78 = 5,7,9,10 = [23/25/43/45/28], no 3 in R4C7
48c. R4C78 = [25] -> no 4 in R2C8 because 37(6) cage cannot contain both of 2,4
or R4C78 = [45] -> no 4 in R2C8
-> no 4 in R2C8, clean-up: no 7 in R2C7 (step 48)
48d. 7 in 37(6) cage at R2C6 only in R234C6, locked for C6

[Now, at last, I’m just about back to my original solving path.]

49. 4 in N3 only in R3C789, locked for R3
49a. R2C2 = 4 (hidden single in N1), clean-up: no 6 in R56C2
[Make sure all eliminations are made for 28(6) cage at R2C2.]

50. R2C23 = R4C2 + 4 (step 5), R2C2 = 4 -> R2C3 = R4C2, no 2,7 in R2C3
50a. R2C9 = 2 (hidden single in R2), R1C9 = 9, R12C1 = [69]
50b. 7 in R2 only in R2C46, locked for N2
50c. 2 in N6 only in R456C7, locked for C7

51. 6 in C2 only in R78C2, locked for N7 -> R7C3 = 9, R7C1 = 1 (step 2), clean-up: no 3 in R7C9 (step 3)

52. R56C2 = {19} (hidden pair in N4), locked for C2, clean-up: no 7 in R1C3
52a. 9 in R4 only in R4C456, locked for N5

53. 18(4) cage at R3C1 (step 45) = {1278/2358}
53a. 1 of {1278} must be in R3C3 -> no 7 in R3C3
53b. 7 in N1 only in R13C2, locked for N2

54. 4 in N3 only in 20(4) cage at R3C7 (step 37c) = {1478/3458}, no 6
54a. R2C8 = 6 (hidden single in N3)
[Make sure all eliminations are made for 37(6) cage at R2C6.]
54b. 4 of {3458} must be in R3C9 -> 5 of {3458} must be in R3C7 (R34C8 = {35/58} clash with R3C9 + R9C8 = [45], step 7) -> no 3 in R3C7, no 5 in R34C8
54c. R2C78 = R4C8 + 6 (step 6), R2C8 = 6 -> R2C7 = R4C8, no 5 in R2C7

55. 20(4) cage at R4C9 (step 22) = {1568/3458}, no 7, clean-up: no 5 in R7C7 (step 3)

56. 20(4) cage at R3C7 (step 37c) = {1478/3458}
56a. 8 of {1478} must be in R4C8, 4,5 of {3458} must be in R3C79 with 3,8 in R34C8 -> 8 locked for C8, no 8 in R3C7, clean-up: no 7 in R3C9 (step 7)
56b. 5,7 only in R3C7 -> R3C7 = {57}

57. 7 in C9 only in R89C9, locked for N9 -> R7C7 = 4, R7C9 = 8 (step 3)
57a. R4C78 = R7C9 + 2 (step 32)
57b. R7C9 = 8 -> R4C78 = 10 = [28], R2C7 = 8 (step 54c), clean-up: no 3,5 in R56C7

58. Naked pair {16} in R56C7, locked for C7 and N6

59. Naked triple {345} in R456C9, locked for C9 -> R3C9 = 1, R9C8 = 2 (step 7), clean-up: no 7 in R1C7, no 2 in R3C1 (step 17a)

60. Naked pair {35} in R1C78, locked for R1 and N3 -> R1C23 = [71], R3C78 = [74]

61. R4C23 (step 27a) = {56} (only remaining combination) -> R4C2 = 5, R4C3 = 6, clean-up: no 4 in R56C3
61a. R2C3 = 5 (hidden single in N1)
[Make sure all eliminations are made for 28(6) cage at R2C2.]

62. Naked pair {37} in R2C46, locked for N2 -> R3C4 = 9, R3C6 = 5, R3C5 = 6

63. R4C4 = 1 (hidden single in R4) -> R2C4 = 3 (cage sum), R2C6 = 7, R4C6 = 9 (cage sum)

64. 5 in C1 only in R89C1 -> 15(3) cage in N7 (step 17c) = {258/357}, no 4
64a. R9C2 = {38} -> no 3,8 in R89C1

65. R7C37 = R7C5 + 8 (step 12)
65a. R7C37 = [94] = 13 -> R7C5 = 5, R7C8 = 3, R1C78 = [35], R8C8 = 1

66. R7C3 = 9 -> R78C4 = 11 = [74]
66a. R7C7 = 4 -> R78C6 = 10 = [28], R1C6 = 4, R5C6 = 3, R6C6 = 6, R7C2 = 6, R9C4 = 6, R9C6 = 1, R89C9 = [67], R9C1 = 5, R89C7 = [59], R89C5 = [93], R9C2 = 8, R8C1 = 2 (step 64), clean-up: no 7 in R6C3

67. 2 in N4 only in R56C3 = {28}, locked for C3 and N4

68. R5C4 = 5 (hidden single in C4), R3C5 = 6, R5C6 = 3 -> R45C5 = 29 - 3 - 5 - 6 = 15 = [78]

and the rest is naked singles.

As then there was the V5 which, quite honestly, is only for masochists. After early steps similar to the other variants I could only manage to make progress by using a series of "brute force" contradition moves, doing the same sort of step repeatedly in the same area, so not an interesting puzzle. It ended up with a shorter walkthrough than the V4 but probably only because I'd used so many contradiction moves.


I don't recommend anyone working through my walkthrough

If anyone wants to have a try to solve this without using contradiction moves (forcing chains would be better), please feel free to have a try and post a walkthrough. I'll be interested to see what I may have missed.

Finally a few general comments on the A142 variants; these are based on the last three, the ones I did this month, but probably also apply to the original Assassin and to V2.

It's an interesting cage pattern which leads immediately to a number of IOUs, where the innie-outie differences are between 1 and 9 (a few are larger and lead to different eliminations). At the same time the wrapped-round 6-cells cages at R2C2 and R2C6 restrict the main solving areas to the first three columns and the last three columns.

There were a lot of interesting and useful hidden killers in the variants; I used them most in my solving path for V4.

V5 was different because, after similar starting steps, the main work was concentrated in one small area and, after finding a step which gave an elimination (either candidate or combination) I found myself repeating the same type of step so this puzzle wasn't interesting; one variant too many.