Almost two years ago I wrote:
Thanks Ed. Great timing. I'd reached the stage with what is now the V2 where after prelims and 18 steps I was looking for contradiction moves to make progress. I don't like using those moves or forcing chains, which I rarely find, so early in a puzzle.
That was when Ed posted what is now the V1.
At the time Ed and I discussed my first 18 steps and discovered that I'd missed a CPE which I've now added to step 6; that led to some changes to step 13. Then after two or three more steps I moved on to other puzzles and didn't come back to A120 V2 until this week. It took me some time to find the first key breakthrough in step 29a (Afmob's step 6b) but after that I was making progress although I still needed a contradiction move for my final breakthrough.
Afmob wrote:
What a challenging Killer!
It certainly was!
Rating Comment. I'll rate my walkthrough for A120 V2 at 1.75 because I found this puzzle almost as hard as A108 V2 which I finished recently. For A120 V2 I used combination and permutation analysis, a "ruudiculous" ALS block using a combined cage for the first key breakthrough and a fairly short contradiction move for my final breakthrough.
Here is my walkthrough for A120 V2.
Prelims
a) R23C1 = {49/58/67}, no 1,2,3
b) R2C34 = {19/28/37/46}, no 5
c) R23C9 = {15/24}
d) R78C1 = {16/25/34}, no 7,8,9
e) R78C9 = {39/48/57}, no 1,2,6
f) R8C67 = {19/28/37/46}, no 5
g) R1C456 = {389/479/569/578}, no 1,2
h) R456C3 = {128/137/146/236/245}, no 9
i) R456C7 = {128/137/146/236/245}, no 9
j) R6C456 = {128/137/146/236/245}, no 9
k) R9C456 = {126/135/234}, no 7,8,9
1. 45 rule on R1 2 outies R2C28 = 6 = {15/24}
2. 45 rule on R123 2 outies R4C28 = 11 = {29/38/47/56}, no 1
3. 45 rule on R789 2 outies R6C28 = 11 = {29/38/47/56}, no 1
4. 45 rule on R9 2 outies R8C28 = 10 = {19/28/37/46}, no 5
5. 45 rule on N4 2 innies R46C2 = 12 = {39/48/57}, no 2,6
5a. 45 rule on N6 2 innies R46C8 = 10 = {28/37/46}, no 5,9
6. 45 rule on N3 3 innies R2C7 + R3C78 = 24 = {789}, locked for N3, CPE no 7,8,9 in R3C5
6a. 15(4) cage in N3 = {1356/2346}, 3,6 locked for R1
[I’ve added the CPE to step 6 after Ed pointed it out in discussions about my first 18 moves, at the time I originally got stuck.]
7. R1C456 = {479/578}, 7 locked for R1 and N2, clean-up: no 3 in R2C3
7a. 3 in R2 locked in R2C456, locked for N2
8. 21(4) cage at R3C6 contains two of 7,8,9 in R3C78 = {1389/1479/2379/2478} (cannot be {1578} because 1,5 only in R3C6, other combinations only have one of 7,8,9), no 5,6, clean-up: no 5 in R4C2 (step 2), no 7 in R6C2 (step 5), no 4 in R6C8 (step 5a)
8a. R3C78 = {789} -> no 7,8,9 in R3C6 + R4C8, clean-up: no 3,4 in R4C2 (step 2), no 8,9 in R6C2 (step 5), no 2,3 in R6C8 (step 5a)
9. 45 rule on N9 3 innies R7C78 + R8C7 = 9 = {126/135/234}, no 7,8,9, clean-up: no 1,2,3 in R8C6
10. 45 rule on N9 3 outies R6C8 + R78C6 = 19, max R6C8 + R8C6 = 17 -> min R7C6 = 2
10a. R6C8 + R8C6 cannot total 13 (because R7C6 “sees” both of R6C8 + R8C6 and there’s no 4,5,9 in R6C8 and no 5 in R8C6) -> no 6 in R7C6
11. 45 rule on N89 2 outies R6C8 + R8C3 = 1 innie R7C4 + 3
11a. Min R6C8 + R8C3 = 7 -> min R7C4 = 4
11b. Max R7C4 = 9 -> max R6C8 + R8C3 = 12, max R8C3 = 6
12. 45 rule on N1 3 outies R23C4 + R4C2 = 14
12a. R4C2 = {789} -> R23C4 = 5,6,7, no 8,9, clean-up: no 1,2 in R2C3
13. 25(4) cage at R2C5 can only contain 7 if it also contains 3 (because R2C34 = [73] is the only other way to place 3 in R2) -> 25(4) cage = {2689/3589/3679} (cannot be {1789/4579/4678} which contain 7 but not 3), no 1,4, 9 locked in R2C567, locked for R2, clean-up: no 1 in R2C4, no 4 in R3C1
13b. 1 in N2 only in R3C46, locked for R3, clean-up: no 5 in R2C9
13c. 5 of {3589} must be in R3C5 -> no 5 in R2C56
[Before I added the CPE to step 6, I used hidden killer quint 1,2,3,4,5 in R2C156, R2C2, R2C34, R2C8 and R2C9 for R2 -> R2C156 can only contain one of 1,2,3,4,5 to make step 13c work. This step is simpler after the CPE. Also I think that 9 locked ... in step 13 and the whole of step 13b require that there to be no 9 in R3C5.]
14. 16(4) cage in N1 = {1249/1258}, 2 locked for N1
15. 45 rule on N2 3 innies R23C4 + R3C6 = 1 outie R2C7
15a. R2C7 = {789} -> R23C4 + R3C6 (must contain 1, step 13b) = 7,8,9 = {124/125/134/126} (cannot be {135} because R23C4 must total 5,6,7 (step 12a) and 3 only in R2C4)
15b. 4 of {124/134} must be R23C4 because R23C4 = 5,6,7 (step 12a) -> no 4 in R3C6
15c. 2 of {126} must be in R3C6 (R23C4 cannot be {26} because R23C4 = 5,6,7) -> {126} must be [612], no 6 in R3C4
16. 25(4) cage at R2C5 (step 13a) = {2689/3589/3679}
16a. R23C4 + R3C6 (step 15a) = {124/134/126} (cannot be {125} which clashes with {2689/3589} and for {3679} R2C7 = 7 => R23C4 + R3C6 = {124} using step 15) -> no 5 in R3C4
[Ed suggested that simpler is R2C7 = 8 => R23C4 + R3C6 (step 15) = 8. R2C7 = 8 => no 8 in R2C3, no 2 in R2C4 => min R2C4 = 3 => max R3C46 = 5, no 5.
Alternatively R2C7 = 8 => R23C4 + R3C6 (step 15) = 8. R2C7 = 8 => no 8 in R2C3, no 2 in R2C4 => R23C4 + R3C6 cannot be {125}, no 5.]
17. 45 rule on R12 3 outies R3C159 = 15 = {258/267/456} (cannot be {249} which clashes with R3C46, ALS block), no 9, clean-up: no 4 in R2C1
18. 21(4) cage at R3C6 (step 8) = {1389/1479/2379/2478}
18a. R3C6 = {12} -> no 2 in R4C8, clean-up: no 9 in R4C2 (step 2), no 3 in R6C2 (step 5), no 8 in R6C8 (step 5a)
18b. Max R6C2 = 5 -> min R7C234 = 19, no 1
18c. R6C8 + R78C6 = 19 (step 10), max R6C8 = 7 -> min R78C6 = 12, no 2 in R7C6
19. R456C3 = {128/137/146/236} (cannot be {245} which clashes with R6C2), no 5
20. R6C456 = {128/137/236} (cannot be {146} which clashes with R6C28, cannot be {245} which clashes with R6C2), no 4,5
20a. R4C456 = {189/369/459/567} (cannot be {279} which clashes with R6C456, cannot be {378} which clashes with R4C2, cannot be {468} which clashes with R4C28), no 2
21. 3 in R3 only in R3C23 -> 20(4) cage at R3C2 = {1379/2378/3458/3467} (cannot be {2369} because R4C2 only contains 7,8)
21a. 4 of {3458/3467} must be in R3C4 -> no 4 in R3C23
22. 3 in R3 only in R3C23
22a. 45 rule on N1 3 innies R2C3 + R3C23 = 16 = {349/358/367}
22b. 8 of {358} must be in R2C3 -> no 8 in R3C23
23. 45 rule on C89 4 innies R3467C8 = 2 outies R19C7 + 9
23a. R46C8 = 10 (step 5a) -> R19C7 = R37C8 + 1
23b. Min R37C8 = 8 -> min R19C7 = 9, no 1,2 in R9C7
23c. Min R37C8 = 8 = [71] => R78C7 = {26/35} (step 9) -> min R19C7 = 9 but cannot be [63] which clashes with R78C7 -> no 3 in R9C7
23d. Max R19C7 = [69] = 15 -> max R37C8 = 14 = [95] (because 9 in R9C7 forces 9 in R3C8)
Max R19C7 without using 9 = [68] = 14 -> max R37C8 = 13 = [85] (9 in N3 must be in R23C7 and R27C8 cannot be [76] which clashes with R6C8)
-> no 6 in R7C8
24. R4C28 (step 2) = [74/83], R46C2 (step 5) = [75/84] -> R4C28 + R6C2 = [745/834], CPE no 4 in R4C13 + R6C79
25. R4C28 (step 2) = [74/83], R46C8 (step 5a) = [37/46] -> R4C28 + R6C8 = [746/837], CPE no 7 in R4C79 + R6C13
26. 9 in R6 only in R6C19
26a. 45 rule on R6789 4 innies R6C1379 = 23 = {1589/2489/3479/3569} (cannot be {1679} which clashes with R6C8, cannot be {2579} which clashes with R6C28)
26b. 4 of {2489/3479} must be in R6C13
{1589/3569}, 5 locked for R6 -> R6C2 = 4
-> 4 locked in R6C123, locked for N4
27. 9 in N6 only in 24(4) cage = {1689/2589/3579/4569} (cannot be {2679} which clashes with R6C8, cannot be {3489} which clashes with R4C8)
27a. R23C9 = [15]/{24}, R78C9 = {39/48/57} -> combined cage R2378C9 = [15]{39}/[15]{48}/{24}{39}/{24}{57}
27b. 24(4) cage = {1689/3579} (cannot be {2589/4569} which clash with R2378C9, ALS block), no 2,4
27c. Killer pair 6,7 in 24(4) cage and R6C8, locked for N6
28. 2 in N6 only in R456C7, locked for C7, clean-up: no 8 in R8C6
28a. R456C7 = {128/245}, no 3
29. R7C78 + R8C7 (step 9) = {126/135/234} -> R7C78 = {12/15/23/24/26/35} (cannot be {13/16/34} because no 2,5 in R8C7)
29a. 18(4) cage at R6C8 = {1269/1278/2367/2457/3456} (cannot be {1359/1458/2349/2358} because R6C8 only contains 6,7, cannot be {1368/1467} which don’t contain any combinations for R7C78)
29b. R7C78 = {12/23/24/26/35} (cannot be {15} because no combinations for 18(4) cage contain both of 1,5)
29c. 2 of {12/24} must be in R7C8 -> no 1,4 in R7C8
30. R7C78 + R8C7 (step 9) = {126/135/234}
30a. 5 of {135} must be in R7C8 (R78C7 cannot be [51] which clashes with R456C7), no 5 in R7C7
30b. R78C7 = {13/16/34}
30c. Killer pair 1,4 in R456C7 and R78C7, locked for C7
31. R78C7 (step 30b) = {13/16/34}
31a. 24(4) cage in N6 (step 27b) = {1689/3579} cannot be {3579}, here’s how
{3579} => R78C9 = {48} (cannot be {39/57} which clash with {3579}, ALS block) => R78C7 = {13/16} => R456C7 (step 28a) = {245} clashes with {3579}
[Almost a complete loop around N69.
]
31b. -> 24(4) cage = {1689}, locked for N6 -> R456C7 = {245}, locked for C7 and N6, R46C8 = [37], R4C2 = 8 (step 2), R6C2 = 4 (step 3), clean-up: no 2 in R2C8 (step 1), no 3,7 in R8C2 (step 4), no 6 in R8C6, no 2,6 in R8C8 (step 4)
31c. 7 in N3 only in R23C7, locked for C7
32. 15(4) cage in N3 (step 6a) = {1356/2346}
32a. 4 of {2346} must be in R2C8 -> no 4 in R1C89
33. 18(4) cage at R6C8 (step 29a) = {1278/2367} (cannot be {2457} because R7C7 only contains 1,3,6) -> R7C8 = 2, clean-up: no 5 in R8C1
33a. R7C67 = [36/81], no 4,5,9 in R7C6, no 3 in R7C7
34. R7C78 + R8C7 (step 9) = {126} (only remaining combination) -> R78C7 = {16}, locked for C7 and N9 -> R1C7 = 3, clean-up: no 9 in R8C2 (step 4), no 7 in R8C6
35. R23C4 + R4C2 = 14 (step 12), R4C2 = 8 -> R23C4 = 6 = {24}, locked for C4 and N2 -> R3C6 = 1, clean-up: no 9 in R1C456 (step 7), no 4,7 in R2C3
36. Naked triple {578} in R1C456, locked for R1 and N2 -> R3C5 = 6, clean-up: no 7 in R2C1
37. 25(4) cage at R2C5 (step 13) = {3679} (only remaining combination) -> R2C7 = 7
37a. Naked pair {89} in R3C78, locked for R3, clean-up: no 5 in R2C1
38. 16(4) cage in N1 (step 14) = {1249} (only remaining combination), no 5, clean-up: no 1 in R2C8 (step 1)
39. R2C8 = 5 (hidden single in R2), R2C2 = 1 (step 1), clean-up: no 9 in R8C8 (step 4)
39a. Naked pair {24} in R23C9, locked for C9, clean-up: no 8 in R78C9
40. Naked triple {489} in R389C8, locked for C8
40a. Naked triple {489} in R8C8 + R9C78, locked for N9, clean-up: no 3 in R78C9
41. R9C9 = 3 (hidden single in N9)
42. R9C456 = {126} (only remaining combination), locked for R9 and N8
43. R6C456 (step 20) = {128/236}, 2 locked for R6 and N5 -> R6C7 = 5
44. 24(4) cage at R6C2 = 4{569} (only remaining combination, cannot be 4{389} which clashes with R7C6, cannot be 4{578} which clashes with R7C9), 5,6,9 locked for R7, 6 also locked for N7 -> R78C9 = [75], R7C7 = 1, R8C7 = 6, R8C6 = 4, R7C6 = 8 (step 33a), R7C5 = 3, R7C1 = 4, R8C1 = 3, R8C2 = 2, R8C3 = 1, R8C8 = 8
45. R7C4 = 5 (hidden single in N8)
46. R456C3 (step 19) = {236} (only remaining combination), locked for C3 and N4 -> R2C3 = 8, R2C4 = 2, R2C1 = 6, R3C1 = 7
47. 7 in R4 only in R4C456, locked for N5
47a. R4C456 (step 20a) = {567} (only remaining combination), locked for R4 and N5 -> R4C3 = 2
48. R6C456 (step 43) = {128} (only remaining combination) -> R6C6 = 2, R6C45 = {18}, locked for R6 and N5 -> R6C1 = 9
and the rest is naked singles.