Many thanks Afmob for a really challenging V2. I can recommend anyone who hasn't yet done so to try it!
udosuk's walkthrough has some very impressive permutation analysis. After going through the walkthrough, but skipping some of the detail in the analysis, it looked like 1.5 (Hard) rather than 1.5 (Easy) to me.
Thanks udosuk for giving eliminations along the diagonals! I've also done that. It makes it much easier for those of us who do manual eliminations.
Afmob wrote:
Here is the easier path I've found with step 6a being the most important move.
IMHO step 7a is equally important. I missed that in my solution path and had to work really hard before my first placement.
Before that my path was very similar to Afmob's. After my step 14 I noticed the first part of Afmob's step 6a but missed the implication of R7C3 being {123}. However I got the same result, and a bit more, with my step 15.
I'll rate A119 V2 at 1.5 the way I solved it because there were times when it was very difficult to find the next move; however none of my steps are harder than 1.25. Maybe there's too much tendency to rate puzzles by hardest move. Mike's original rating definitions appear to me to refer more to puzzle difficulty than to hardest move.
Having said that, Afmob's walkthrough using both important steps 6a and 7a, is definitely 1.25 assuming that he didn't find it hard to spot those two steps.
If Ed's internet problems hadn't delayed him posting A120 I might not have been able to finish this puzzle yesterday evening. It took me some time to find steps 23a and 24a.
Here is my walkthrough for A119 V2. I'll leave it in TT for a couple of days. Step 37 was a fun move, the only one where I managed to use the diagonals for other than eliminations.
Thanks Afmob for pointing out the omission in step 25a and that step 15 needed better explanation; I've added a preliminary part for step 15. While checking this feedback I also found an omission in step 10a.
Prelims
a) R8C23 = {29/38/47/56}, no 1
b) R8C78 = {18/27/36/45}, no 9
c) R9C34 = {14/23}
d) R9C67 = {49/58/67}, no 1,2,3
e) 26(4) cage at R1C6 = {2789/3689/4589/4679/5678}, no 1
f) 26(4) cage in N9 = {2789/3689/4589/4679/5678}, no 1
g) 28(7) cage in N5 = {1234567}, locked for N5
h) 33(5) cage at R3C2 = {36789/45789}, no 1,2
1. Naked pair {89} in R5C46, locked for R5
2. 45 rule on N4 2 innies R46C3 = 12 = {39/48/57}, no 6
2a. 45 rule on N4 3 outies R37C2 + R5C4 = 21 = {489/579/678}, no 3
2b. 33(5) cage at R3C2 = {36789/45789}, CPE no 7 in 18(4) cage in R456C2
3. 45 rule on N6 2 innies R46C7 = 10 = {19/28/37/46}, no 5
3a. 45 rule on N6 3 outies R37C8 + R5C6 = 17 = {179/269/278/359/368/458} (cannot be {467} because R5C6 only contains 8,9)
3b. R5C6 = {89} -> no 8,9 in R37C8
4. 45 rule on R123 2 innies R3C28 = 12 = [57/75/84/93], no 6, no 4 in R3C2, no 1,2 in R3C8
5. 45 rule on R789 2 innies R7C28 = 9 = [45/54/63/72/81], no 9 in R7C2, no 6,7 in R7C8
5a. R37C2 + R5C4 (step 2a) = {489/579/678}
5b. 4,6 of {489/678} must be in R7C2 -> no 8 in R7C2, clean-up: no 1 in R7C8 (step 5)
6. 45 rule on N9 3 innies R7C78 + R9C7 = 10 = {127/136/145/235}, no 8,9, clean-up: no 4,5 in R9C6
6a. 1 of {127/136/145} must be in R7C7 and 5 of {235} must be in R9C7 -> no 4,5,6,7 in R7C7
7. 45 rule on C1 2 outies R19C2 = 8 = {17/26/35}, no 4,8,9
8. 45 rule on C9 2 outies R19C8 = 9 = {18/27/36/45}, no 9, no 8 in R1C8
8a. 9 in N9 locked in R789C9, locked for C9
9. 45 rule on N7 3 innies R7C23 + R9C3 = 17 = {179/269/278/359/368/458/467}
9a. R9C3 = {1234} -> no 1,2,3,4 in R7C23, clean-up: no 5 in R7C8 (step 5)
9b. 5 of {359/458} must be in R7C2 -> no 5 in R7C3
9c. R7C78 + R9C7 (step 6) = {127/136/145/235}
9d. 5,6,7 only in R9C7 -> R9C7 = {567}, clean-up: no 9 in R9C6
10. R37C2 + R5C4 (step 2a) = {579/678}, 7 locked in R37C2 for C2 and for 33(5) cage at R3C2, clean-up: no 1 in R19C2 (step 7), no 5 in R46C3 (step 2), no 4 in R8C3
10a. 8,9 of {579
/678} must be in R5C4 -> no 8,9 in R3C2, clean-up: no 3,4 in R3C8 (step 4)
10b. Naked pair {57} in R3C28, locked for R3
10c. Killer pair 5,6 in R19C2 and R37C2, locked for C2, clean-up: no 5,6 in R8C3
11. 18(4) cage in N4 = {1269/1278/1359/1458} (cannot be {1368/2349/3456} which clash with R46C3, cannot be {1467/2367/2457} because 5,6,7 only in R5C3, cannot be {2358} which clashes with R19C2), 1 locked in R456C2, locked for C2 and N4
11a. 5,6,7 only in R5C3 -> R5C3 = {567}
11b. Killer pair 8,9 in R46C2 and R46C3, locked for N4
12. 45 rule on N1 2 innies R3C23 = 1 outie R1C4 + 9
12a. Max R3C23 = 16 -> max R1C4 = 7
12b. Min R1C4 = 1 -> min R3C23 = 10, no 1,2 in R3C23
12c. R3C23 cannot total 12 -> no 3 in R1C4
13. 45 rule on N3 2 innies R3C78 = 1 outie R1C6 + 3
13a. Min R3C78 = 6 -> min R1C6 = 3
13b. R3C78 cannot total 12 -> no 9 in R1C6
13c. Max R1C6 = 8 -> max R3C78 = 11, no 8,9 in R3C7
14. 45 rule on R12 4 outies R3C1379 = 1 innie R2C5 + 8
14a. Min R3C1379 = 10 -> min R2C5 = 2
15. R7C78 = {12/13/14/23} (steps 9c and 9d)
15a. R7C456 cannot total 8,9 (because {125/134/126/135/234} clash with R7C78) -> no 8,9 in R8C5
16. 9 in N8 locked in R7C456 + R8C46 + R9C5, CPE no 9 in R7C3, clean-up: no 1 in R9C3 (step 9), no 4 in R9C4
16a. 1 in N7 locked in R789C1, locked for C1
17. 1 in N1 locked in R12C3, locked for 15(4) cage at R1C3, no 1 in R1C4
17a. R3C23 = R1C4 + 9 (step 12), min R1C4 = 2 -> min R3C23 = 11, min R3C3 = 4
18. R7C23 + R9C3 (step 9) = {368/458/467} (cannot be {278} which clashes with R8C23), no 2, clean-up: no 3 in R9C4
18a. Killer pair 3,4 in R46C3 and R9C3, locked for C3, clean-up: no 8 in R8C2
19. Hidden killer pair 8,9 in C1, 21(4) cage in N1 and 17(4) cage can each only contain one of 8,9 -> R123C1 and R789C1 must each contain one of 8,9
20. 17(4) cage in N7 = {1259/1358} (cannot be {1349} which clashes with R9C3, cannot be {1268} which clashes with R7C23 + R9C3, cannot be {1367/1457} which don’t contain 8 or 9), no 4,6,7, 5 locked for N7, clean-up: no 2 in R1C2 (step 7), no 4 in R7C8 (step 5)
20a. 6 in N7 locked in R7C23, locked for R7
21. R37C8 + R5C6 (step 3a) = {278/359}
21a. R46C7 (step 3) = {19/28/46} (cannot be {37} which clashes with R37C8 + R5C6), no 3,7
22. 17(4) cage in N8 = {1349/1358/1367/1457/2348/2357/2456} (cannot be {1259/1268} which clashes with R9C4)
22a. Killer pair 1,2 in 17(4) cage and R9C4, locked for N8
23. 45 rule on C123 3 outies R159C4 = 2 innies R37C3
23a. Min R37C3 = 13 -> min R159C4 = 13 cannot be [292] -> no 2 in R1C4
24. Min R1C4 = 4 -> max R1C3 + R2C23 = 11, no 9
24a. 15(4) cage at R1C3 = {1248/1257/1347} (cannot be {1356} which clashes with R1C2 which “sees” all cells of the 15(4) cage), no 6
24b. 4 of {1248} must be in R1C4, 3 of {1347} must be in R2C2 -> no 4 in R2C2
25. 4 in N1 locked in R123C1, locked for C1
25a. R456C1 = {267/357},
7 locked for C1 and N4
26. 1 in N2 locked in R1C5 + R2C46 + R3C456, CPE no 1 in R3C7
26a. R3C78 = R1C6 + 3 (step 13), min R3C78 = 7 -> min R1C6 = 4
27. R3C1379 = R2C5 + 8 (step 14), min R3C1379 = 12 -> min R2C5 = 4
27a. 2,3 in N2 locked in R1C5 + R2C46 + R3C456, CPE no 2,3 in R3C7
28. R3C78 = R1C6 + 3 (step 13)
28a. Min R3C78 = 9 -> min R1C6 = 6
28b. R3C78 cannot total 10 (it must be an odd number) -> no 7 in R1C6
28c. R1C6 = {68} -> R3C78 = 9,11 = [45/47/65]
29. 1 in N3 locked in 16(4) cage = {1258/1267/1348} (cannot be {1357} which clashes with R3C8, cannot be {1456} which clashes with R3C7)
29a. Killer pair 4,5 in 16(4) cage and R3C78, locked for N3
[Note that when 16(4) cage = {1267}, R3C78 = [45] provides both candidates for the killer pair.]
30. Min R3C37 = 10 -> max R1C5 + R2C46 = 10, no 8,9
31. 9 in N2 locked in 25(4) cage = {1789/3589/3679} (cannot be {2689} which clashes with R1C6, cannot be {4579} because 5,7 only in R2C5), no 2,4
31a. 5,7 only in R2C5 -> R2C5 = {57}
31b. Killer pair 6,8 in 25(4) cage and R1C6, locked for N2
31c. 9 in N2 locked in R3C456, locked for R3
32. R3C1379 = R2C5 + 8 (step 14)
32a. 2,4 in R3 locked in R3C179
32b. R2C5 = {57} -> R3C1379 = 13,15 = {1246/1248/2346}
32c. R3C3 = {68} -> no 6,8 in R3C179 -> R3C7 = 4, locked for D/, clean-up: no 6 in R46C7 (step 3), no 7 in R8C3, no 5 in R8C8
32d. R1C4 = 4 (hidden single in N2), R2C1 = 4 (hidden single in R2), R1C1 = 9 (hidden single in N1), locked for D\
33. 16(4) cage in N3 (step 29) = {1258/1267}, no 3, 2 locked for N3
33a. Killer pair 5,7 in 16(4) cage and R3C8, locked for N3
34. 17(4) cage in N7 (step 20) = {1358}, locked for N7 -> R9C3 = 4, R9C4 = 1, clean-up: no 6 in R1C2 (step 7), no 8 in R46C3 (step 2)
35. R1C12 = [94] = 13 -> R1C2 + R3C1 = 8 = [53], R9C2 = 3, R3C2 = 7, R7C2 = 6, R7C3 = 7, locked for D/, R5C4 = 8 (step 10), R5C6 = 9, R3C8 = 5, R7C8 = 3 (step 5), clean-up: no 5 in R456C1 (step 25a), no 1 in R46C7 (step 3), no 6 in R8C78
36. Naked pair {28} in R46C7, locked for C7 and N6 -> R7C7 = 1, locked for D\, R9C7 = 6 (step 6), R9C6 = 7, R12C7 = [39], clean-up: no 7,8 in R8C8
37. 3 on D\ locked in R4C4 + R5C5 + R6C6, 3 on D/ locked in R4C6 + R5C5 + R6C4 -> R5C5 = 3
38. Naked pair {39} in R46C3, locked for C3 and N4 -> R8C3 = 2, R8C2 = 9, R8C8 = 4, locked for D\, R8C7 = 5
39. Naked pair {18} in R12C3, locked for N1 -> R2C2 = 2, R3C3 = 6, both locked for D\, R5C37 = [57], R6C6 = 5, R4C4 = 7, R9C9 = 8, R9C1 = 5, R9C58 = [92], R1C8 = 7 (step 8)
40. R2C8 = 8 (hidden single on D/), R1C6 = 6, R6C4 = 6 (hidden single on D/)
41. R3C37 = [64] = 10 -> R1C5 + R2C46 = 10 = [253], R1C9 = 1, locked for D/
and the rest is naked singles.