Mike wrote:This was a difficult puzzle, because there are 2 critical moves (Afmob's steps 5a and 8a = my steps 1a and 8a) that are difficult to spot. If you don't get both of them, you'll have an exceedingly hard time solving this Assassin!
I'll agree with most of that. It took me a very long time to spot my step 2a, which is clearly the key move for this puzzle. Congratulations to Ruud for creating a puzzle that required this key move! I didn't spot the other move that Mike identified. It clearly speeds up the solution, particularly when used with his step 3, but doesn't seem to be critical.
In Afmob's walkthrough steps 4a and 4b speeded up the solution.
In his introduction to A73V1.5, Mike wrote:The A86 is not very suitable for the creation of variants, so let's wind the clock back a little...
Having spent a lot of time before I found step 2a, I know that it would have been very difficult to solve A86 without that step. Therefore any variant would almost certainly need to provide a similar step or it would risk becoming an Unsolvable.
Here is my walkthrough for A86. Not one of my best. I only modified it as required after finding step 2a. As well as taking a long time to find that step, I'd also taken time to find step 19 and a modified version of step 22, both before I found step 2a.
Thanks Mike for the comments, which I've added below, and for corrections to steps 9 and 22; I've also corrected a typo in step 32a.
Mike's comments about some of my steps, in the later message where he analyses Gary's steps, are appreciated!
Prelims
a) R1C23 = {15/24}
b) R1C78 = {18/27/36/45}, no 9
c) R34C5 = {14/23}
d) R67C3 = {29/38/47/56}, no 1
e) R67C7 = {49/58/67}, no 1,2,3
f) R78C2 = {49/58/67}, no 1,2,3
g) R78C8 = {39/48/57}, no 1,2,6
h) R567C5 = {389/479/569/578}, no 1,2
i) 8(3) cage in N9 = 1{25/34}, 1 locked for N9
j) 26(4) cage at R2C6 = {2789/3689/4589/4679/5678}, no 1
k) 26(4) cage at R4C6 = {2789/3689/4589/4679/5678}, no 1
l) 26(6) cage at R1C9 = {123479/123569/123578/124568/134567}, must contain 1
1. 45 rule on C12 2 outies R14C3 = 9 = [18/27]/{45}, no 1,2,3,6,9 in R4C3
2. 45 rule on C89 2 outies R14C7 = 3 = {12}, locked for C7, clean-up: R1C8 = {78}
2a. R1C9 + R2C89 + R3C8 can 'see' R14C7 -> CPE no 1,2 in R1C9 + R2C89 + R3C8
2b. 1 in 26(6) cage locked in R4C78, locked for R4 and N6, clean-up: no 4 in R3C5
2c. R3C9 = {12} (hidden pair R1C7 + R3C9 for N9)
2d. R345C9 = {179/269/278}, no 3,4,5
2e. 2 of {269/278} must be in R345C9 -> no 2 in R45C9
3. Killer pair 1,2 in R1C23 and R1C7, locked for R1
3a. 1 in R789C6 locked in C6, locked for N8
4. 45 rule on C1234 2 innies R19C4 = 12 = {39/48/57}, no 2,6
5. 45 rule on C6789 2 innies R19C6 = 5 = [32/41]
6. 45 rule on R1 2 innies R1C19 = 1 outie R2C5 + 13
6a. Min R1C19 = 14, no 3,4
6b. Max R1C19 = 17 -> max R2C5 = 4
7. 3 in R1 locked in R1C456, locked for N2, clean-up: no 2 in R4C5
7a. 17(4) cage at R1C4 = {1349/1358/1367/2348/2357}
7b. 1,2 only in R2C5 -> R2C5 = {12}
7c. Naked pair {12} in R23C5, locked for C5 and N2
7d. Naked pair {12} in R3C59, locked for R3
8. Hidden killer triple 7,8,9 in R1C19, R1C45 and R1C8 for R1 -> R1C19 must contain one of 7,8,9
8a. R2C6 = {12} -> R1C19 = 14,15 (step 6) = {59/68/69} (cannot be {78} because of step 8), no 7
[Simpler was "cannot be {78} which clashes with R1C8".]
9. 45 rule on R89 2 innies R8C28 =
2 outies R7C46 + 8, max R8C28 = 17 -> max R7C46 = 9, no 9, no 8 in R7C6
10. 6 in N9 locked in R7C79 + R89C7
10a. 45 rule on N9 4 innies R7C79 + R89C7 = 25 = {2689/3679/4678}, no 5, clean-up: no 8 in R6C7
11. 45 rule on N7 4 innies R7C13 + R89C3 = 14 = {1238/1247/1256/1346/2345}, no 9, clean-up: no 2 in R6C3
11a. 18(3) cage in N7 = {189/279/369/378/459/567} (cannot be {468} which clashes with R7C13 + R89C3)
12. 45 rule on N89 5 innies R7C4579 + R8C4 = 33, max R7C4579 = 30 -> min R8C4 = 3
13. 45 rule on N9 2 innies R7C79 = 2 outies R78C6 + 8
13a. Min R78C6 = 5 (cannot be {12} which clashes with R9C6, cannot be {13} which clashes with R19C6) -> min R7C79 = 13, no 2,3 in R7C9
14. 2 in N9 locked in 8(3) cage = {125}, locked for N9, clean-up: no 7 in R78C8
15. Naked triple {125} in R9C689, locked for R9, clean-up: no 7 in R1C4 (step 4)
15a. 18(3) cage in N7 (step 11a) = {189/279/369/378/459/567}
15b. 5 of {459} must be in R8C1 -> no 4 in R8C1
16. Naked triple {125} in R389C9, locked for C9
[Mike: At this point, 5 of both R9 and C9 is locked in 8(3)N9 -> R9C9 = 5.
Another example of my "killer brain" being blind, as well as the very long time I took before I saw step 2a.]
17. 17(4) cage at R1C4 (step 7a) = {1349/1358/2348/2357} (cannot be {1367} because 6,7 only in R1C5), no 6
17a. 6 in R1 locked in R1C19 = {68/69} (step 8a), no 5
[Alternatively Killer pair 4,5 in R1C23 and R1C456, locked for R1]
17b. 17(4) cage at R1C4 = {1349/2348/2357} (cannot be {1358} which clashes with R1C19 which must be {68} when R2C5 = 1)
17c. 7 of {2357} must be in R1C5 -> no 5 in R1C5
18. 17(4) cage at R7C6 = {1349/1358/1367/1457/2348/2357/2456} (cannot be {1259/1268} which clash with R9C6)
18a. Cannot be {1358} because {13} in R78C6 clashes with R19C6 and {38} in R89C7 clashes with R7C79 + R89C7
18b. -> 17(4) cage at R7C6 = {1349/1367/1457/2348/2357/2456}
18c. 8,9 of {1349/2348} must be in R89C7 because R89C7 cannot be {34} (step 10a) -> no 8,9 in R8C6
19. Killer pair 1,2 in R78C6 and R9C6, locked for C6 and N8, clean-up: no 7 in R7C6 (step 9)
20. R7C79 + R89C7 (step 10a) = {3679/4678}
20a. R789C7 cannot be {367} which clashes with R6C7 -> no 9 in R7C9, clean-up: no 4 in R7C7 (step 13a), no 9 in R6C7
20b. R789C7 cannot be {467} which clashes with R6C7 -> no 8 in R7C9
20c. No 6 in R7C6, CPE R7C6 'sees' all cells of R7C79 + R89C7
[That has been there since step 13a, when it would have eliminated 6,7, but I only spotted it here.]
21. 19(4) cage at R7C4 = {1369/1378/1459/1468/1567/2359/2368/2458/2467/3457} (cannot be {1279} because 1,2 only in R8C3)
21a. Only combination without 1,2 is {3457} -> no 6,8 in R8C3
22. 1,2 in N5 locked in R123C4, locked for 16(4) cage -> no 1,2 in R5C3
22a. 16(4) cage at R4C4 = {1249/1258/1267}, no 3
23. R7C79 + R89C7 (step 20) = {3679/4678}
23a. 17(4) cage at R7C6 (step 18b) = {1349/1367/1457/2348/2357/2456}
23b. R89C7 cannot be {67} because R78C6 = {13} clashes with R19C6
23c. -> R89C7 must contain 3 or 4 -> no 4 in R7C9
24. Naked quad {6789} in R1457C9, locked for C9
25. 7 in C9 locked in R457C9, CPE no 7 in R56C8
26. Killer pair 6,7 in R45C9 and R7C9, locked for C9
26a. R1C1 = 6 (hidden single in R1)
[Mike: You could have dispensed with step 25 and had KP on {67} eliminating 6,7 from R56C8, too.
Wow! A CPE killer pair. I’ve never thought of that and don’t remember it ever appearing on the forum.]
27. 26(6) cage at R1C9 = {123479/123569/123578/124568} (cannot be {134567} because R1C9 only contains 8,9)
27a. 1,2 only in R4C78 -> R4C78 = {12}, locked for R4 and N6
27b. R1C9 = {89} -> no 8,9 in R23C8
27c. 7 in C8 locked in R123C8, locked for N3
28. 16(4) cage at R4C4 (step 22a) = {1249/1258/1267}
28a. 1,2 only in R56C4 -> R56C4 = {12}
29. 21(4) cage at R5C8 = {3468/3567} (cannot be {3459} because R7C9 only contains 6,7), no 9
30. 9 in C8 locked in R78C8 -> R78C8 = {39}, locked for C8 and N9, clean-up: no 4 in R6C7
31. 21(4) cage at R5C8 = {3468/3567} -> R6C9 = 3, R2C9 = 4, clean-up: no 8 in R7C3
32. 26(6) cage at R1C9 (step 27) = {124568} (only remaining combination) -> R1C9 = 8, R1C78 = [27], R3C9 = 1, R4C78 = [12], R34C5 = [23], R2C5 = 1, clean-up: no 4 in R1C23, no 6 in R45C9 (step 2d)
32a. Naked pair {56} in R23C8, locked for C8 and N3 ->
R9C8 = 1, R9C6 = 2, R89C9 = [25]
32b. Naked pair {79} in R45C9, locked for C9 and N6 -> R7C9 = 6, clean-up: no 5 in R6C3, no 7 in R8C2
32c. Naked pair {48} in R56C8, locked for N6
33. 1,4 locked in 17(4) cage at R7C6 (step 18) = {1457} (only remaining combination), no 3,6,8
33a. Naked pair {47} in R89C7, locked for C7 and 17(4) cage -> R7C7 = 8, R6C7 = 5, R5C7 = 6, clean-up: no 7 in R4C4 (step 22a), no 5 in R8C2
33c. Naked pair {15} in R78C6, locked for C6 and N8
34. R5C7 = 6 -> R456C6 = 20 = {479} (only remaining combination), locked for C6 and N5 -> R1C6 = 3, clean-up: no 4,9 in R5C3 (step 22a)
35. Naked pair {15} in R1C23, locked for R1 and N1
35a. Naked pair {49} in R1C45, locked for N2
35b. Naked pair {68} in R23C6, locked for N2
35c. Naked pair {57} in R23C4, locked for C4 and 23(4) cage, clean-up: no 8 in R5C3 (step 22a)
36. R5C5 = 5 (hidden single in C5), R5C3 = 7, R4C4 = 6 (step 22a), R45C9 = [79], R5C6 = 4, R46C6 = [97], R56C8 = [84], R6C5 = 8, R7C5 = 7 (cage sum), clean-up: no 3,4 in R7C3, no 6 in R8C2
37. 6 in C5 locked in R89C5, 20(4) cage in N8 = {2369/2468}
37a. Only other 8,9 in R8C4 -> R8C4 = {89}
38. R23C4 = {57} = 12 -> R23C3 = 11 = [29]/{38}, no 4, no 9 in R2C3
39. 19(4) cage at R7C4 (step 21) = {1369/1468} (cannot be {1459} because 1,5 only in R8C3) -> R8C3 = 1
39a. R7C4 = {34} -> no 3,4 in R9C3
40. R78C6 = [15], R1C23 = [15], R7C3 = 2, R6C3 = 9
41. Naked pair {38} in R23C3, locked for C3 and N1 -> R4C3 = 4, R9C3 = 6, R4C2 = 5 (cage sum), R4C1 = 8, clean-up: no 8 in R8C2
42. R4C1 = 8 -> R35C1 = 6 = [42], R56C4 = [12], R5C2 = 3, R6C12 = [16], R7C1 = 5
43. Naked pair {49} in R78C2, locked for C2 and N7 -> R23C2 = [27], R2C1 = 9, R9C2 = 8, R23C7 = [39], R23C3 = [83], R23C6 = [68], R23C4 = [75], R23C8 = [56]
44. R8C4 = 8 (hidden single in C4), R7C4 = 4 (step 39)
and the rest is naked singles