Assassin 115 deserves more than one walkthrough.
I struggled and had to resort to step 21 that's somewhere between a chain and a hypothetical. There's also some combination analysis but not too heavy; step 34 doesn't use hypotheticals, I've just written it that way to avoid a long string of permutations which would only lead to one candidate elimination.
I'll rate A115 as 1.5.
Here is my walkthrough. Thanks Afmob for the alternative (better) way to do step 35 and to Ed for an interesting (fun) alternative to step 22.
Prelims
a) R23C1 = {79}, locked for C1 and N1
b) R23C9 = {29/38/47/56}, no 1
c) R4C12 = {16/25/34}, no 7,8,9
d) R4C89 = {49/58/67}, no 1,2,3
e) R6C12 = {16/25/34}, no 7,8,9
f) R6C89 = {39/48/57}, no 1,2,6
g) R78C1 = {18/36/45}, no 2
h) R78C9 = {19/28/37/46}, no 5
i) 7(3) cage in N1 = {124}, locked for N1
j) 11(3) cage at R3C3 = {128/137/146/236/245}, no 9
k) 22(3) cage in N7 = {589/679}, 9 locked for N7
l) 9(3) cage in N9 = {126/135/234}, no 7,8,9
m) R1C6789 = {1238/1247/1256/1346/2345}, no 9
n) 14(4) cage in N8 = {1238/1247/1256/1346/2345}, no 9
o) R9C6789 = {2789/3689/4589/4679/5678}, no 1
p) 42(7) cage at R3C5 = {3456789}, no 1,2
1. 45 rule on R1 1 innie R1C5 = 9
1a. 9 in 42(7) cage locked in R5C46, locked for R5 and N5
1b. 9 in N4 locked in R46C3, locked for C3
2. 45 rule on R9 1 innie R9C5 = 1
3. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 8 -> R1C6 = 1, R3C7 = 9, R23C1 = [97], clean-up: no 2,4 in R2C9, no 2 in R3C9
4. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 -> R1C4 = {3568}
4a. Naked quad {3568} in R1C1234, locked for R1
4b. Naked triple {247} in R1C789, locked for N3
5. R3C7 = 9 -> R34C6 = 7 = {25/34}, no 6,7,8
6. 45 rule on R123 1 innie R3C5 = 2 outies R4C46 + 3
6a. Min R4C46 = 3 -> min R3C5 = 6
6b. Max R3C5 = 8 -> max R4C46 = 5, no 5,6,7,8, no 4 in R4C4, clean-up: no 2 in R3C6 (step 5)
7. 45 rule on R1234 4(3+1) innies R3C5 + R4C357 = 28 = 6{589}/6{679}/8{389}/8{479}/8{569}/8{578}, no 1,2 in R4C37
8. 9 in N6 must be in R46C89
8a .Combined cage R46C89 = 25 = {3589/3679/4579}
8b. 3,9 of {3589} must be in R6C89, 4,9 of {4579} must be in R4C89 -> no 4,8 in R6C89
9. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 4, no 6,7,8 in R7C3, no 2,3,4 in R9C4
10. Max R7C3 = 5 -> min R67C4 = 11, no 1 in R6C4
10a. R4C4 = 1 (hidden single in C4), clean-up: no 6 in R4C12
10b. R4C4 = 1 -> R3C34 = 10 = [64/82], R1C4 = {68} (step 4)
10c. Naked pair {68} in R3C35, locked for R3, clean-up: no 3,5 in R2C9
10d. Naked pair {68} in R1C4 + R3C5, locked for N2, CPE no 6,8 in R5C4
11. 45 rule on R12 3 remaining outies R3C289 = 8 = {125/134}
11a. 2,4 only in R3C2 -> R3C2 = {24} -> R3C8 = 1
11b. Naked pair {24} in R3C24, locked for R3, clean-up: no 3 in R4C6 (step 5)
11c. Killer pair 2,4 in R4C12 and R4C6, locked for R4, clean-up: no 9 in R4C89
12. R4C3 = 9 (hidden single in R4)
12a. 9 in N6 locked in R6C89 = {39}, locked for R6 and N6, clean-up: no 4 in R6C12
12b. 3 in N5 locked in R4C5 + R5C456 for 42(7) cage, no 3 in R7C5
13. 45 rule on C6789 3 innies R258C6 = 20 = {389/479/569/578}, no 2
13a. 3 of {389} must be in R2C6 -> no 3 in R58C6
13b. 9 of {479/569} must be in R5C6 -> no 4,6 in R5C6
13c. 6 in 42(7) cage locked in R34567C5, locked for C5
14. 45 rule on C1234 3 innies R258C4 = 14 = {239/257/347/356} (cannot be {248} which clashes with R3C4), no 8
15. 45 rule on N7 3 outies R679C4 = 20 = {389/479/569/578}, no 2
15a. 2 in N5 locked in R46C6, locked for C6
16. 45 rule on N9 1 outie R9C6 = 1 innie R7C7, no 1,2 in R7C7, no 9 in R9C6
16a. 45 rule on N9 3 outies R679C6 = 17 = {269/278/368/467} (cannot be {359/458} which clash with R258C6), no 5, clean-up: no 5 in R7C7 (step 16)
17. R3C5 + R4C357 (step 7) = 6{589}/6{679}/8{389} (cannot be 8{569} which clashes with R4C89)
17a. 6{679} must be 6[976] -> no 6 in R4C5, no 7 in R4C7
17b. 6 in R4 locked in R4C789, locked for N6
18. 9(3) cage in N9 = {126/135/234}
18a. 1 of {126/135} must be in R8C7 -> no 5,6 in R8C7
19. 14(4) cage in N8 = {1238/1247/1256/1346}
19a. 8 of {1238} must be in R8C6 -> no 8 in R8C5
19b. 8 in C5 locked in R34567C5, locked for 42(7) cage, no 8 in R5C6
20. R258C6 (step 13) = {389/479/569/578}
20a. 6,8 of {569/578} must be in R8C6 -> no 5 in R8C6
21. R3C5 = R4C46 + 3 (step 6), R4C4 = 1 -> R3C5 = R4C6 + 4 -> R3C5 + R4C6 = [62/84]
21a. 45 rule on R789 2 outies R6C46 = 1 innie R7C5 + 6
21b. If R3C5 = 6 => no 6 in R567C6 => R6C46 must contain 6 => R6C46 = {46/67/68} (cannot be {56} which clashes with R6C12) => no 5,6 in R7C5
21c. If R3C5 = 8 => R4C6 = 4 => R7C5 = 4 (only remaining position in 42(7) cage)
21d. -> no 5,6 in R7C5
21e. 5 in 42(7) cage locked in R4C5 + R5C456 + R6C5, locked for N5
22. 45 rule on R
6789 4(3+1) innies R6C357 + R7C5 = 20 = 4{178}/4{457}/7{148}/8{147} (cannot be 4{268}/7{157}/7{256}/8{156}/8{246} which clash with R6C12, cannot be 7{247} because R6C12 then cannot be {25} so 5 cannot be placed in R6), no 2,6
[Ed pointed out an interesting alternative.
R3C5 = 6 => 6 in N5 must be in R6C46
R3C5 = 8 => R4C6 = 4(step 21) => R6C6 = 2
-> R6C46 must have 2 or 6
Killer pair 2,6 in R6C12 and R6C46, locked for R6.]23. 2 in N6 locked in R5C789, locked for R5
23a. Combined cage R46C12 must contain 2 = {1256/2345}, 5 locked for N4
24. 16(3) cage at R6C4 = {169/178/349/358/457} (cannot be {259} because no 2,5,9 in R6C4, cannot be {268/367} because they would make R679C4 {68}6/{67}7), no 2, clean-up: no 6 in R9C4 (step 9)
24a. 9 of {169} must be in R7C4 -> no 6 in R7C4
25. 2 in N7 locked in R9C123, locked for R9
25a. R9C1234 = {2349/2358/2457} (cannot be {2367} because R7C3 = 3 when R9C4 = 7), no 6
26. 2 in N8 locked in R8C45, locked for R8, clean-up: no 8 in R7C9
26a. 9(3) cage in N9 = {126/135/234}
26b. 6 of {126} must be in R8C8 -> no 6 in R7C8
26c. 2 of {234} must be in R7C8 -> no 4 in R7C8
27. 14(4) cage in N8 = {1238/1247/1256}
27a. 6 of {1256} must be in R8C6 -> no 6 in R8C4
27b. 6 in N8 locked in R789C6, locked for C6
28. R258C4 (step 14) = {239/257/347}
28a. R679C4 (step 15) = {389/569/578} (cannot be {479} which clashes with R258C4), no 4
29. 6 in R9 locked in R9C6789
29a. R9C6 = R7C7 (step 16) -> 6 locked in R7C7 + R9C789, locked for N9, clean-up: no 4 in R78C9
29b. 9(3) cage in N9 = {135/234}, 3 locked for N9, clean-up: no 7 in R78C9, no 3 in R9C6 (step 16)
30. 3 in R9 locked in R9C123, locked for N7, clean-up: no 6 in R78C1, no 7 in R9C4 (step 9)
30a. 6 in N7 locked in 22(3) cage = {679}, locked for N7
31. R6C46 = R7C5 + 6 (step 21a)
31a. R7C5 = {478} -> R6C46 = [64/82/67/68], no 7 in R6C4
31b. Naked pair {68} in R16C4, locked for C4, clean-up: no 4 in R7C3 (step 9)
32. Killer pair 1,5 in R78C1 and R7C3, locked for N7
33. 16(3) cage at R6C4 (step 24) = {169/178/358}
33a. 5 of {358} must be in R7C3 -> no 5 in R7C4
34. R46C12 (step 23a) = {1256/2345}
34a. R46C1 cannot be {25} which clashes with R4C12 and R6C12, R46C1 cannot be [35] because R46C2 = [42] clashes with R3C2, R46C1 cannot be [56] because R46C2 = [21] clashes with R23C3
34b. 45 rule on C1 5 innies R14569C1 = {12368/23456}
34c. 6 of {12368} must be in R1C1 (R56C1 cannot be {16} which clashes with R6C12) -> no 8 in R1C1
Consider key points for the permutations of {23456}
34da. If 2 is in R46C1 => no 5 in R4C1 (or R6C1)
34db. If 2 is in R9C1 and 3 in R1C1, R46C1 cannot be [56] => no 5 in R4C1
34dc. If 2 is in R9C1 and 6 in R1C1 => R6C1 = 5
34dd. If 5 is in R1C1 => no 5 in R4C1
34e. -> no 5 in R4C1, clean-up: no 2 in R4C2
35. 45 rule on C9 5 innies R14569C9 = 24 = {13479/13569/23478} (cannot be {12678/14568/24567} because R6C9 only contains 3,9, cannot be {12489/12579/13578/23469} which clash with R78C9, cannot be {23568} which clashes with R23C9), 3 locked for C9 -> R3C9 = 5, R2C9 = 6, R3C6 = 3, R4C6 = 4 (step 5), clean-up: no 3 in R4C12, no 7,8 in R4C8
[This step has been available for a long time but I only spotted it after step 34b and then only realised how powerful it was after I’d eliminated the non-valid combinations, some of which only worked after R78C9 had been reduced to two combinations in step 29.]
[Afmob pointed out that an alternative here is hidden killer triple 3,8,9 for C9. R6C9 = {39}, R78C9 must contain one of 8,9 -> R23C9 cannot be [83] -> R23C9 = [65]. This is simpler than my step but possibly harder to spot.]36. R4C12 = [25] -> R4C789 = [867], R4C5 = 3, R2C78 = [38]
36a. Naked pair {16} in R6C12, locked for R6 and N4 -> R6C4 = 8, R1C4 = 6, R3C5 = 8, R3C3 = 6, R3C4 = 4 (step 10b), R3C2 = 2, R8C3 = 7, R6C3 = 4, R6C7 = 5, R6C56 = [72], R7C5 = 4, R2C23 = [41], R7C3 = 5, R7C4 = 3 (step 33), R9C4 = 9 (step 9), R5C456 = [569], R8C45 = [25], R2C456 = [725], R8C6 = 6 (step 27), R78C2 = [69], R6C12 = [61], R7C67 = [87], R9C6 = 7, R7C1 = 1, R7C89 = [29], R8C1 = 8
and the rest is naked singles
udosuk wrote:
According to SS it is as hard as
Para's Assassin 113. But there is a key move in the puzzle which if you can spot will make a huge breakthrough in the solving. And then I've found 2 different approaches to the "critical step", which I'll be glad if you guys can find a 3rd.
I'll be interested to see what that is. I don't think I found it.
Now to try V0.9 and see if it is easier. Don't know whether I'll try V2, certainly not before I try A113 and/or A113-Lite which I haven't yet started.