Thanks Ed for a challenging Anniversary Assassin!
As you will see from the comments in my walkthrough, I was stuck for a while, partly because I was trying to avoid two directional combination and permutations clashes in N5.
Then I eventually managed to find the key IOUs in N5. It's a neat cage pattern because outies for R6789 and innies for R6789 are obvious but IMHO outies-innies for R6789 are very hard to spot!
Once I'd found the key IOUs the rest was easy.
It should be noted that there is an alternative which gives the same result as the IOUs; CCCs were used in manu's walkthrough.
I'll rate A157 as Easy 1.25 although I was tempted to go higher because of the difficulty of spotting the key IOUs.
Here is my walkthrough. The later stages are possibly more optimised than many of my walkthroughs. That's because several times I had to rework after realising that I'd forgotten to do eliminations on the diagonals or spotted hidden singles that I ought to have been seen earlier.
Prelims
a) R1C12 = {14/23} b) R1C89 = {29/38/47/56}, no 1 c) R56C4 = {29/38/47/56}, no 1 d) R56C5 = {18/27/36/45}, no 9 e) R56C6 = {29/38/47/56}, no 1 f) R67C1 = {12} locked for C1, clean-up: no 3,4 in R1C2 g) R67C9 = {69/78} h) 8(3) cage in N9 = {125/134}, 1 locked for N9 i) 31(5) cage in N4 must contain 9, locked for N4
1. 45 rule on R9 2 outies R8C19 = 4 = [31], R1C1 = 4, placed for D\, R1C2 = 1, clean-up: no 7 in R1C89, no 7 in R5C6, no 5 in R6C5 1a. R8C1 = 3 -> R9C12 = 15 = {69/78} 1b. R8C1 = 1 -> R9C89 = 7 = [25/43/52], no 3 in R9C8
2. 45 rule on N5 2 outies R3C46 = 13 = {49/58/67}, no 1,2,3
3. 45 rule on N25 2 outies R1C37 = 15 = {69/78} 3a. 45 rule on N2 4 innies R12C46 = 14 = {1238/1247/1256/1346/2345}, no 9 3b. 1,4 of {1247} must be in R2C46 -> no 7 in R2C46
4. 45 rule on R1 3 remaining innies R1C456 = 14 = {239/257/356}, no 8
5. 45 rule on R1 2 outies R2C46 = 1 innie R1C5 5a. Min R2C46 = 3 -> min R1C5 = 3
6. 45 rule on N1 1 innie R1C3 = 1 outie R4C2 + 4, no 6,7,8 in R4C2
7. 45 rule on N3 1 innie R1C7 = 1 outie R4C8 + 2, R4C8 = {4567}
8. 45 rule on N1235 2 outies R4C28 = 9 = [27/36/45/54] [No immediate eliminations but used later for clean-up.]
9. Min R1C3 = 6 -> max R12C4 = 6, no 6,7,8, no 5 in R2C4
10. Min R34C2 = 5 -> max R3C1 = 7 10a. Min R3C12 = 7 -> max R4C2 = 4 (because 12(3) cage cannot be {25}5), clean-up: no 9 in R1C3 (step 6), no 6 in R1C7 (step 3), no 4 in R4C8 (step 8) 10b. Min R3C1 + R4C2 = 7 -> no 5,6,7,8,9 in R3C2 (because 12(3) cage cannot be {55}2) [Afmob’s step 2a gave these eliminations much more directly.]
11. 45 rule on C123 3 outies R127C4 = 1 innie R9C3 + 4 11a. Min R127C4 = 6 -> min R9C3 = 2 11b. 1 in N7 locked in R7C13, locked for R7
12. 1 on D\ locked in R4C4 + R5C5, locked for N5, clean-up: no 8 in R5C5
13. 45 rule on N3 3 innies R1C7 + R3C89 = 16 13a. Min R1C7 + R3C9 = 9 -> max R3C8 = 6 (hidden 16(3) cannot be [772]) 13b. Min R1C7 + R3C8 = 8 -> max R3C9 = 8
14. 45 rule on C789 2 innies R19C7 = 1 outie R7C6 + 5 14a. Min R19C7 = 9 -> min R7C6 = 4 14b. Max R7C6 = 9 -> max R19C7 = 14, max R9C7 = 6 (R19C7 cannot be [77])
[I’ve deleted two heavy combination analysis steps and an ALS block step, my original steps 15, 16 and 17, which were unnecessarily complicated.]
15. 45 rule on R6789 2 outies R5C46 = 1 innie R6C5 + 5, IOU no 5 in R5C46, clean-up: no 6 in R6C46 15a. 45 rule on R6789 2 outies R5C45 = 1 innie R6C6 + 3, IOU no 3 in R5C45, clean-up: no 8 in R6C4, no 6 in R6C5 15b. 45 rule on R6789 2 outies R5C56 = 1 innie R6C4 + 3, IOU no 3 in R5C6, clean-up no 8 in R6C6
[Before I found step 15, I’d looked at the combinations for R4C456 and found that I could eliminate a couple of them which clashed with three combinations for the two 11(2) cages in N5. I didn’t use these eliminations because I hoped for something simpler.]
16. 45 rule on R6789 3 innies R6C456 = 17 = {278/359/458} 16a. 8 of {278/458} must be in R6C5 -> no 2,4,7 in R6C5, clean-up: no 2,5,7 in R5C5 16b. 3 of {359} must be in R6C5 -> no 3 in R6C46, clean-up: no 8 in R5C46
17. 45 rule on R6789 3 outies R5C456 = 14 = {149/167} -> R5C5 = 1, placed for D/, R6C5 = 8, R5C46 = [49/76/94], no 2,6 in R5C4, no 2 in R5C6, clean-up: no 5,9 in R6C4, no 9 in R6C6, no 7 in R7C9 17a. 9 in R6 locked in R6C789, locked for N6
18. R7C1 = 1 (hidden single in R7), R6C1 = 2, clean-up: no 6 in R1C3 (step 6), no 9 in R1C7 (step 3), no 7 in R4C8 (step 8), no 9 in R5C46, no 7 in R6C46 (step 16) 18a. R6C4 = 4, placed for D/, R6C6 = 5, placed for D\, R5C46 = [76], clean-up: no 8 in R3C4, no 9 in R3C6 (both step 2), no 2 in R9C8 (step 1b) 18b. Naked pair {78} in R1C37, locked for R1, clean-up: no 3 in R1C89
19. Naked triple {239} in R4C456, locked for R4 and 27(5) cage at R3C4 -> R4C2 = 4, R1C3 = 8 (step 6), R1C7 = 7, R4C8 = 5 (step 8), R9C8 = 4, R9C9 = 3 (step 1b), placed for D\, clean-up: no 6 in R1C9, no 4 in R3C6 (step 2)
20. R1C3 = 8 -> R12C4 = 4 = [31], R1C6 = 2, R2C6 = 8 (step 3a), R3C6 = 7,. R3C4 = 6 (step 2), R3C1 = 5, R3C2 = 3 (cage sum), clean-up: no 9 in R1C89 20a. R1C8 = 6, R1C9 = 5, placed for D/, R123C5 = [954]
21. 45 rule on N4 2 remaining innies R6C23 = 8 = [71], R4C13 = [86], R4C79 = [17], R5C123 = [953], clean-up: no 8 in R7C9, no 6 in R9C2 (step 1a) 21a. Naked pair {69} in R67C9, locked for C9
22. R4C8 = 5 -> R3C89 = 9 = [18]
23. R9C6 = 1 (hidden single in R9), R4C6 = 3 (hidden single in C6), placed for D/, R4C5 = 2, R4C4 = 9, placed for D\, R3C3 = 2, R2C2 = 6, both placed for D\, R2C13 = [79], R2C8 = 2, R3C7 = 9, both placed for D/
and the rest is naked singles
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