I got stuck on this puzzle when I first tried it and came back to it yesterday after getting a hint from Peter.
“Got stuck? Here is a vague hint in tiny font.
Hint: Forget about innies/outies (for now). The key is in another approach.
Not enough? Here's a more specific hint.
Hint: Look at N4 and N6 and there at c1 and c9. Look at the possible combinations and possible killer pairs.”
That solved it. The extra work prompted by these hints is mainly in steps 24 and 31. I don’t think I’d been making full use of R4C19 and R6C19 before. Thanks Peter!
I think the key step that I used to break it open is sufficiently different from Ed's method so here is my walkthrough.
Clean-up is used in various steps, using the combinations in steps 1 to 10 for further eliminations from these two cell cages; it is also used for the two cell split sub-cages that are produced by applying the 45 rule to R4, R6, N1, N3, N7 and N9. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.
1. R1C23 = {49/58/67}, no 1,2,3
2. R1C78 = {29/38/47/56}, no 1
3. R34C5 = {16/25/34}, no 7,8,9
4. R5C23 = {59/68}
5. R5C78 = {29/38/47}(not {56} which would clash with R5C23), no 1,5,6
6. R67C5 = {49/58/67}, no 1,2,3
7. R9C23 = {14/23}
8. R9C78 = 10(2), no 5
9. 9(3) cage in N1 = {126/135/234}, no 7,8,9
10. 19(3) cage in N12, no 1
11. 20(3) cage in N3 = {389/479/569/578}, no 1,2
12. 10(3) cage in N254 = {127/136/145/235}, no 8,9
13. 9(3) cage in N458 {126/135/234}, no 7,8,9
14. 10(3) cage in N658 = {127/136/145/235}, no 8,9
15. 21(3) cage in N7 = {489/579/678}, no 1,2,3
16. 20(3) cage in N89 = {389/479/569/578}, no 1,2
17. 13(4) cage in N2 = 1{237/246/345}, no 8,9, 1 locked for N2, clean-up no 6 in R4C5
18. 25(4) cage in N8 = {1789/2689/3589/3679/4579/4678}, must contain at least two of 7,8,9 and contains three of 5,6,7,8,9
19. 45 rule on R5 2 innies R5C19 = 3 = {12}, locked for R5, clean-up: no 9 in R5C78
20. 45 rule on R1234 2 innies R4C19 = 14 = {59/68}
21. Valid combinations for R456C1 with R5C1 = {12} are {159/168/249/267} (cannot be {258} which would clash with R5C23), no 1,2,3 in R6C1, R456C1 must contain one of 4,5,6 and one of 7,8,9
22. Valid combinations for R456C9 with R5C9 = {12} are {139/157/238/247/256} (cannot be {148} which would clash with R5C78), no 1,2 in R6C9, R456C9 must contain one of 3,4,5 and one of 6,7,8,9
23. 12(3) in R789C9 must have 1 or 2, valid combinations are {138/147/156/237/246} ({129} not valid because it has both 1 and 2), no 9, must contain one of 3,4,5 and one of 6,7,8
24. 45 rule on R6789 2 innies R6C19 = 11 = {38/47/56}, clean-up: no 9 in R6C1, no 8,9 in R6C9
24a. Only combination in R456C1 with 5 is {159} -> no 5 in R4C1, clean-up: no 9 in R4C9 -> no {139} in R456C9
24b. From step 22, no {247} in R456C9 because 4,7 both in same cell -> no 4 in R6C9, clean-up: no 7 in R6C1
24c. R456C1 = {159/168/249} [8/9], R5C23 = {59/68} [8/9], killer pair 8,9 for N4
24d. 45 rule on N4 4 innies R4C23 + R6C23 = 16, must contain 1 or 2 and must contain 3,7 -> no 6 in R4C23 + R6C23 [Edited. 3 added]
25. 45 rule on N1 2 outies R2C4 + R4C2 = 13 = {49/58/67}, no 1,2,3
25a. Clean-up: no 4,5,7 in R2C4
26. 45 rule on N3 2 outies R2C6 + R4C8 = 17 = {89} -> no 8,9 in R2C8 and R4C6
26a. Killer pair 8,9 in R4C189 for R4
27. 45 rule on N7 2 outies R6C2 + R8C4 = 8 = {17/26/35/44}, no 8,9, clean-up: no 2 in R8C4
28. 45 rule on N9 2 outies R6C8 + R8C6 = 14 = {59/68/77}, no 1,2,3,4
29. 9 in C9 locked in R123C9 = 9{38/47/56}, clean-up: no 2 in R1C78
30. 9 in N6 locked in R46C8, locked for C8, clean-up: no 1 in R9C7
31. Remaining combinations for R456C9 are {157/238/256}. It looks like the 3 cages in C9 are meant to be effectively 9-11, 2-11 and 1-11 so let’s look at a contradiction move to see if {157} can be eliminated.
If R456C9 = {157}, R4C9 = 5, R5C9 = 1, R6C9 = 7 => R4C1 = 9, R5C1 = 2, R6C1 = 4 => R5C23 = {68} which clashes with R5C78 = {38} so R456C9 cannot be {157}
31a. R5C9 = 2, R5C1 = 1
31b. R46C1 = {68}/[95]
31c. R46C9 = {56}/[83]
31d. R123C9 = 9{38/47/56}
31e. R789C9 = 1{38/47/56}, 1 locked for N9, clean-up: no 9 in R9C7
32. R123C1 = {234}, locked for C1 and N1, clean-up: no 9 in R1C23
33. 7 in C1 locked in R789C1, locked for N7, R789C1 = 7{59/68}
34. 1 in N1 locked in R23C2, locked for C2, clean-up: no 4 in R9C3
34a. Only valid combination for R234C2 = {179} -> R4C2 = 7, R23C2 = {19}, locked for C2 and N1, clean-up: no 6 in R1C3, no 5 in R5C3
35. R9C2 = {234} -> R678C2 must contain two of 2,3,4 -> only possible combination = {348}, locked for C2 with 8 in R78C2, locked for N7 -> R9C2 = 2, R9C3 = 3, clean-up: no 5 in R1C3, no 6 in R5C3, no 6 in R789C1 = {579}, locked for C1 and N7, no 7,8 in R9C78 -> R9C78 = {46}, locked for R9 and N9, clean-up: no 5,7 in R789C9
36. R46C1 = {68}, locked for N4 -> R5C23 = [59], R1C23 = [67], clean-up: no 4,5 in R1C78 -> R1C78 = {38}, locked for R1 and N3
37. 1 in N2 locked in R1C456
38. R1C9 = 9 (hidden single in R1)
39. R789C9 = {138}, locked for C9 and N9 -> R46C9 = {56}, locked for C9 and N6 -> R23C9 = {47}, locked for N3
40. R78C2 = {48}, locked for C2 and N7 -> R6C2 = 3
40a. R78C3 = {16}, locked for C3 -> R8C4 = 5, clean-up: no 8 in R6C5
41. 6 in N5 locked in R456C5 = 6{38/47}, 6 locked for N5, clean-up: no 7 in R7C5
[Alternatively could have used X-wing with 6s in R46C19]
42. R23C3 = {58}, locked for C3 -> R2C4 = 6, clean-up: no 1 in R4C5 [Edited. "locked for C3" added for clarity"]
43. Only remaining combination for 9(3) cage in N458 = {234} -> R6C34 = {24}, locked for R6, R7C4 = 3, clean-up: no 9 in R7C5
44. Only remaining combination for 10(3) cage in N254 is R3C4 = 7, R4C34 = [21]
-> R6C34 = [42], R23C9 = [74], R1C4 = 4, clean-up: no 5 in R3C5, no 3 in R4C5
45. R1C1 = 2, R3C1 = 3, R2C1 = 4, R1C56 = {15}, locked for N2 -> R2C5 = 3, R3C5 = 2, R4C5 = 5, R1C56 = [15], R46C9 = [65], R46C1 = [86], R4C8 = 9, R23C6 = {89}, locked for C6 [Edited. "locked for N2" added for clarity"]
and the rest is naked and hidden singles, naked pairs, simple elimination and cage sums
Ruud wrote:Brace for impact. Here is an Assassin that is helping me teach SumoCue some new tricks. You already know these tricks... don’t you?
nd wrote:I'm still wondering what Ruud has in mind for the solving-path--I can't see any elegant way of dealing with this puzzle. It's solvable but not pretty.
Was the intended solving path to use one or more contradiction moves or is there something else that we've all missed?