SudokuSolver Forum

3x3::k:2048:2817:2817:3075:3075:3075:4870:4870:4870:2048:10:4619:4619:13:3854:4870:16:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:37:3366:3366:40:2857:2857:43:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:64:3905:5433:67:3643:3643:70:6727:3905:3905:3905:3915:3915:3915:6727:6727:6727:

(note: updated Oct 08)

1. N123
a) Innies N12 = 14(2) = {59/68}
b) Innies N3 = 15(2) = [78/87/96]
c) 9(2): R4C7 = (123)
d) Outies R1 = 5(2) = [14/23/32]
e) 8(2): R1C1 = (567)
f) Innies N1 = 14(3): R23C3 <> 9 because R2C2 >= 5

2. N789
a) 24(3) = {789} locked for N7
b) 15(4) = 36{15/24} -> 3,6 locked for N7
c) 4(2) = {13} locked for R7+N9
d) Innies N7 = 6(2) = [24/51]
e) Innies N8 = 9(3) <> 7,8,9
f) Innies N89 = 10(2) = [19/28/37/46/64]
g) 8(2) = [35/62]
h) Outies R9 = 11(2) = [29/38/47/56/65]

3. R46
a) Innies+Outies R6: -2 = R5C1 - R6C3: R5C1 = (14) because R6C3 = (36)
b) Innies R4 = 11(3) <> 9
c) Innies+Outies R4: 6 = R5C9 - R4C8: R5C9 = (789) because R4C8 = (123)

4. N8
a) 14(4) = 2{147/156/345} <> 8 because (13) only possible @ R8C6
b) 14(4) must have 1 xor 3 and it's only possible @ R8C6 -> R8C6 = (13)

5. R5789 !
a) Innies R5 = 21(5): R5C258 <> 9 because R5C9 >= 7
b) ! 4,6 in R7 can only be in 21(3) and 14(4) and none of them can have both
-> 21(3) = 8{49/67} -> 8 locked for N8
-> R8C47 <> 4,6
c) 8 locked in R9C789 for N9 @ 26(4) = 8{279/459/567}
d) Innies N89 = 10(2) <> 2
e) Outies R9 = 11(2) <> 3
f) 3 locked in R9C123 for R9
g) Killer pair (69) locked in 21(3) + 15(3) for N8
h) Naked triple (134) locked in R8C256 for R8
i) Outies R9 = 11(2) <> 7

6. N789 !
a) ! Innies = 16(1+3) = R7C3+R8C258 = 2+4[19/37] / 5+1[37/46] because R8C58 = 10(2)
- But 5+[137] blocked by R8C6 = (13)
-> 16(1+3) = 2+4[19/37]/5+[146]
-> Both combos force R6C8+R8C3 <> 6 because either R6C3 = 6 or R8C8 = 6
b) Naked pair (25) locked in R78C3 for C3+N7
c) Outies R9: R8C9 <> 5
d) 6 locked in 15(4) for R9
e) 15(3) = 9{15/24} -> 9 locked for R9+N8
f) 21(3) = {678} -> 6 locked for R7
g) 7 locked in 26(4) @ R9 = 78{29/56} -> 7 locked for N9
h) Hidden Killer pair (25) in R7C6 for N8 -> R7C6 <> 4
i) 14(4) = {2345} -> R7C7 = 4, R8C6 = 3
j) Naked pair (69) locked in R8C89 for R8

7. R123 !
a) Outies R1 = 5(2) = {23} locked for R2
b) 8(2) <> 7
c) 1 locked in Innies N1 = 14(3) = 1{49/58/67} <> 3; 1 locked for C3+18(4)
d) Innies N1 = 14(3): R23C3 <> 6 because R2C2 <> 1,7; R2C2 <> 8 because 5 only possible there
e) Innies N12 = 14(2): R2C5 <> 6
f) Innies N2 = 18(3): R23C4 <> 8 because R23C4 <> 1
g) ! 18(4): R23C3 <> 4 because {67} @ R23C4 blocked by R78C4 = (678)
h) Innies N1 = 14(3) = 1{58/67}
i) Naked pair (56) locked in R1C1+R2C2 for N1
j) 12(2) <> 7
k) 11(2): R1C2 <> 9
l) Innies N12 = 14(2): R2C5 <> 5

8. R123 !
a) 18(4) = 1{278/368/458/467} <> 9 because R23C3 = 1{7/8}; R3C4 <> 7 because 2 only possible there
b) 11(2) @ N1 <> 3,8 because (38) is a Killer pair of 12(2)
c) 12(3) <> 6 because R1C1 = (56) and (24) is a Killer pair of 11(2)
d) 12(3) <> {237} because (27) is a Killer pair of 11(2)
e) 19(4) = {1369/1378/2359/2458/3457} since R2C7 = (23) and other combos blocked by Killer pairs of Innies N3
f) ! Killer pair (15) locked in 19(4) + 12(3) for R1
g) R1C1 = 6 -> R2C1 = 2, R2C2 = 5, R2C7 = 3
h) 11(2) = {47} locked for R1+N1
i) 19(4) = {2359} -> {259} locked for R1+N3
j) Innies N12 = 14(2) = [59] -> R2C5 = 9
k) 12(3) = {138} locked for N2
l) 18(4) = 18{27/45}; R3C4 <> 4

9. R456 !
a) 9 locked in R45C3 @ C3 for N4
b) ! 21(4) <> {1389} because together with R7C3 it would build a Killer triple (136) for 8(2) @ N5
c) Hidden Killer pair (89) in 14(3) + 21(4) for R6
-> 14(3) = 8{15/24} -> 8 locked for R6+N4; R6C12 <> 1,4
d) 13(2) <> 5,8
e) 11(2) = {29/56} because (47) is a Killer pair of 13(2)
f) Killer pair (69) locked in 13(2) + 11(2) for R5
g) 9(2) @ N6 <> 6
h) 1,6 locked in R456C7 @ C7 for N6
i) 17(3) = 8{27/45} -> 8 locked for N6
j) ! Hidden Killer pair (36) in 9(2) + 25(4) for R4 -> One of them must have both (because of 9(2))
-> 25(4) = 79{18/36/45} <> 2; 7 locked for R4
k) 9(2) @ N5 <> 2; R4C5 <> 6

10. C6789 !
a) 2 locked in R4C789 @ R4 for N6
b) 11(2): R5C6 <> 9
c) 15(3) @ R2C6 = 6{27/45}
d) ! Killer triple (256) locked in 15(3) + R57C6 for C6
e) 9(2) @ N5: R4C5 <> 3,4

11. R456+N2
a) 3 locked in 25(4) @ R4 = {3679}
b) Hidden Single: R6C1 = 5 @ N4 -> R5C1+R6C2 = 9(2) = [18] -> R5C1 = 1, R6C2 = 8
c) Hidden Single: R5C2 = 2 @ N4, R5C3 = 4 @ N4 -> R5C4 = 9
d) 21(4) = {1479} locked for R6
e) 8(2) @ N5 = {26} locked for R6+N5
f) R5C6 = 5
g) 9(2) @ N5 = {18} locked for R4+N5
h) R4C7 = 2 -> R3C7 = 7
i) Hidden Single: R6C6 = 4 @ N5
j) 15(3) = {267} because (45) only possible @ R3C5 -> R2C6 = 7, {26} locked for R3

12. Rest is singles.


Rating: 1.75

I first tried Ed's challenging CDK V3 soon after it first appeared in early 2007. At times I was struggling and without Ed's help, as discussed below, I would probably have given up. I thought I had finished it early last year and had intended to post my walkthrough but on checking I found that I'd incorrectly eliminated a combination in step 29. I've recently reworked the later steps because of that.

It's good to see that Afmob solved it using very different methods that I did.

It's hard to rate this puzzle the way I did it, particularly since some of the steps were done over a year ago, so I'll just agree with Afmob's rating of 1.75.

Here is my walkthrough, including Ed's hints and some discussion between us. A couple of steps use detailed analysis of remaining combinations. If you don't want to work through this analysis, I've provided summaries after these steps. I won't say enjoy, some of it is heavy going.

As with V2, the centre dot cells don’t necessarily form a remote nonet so there is no elimination between the centre dot cells except for ones in the same row/column.

Many thanks to Ed for his feedback on my earlier steps and the hint he gave me after step 27. In a couple of cases, steps 14a and 25, the feedback has been included and forms part of the walkthrough.

1. R12C1 = {17/26/35}, no 4,8,9

2. R1C23 = {29/38/47/56}, no 1

3. R3C12 = {39/48/57}, no 1,2,6

4. R34C7 = {18/27/36/45}, no 9

5. R4C56 = {18/27/36/45}, no 9

6. R5C34 = {49/58/67}, no 1,2,3

7. R5C67 = {29/38/47/56}, no 1

8. R67C3 = {17/26/35}, no 4,8,9

9. R6C45 = {17/26/35}, no 4,8,9

10. R7C89 = {13}, locked for R7 and N9, clean-up: no 5,7 in R6C3

11. 11(3) cage in N3, no 9

12. 24(3) cage in N7 = {789}, locked for N7, clean-up: no 1 in R6C3

13. 21(3) cage in N8 = {489/579/678}, no 1,2,3

14. 14(4) cage in N89, no 9; only remaining 1,3 in same cell -> no 8
14a. 14(4) must have 1/3 -> R8C6 = {13} (thanks Ed)
[I’d only got “Min R7C67 + R8C7 = 11 -> max R8C6 = 3”. I’d missed Ed’s better move because I hadn’t listed the combinations for the 14(4) cage.]

15. 45 rule on R1 1 innie R1C1 = 1 outie R2C7 + 3 -> R1C1 = {567}, R2C7 = {234}, clean-up: R2C1 = {123}

16. 45 rule on R9 2 outies R8C39 = 11 = [29/38/47/56/65], no 1 in R8C3, no 2,4 in R8C9

17. 45 rule on R4 1 outie R5C9 = 1 innie R4C7 + 6 -> R4C7 = {123}, R5C9 = {789}, clean-up: R3C7 = {678}
17a. 45 rule on N3 1 innie R2C8 = 1 outie R4C7 + 6 -> R2C8 = R5C9 = {789}
[Alternatively 17b. 45 rule on N3 2 innies R2C8 + R3C7 = 15]

18. 45 rule on R6 1 innie R6C3 = 1 outie R5C1 + 2 -> R6C3 = {36}, R5C1 = {14}, clean-up: no 6 in R7C3
18a. 45 rule on N7 1 outie R6C3 = 1 innie R8C2 + 2 -> R8C2 = R5C1 = {14}
[Alternatively 18b. 45 rule on N7 2 innies R7C3 + R8C2 = 6]

19. 45 rule on N8 3 innies R7C6 + R8C56 = 9 = {126/135/234}, no 7,8,9
19a. 5 of {135} must be in R7C6 -> no 5 in R8C5

20. 21(3) cage cannot have 4,5,6 in R8C4 because {89/79/78} would clash with R7C12 -> R8C4 = {789}
20a. Killer triple 7,8,9 in R7C1245, locked for R7

21. Only valid combinations for 15(4) cage in N7 are {1356/2346} = 36{15/24}

22. 45 rule on N12 2 innies R2C25 = 14 = {59/68}

23. 45 rule on N89 2 innies R8C58 = 10 = [19/28/37/46/64]

24. 45 rule on N1 3 innies R2C23 + R3C3 = 14, min R2C2 = 5 -> max R23C3 = 9, no 9 in R23C3

25. 45 rule on N4 3 innies R5C23 + R6C3 – 6 = 1 outie R4C4, min R5C23 + R6C3 = 9 (cannot be {124} because R6C3 only contains 3,6, cannot be {134} which would clash with R5C1, thanks Ed) -> min R4C4 = 3

26. 45 rule on R4 3 innies R4C789 = 11 -> no 9 in R4C89
26a. 9 in R4 locked in 25(4) cage = 9{178/268/358/367/457}

[While reviewing the early steps, Ed commented
Just noticed a nice elim from this. I'll put it into tt since it ends up being potentially very helpful.

9 in r4c4 -> 9 cannot be in r5c23. Here's how.
a. 9 in r4c4 -> from step 25: R5C23 + R6C3 = 15.
i. 3 in r6c3 -> r5c23 = 12 but cannot be {39} -> no 9 in r5c23
ii. 6 in r6c3 -> r5c23 = 9 -> cannot have 9
b. 9 elsewhere in 25(4) must be in n4 -> no 9 in r5c23]

27. 45 rule on N47 2 outies R45C4 – 9 = 2 innies R58C2, max R45C4 = 17 -> max R58C2 = 8 -> max R5C2 = 7

At this stage I was struggling. Ed reviewed my earlier steps, including the first part of step 28, and then added
“Now, in case these things above don't unlock it, here's a big hint. The way to unlock this puzzle is combining steps 15 and 17b and seeing what this means for R1. Easy. If you want a harder way, do a similar thing for R9! If you want to make it a really easy puzzle, do both.”

Many thanks for the hint. A typical hint from Ed, just enough to provide help but still make one work to make progress. That’s how good hints ought to be! Not sure about the last sentence. There was still a lot of hard work.

28. 19(4) cage in N3 must contain {234} in R2C7, valid combinations at this stage are {1279/1369/1378/1459/1468/2359/2368/2458/2467/3457}
[When Ed reviewed this he told me that I had too many combinations, leaving me to work out which ones weren’t valid. I found that was because I hadn’t been looking at the effect of steps 15 and 17b.]
28a. There cannot be any combinations with {67}, {68} or {79} which would clash with R2C8 + R3C7, eliminating {1279/1468/2368/2467}
28b. There cannot be any combinations with 5,6,7 in R1 when 2,3,4 (respectively) are in R2C7 (step 15), eliminating {1369} and also limiting three of the other combinations to having a specific value in R2C7
28c. The remaining valid combinations, with [] indicating the value in R2C7, are {159[4]/178[3]/258[4]/259[3]/457[3]} -> no 3,6 in R1C789, R2C7 = {34}, R1C1 = {67} (step 15), clean-up: R2C1 = {12}

29. Consider each of these combinations and their effect on R1
For {1378}, R2C7 = 3, R1C789 = {178}, R1C1 = 6, R1C23 = {29), R1C456 = {345}
For {1459}, R2C7 = 4, R1C789 = {159}, R1C1 = 7, R1C23 = {38}, R1C456 = {246}
For {2359}, R2C7 = 3, R1C789 = {259}, R1C1 = 6, R1C23 = {38/47}, R1C456 = {138/147}
For {2458}, R2C7 = 4, R1C789 = {258}, R1C1 = 7, R1C23 is blocked
For {3457}, R2C7 = 3, R1C789 = {457}, R1C1 = 6, R1C23 = {29/38}, R1C456 = {138/129}

Summary of step 29: no {2458} combination in 19(4) cage in N3, no 5,6 in R1C23

[Ed said that he’d done similar analysis of hypotheticals but using 4 innies in R1, R1C1789 = 22 together with the restrictions from steps 15 and 17b.]

30. If R2C7 = 3, R34C7 <> [63] => R2C8 + R3C7 => {78} -> 19(4) cage in N3 cannot have 178[3] or 457[3] combinations.
Remaining valid combinations are {159[4]/259[3]} -> R1C789 = {159/259}, no 4,7,8 -> 5,9 locked for R1 and N3, clean-up: no 2 in R1C23, no 6 in R3C7 (step 17b), no 3 in R4C7
30a. R2C8 + R3C7 = {78}, locked for N3
30b. R5C9 = {78} (step 17a)

31. 17(3) cage in N6 must have R5C9 = {78}, valid combinations {278/368/458/467}, no 1
31a. R4C789 = 11 (step 26), R4C7 = {12} -> 17(3) cage combination {278} can only have 7 in R5C9 (cannot have [227] in R4C789) -> no 7 in R4C89

32. R3C12 = {39/57} (cannot be {48} which clashes with R1C23)

33. Killer pair 3,7 in R1C23 and R3C12, locked for N1 -> R1C1 = 6, R2C1 = 2, clean-up: no 8 in R2C5 (step 22)
33a. 1 in N1 locked in R23C3, locked for C3 and 18(4) cage -> no 1 in R23C4

34. R2C7 = 3 (step 15) -> R1C789 = {259} (step 30), locked for R1 and N3, clean-up: no 8 in R5C6
34a. 1 in R1 locked in R1C456, locked for N2
34b. 1 in C7 locked in R46C7, locked for N6

35. 18(4) cage in N12 must contain 1 = 1{278/359/368/458/467} (cannot be {1269} because no 2,6,9 in R23C3)
35a. 3 of {1359} must be in R3C4 -> no 9 in R3C4

36. 9 in C3 locked in R45C3, locked for N4
36a. 14(3) cage in N4 must have R5C1 = {14}, valid combinations are {158/167/248/347}, no 1,4 in R6C12

37. 4,9 in R6 locked in 21(4) cage = 49{17/26/35}, no 8

38. 8 in R6 locked in R6C12, locked for N4 -> 14(3) cage = 8{15/24}, no 3,6,7, clean-up: no 5 in R5C4
38a. R6C123 = 8{26/35} (step 18)

39. 25(4) cage = 9{178/268/358/367/457} (step 26a), any combinations with 8 must have R4C3 = 9, R4C4 = 8 -> cannot be {2689} because no 2,6 in R4C1 -> no 2 in 25(4) cage
39a. 25(4) cage = 9{178/358/367/457}

40. 2 in N4 locked in R56C2, locked for C2

41. Consider 14(4) cage in N89 = {1247/1256/2345} (only combinations because 1,3 only in R8C6)
If R8C6 = 1 => R8C2 = 4 => R7C3 = 2 (step 18b) => R7C67 = {456} => only valid combination for 14(4) cage in N89 = {1256}, no 7
If R8C6 = 3 => only valid combination for 14(4) cage in N89 = {2345}, no 7
-> 14(4) cage in N89 = 25{16/34}, no 7
41a. 2 of {1256} must be in R8C7 -> no 6 in R8C7

42. 45 rule on N9 1 innie R8C8 = 2 outies R78C6 + 1
42a. Min R78C6 = 5 (from combinations in step 41) -> min R8C8 = 6, clean-up: no 6 in R8C5 (step 23)

43. R7C6 + R8C56 (step 19) = {126/135/234}
43a. If {126} => R7C6 = 6, R8C56 = [21], R8C2 = 4, R7C3 = 2 (step 18b), R78C7 = [45] which gives wrong cage total in R78C67 -> R7C6 + R8C56 cannot be {126}
43b. R7C6 + R8C56 = {135/234}, no 6, 3 locked in R8C56 for R8 and N8, clean-up: no 8 in R8C9 (step 16)

44. R8C1 = {789}, R8C4 = {789} -> R8C89 must contain one of 7,8,9 -> R9C789 must contain two of 7,8,9. Combinations for the 26(4) cage in N9 are {2789/4589/4679/5678}, in the case of {5678} either 5 or 6 must be in R8C9

Here’s a discussion with Ed relating to the next step
Andrew “I looked at R9 but could only see how to make progress by doing hypotheticals on the five pairs of values for R8C39 (the discussion took place before I found step 43 which eliminated one pair of values for R8C39). This did provide progress by eliminating at least one of those pairs. Did you use hypotheticals in that way or did you have a more direct way to use r9?”
Ed “Yeah, I used the hypo's you've mentioned, not including the 15(3) in R9”
Andrew “but there is also interaction with R67C3 = [35/62] and of course with R8C39.”
Ed “True.”
Andrew “Since sending yesterday's message I haven't made any more progress and can't see how to proceed except to use those hypotheticals. However some of the steps that I made after doing R1/N3 should help to make the hypotheticals a bit simpler. I had a look at doing two hypotheticals for R7C3 rather than more of them for R8C39 but that looks very messy and appears that it doesn't produce as much useful "output information". ”

The second part of Ed’s hint suggests that a similar approach is needed for R9. The interactions between R8C2 + R7C3 and the 15(4) cage are already built into the latter which must be 36{15/24} (step 21). Other useful interactions are provided by R8C39 = 11 (step 16) and by R67C3 = [35/62]. Values for R9C789 must be consistent with step 44.

45. Consider the combinations for R8C39 and their effect on R9
For R8C39 = [29], R9C123 = {346}, R9C789 = {278}, R9C456 = {159}
For R8C39 = [47], R9C123 = [362], R9C789 is blocked
For R8C39 = [56], R9C123 = [163] (6 cannot be in R9C3 because R67C3 = [62] when R8C3 = 5), R9C789 = {479/578}, R9C456 = {258/249}
For R8C39 = [65], R9C123 = {234} (cannot be {135} because R67C3 = [35] when R8C3 = 6), R9C789 = {678}, R9C456 = {159}

46. Summarising the results of step 45
R8C39 = [29/56/65], no 4 in R8C3, no 7 in R8C9
R9C123 = [163]/{234}/{346}, no 5, no 1 in R9C2
R9C456 = {159/249/258} -> no 6,7 in R9C456
R9C789 = {278/479/578/678}
46a. 5 in N7 locked in R78C3, locked for C3, clean-up: no 8 in R5C4

47. 6 in N8 locked in R7C45, locked for R7
47a. 21(3) cage in N8 (step 13) = {678}, locked for N8
47b. 8 in R9 locked in R9C789, locked for N9, clean-up: no 2 in R8C5 (step 23)

48. 14(4) cage in N89 (step 41) = {2345} (only remaining combination) -> R8C6 = 3, clean-up: no 6 in R4C5, no 8 in R5C7, no 7 in R8C8 (step 23)
48a. Naked pair {14} in R8C25, locked for R8

49. 8 in R9 locked in R9C789 (step 46) = {278/578/678} (cannot be {479} which doesn’t contain 8), no 4,9
49a. 9 in N9 locked in R8C89, locked for R8

50. R7C7 = 4 (hidden single in N9), clean-up: no 7 in R5C6

51. Combined cage R5C3467 = 24 = {2679/4569}, 6,9 locked for R5
51a. R5C67 = {29/56} (cannot be [47] which clashes with combined cage), no 4,7

52. 15(3) cage in N2 = {249/258/267/456} (cannot be {348/357} which clash with R1C456), no 3
52a. 8 of {258} must be in R23C6 (R23C6 cannot be [52] which clashes with R7C6), no 8 in R3C5

53. 45 rule on N2 3 innies R2C45 + R3C4 = 18 = {279/369/459/567} (cannot be {378/468} which clashes with R1C456), no 8
53a. 2,3 of {279/369} must be in R3C4
53b. 6 of {567} must be in R2C5 (R23C4 cannot be {67} which clashes with R78C4)
53c. Combining steps 53a and 53b -> no 6 in R3C4

54. 18(4) cage in N12 (step 35) = 1{278/368/458} (cannot be {1359} because 3,5,9 only in R23C4, cannot be {1467} which clashes with R78C4), no 9
54a. 1,8 of {1458} must be in R23C3 -> no 4 in R23C3
54b. 2 of {1278} must be in R3C4 -> no 7 in R3C4

55. R2C23 + R3C3 = 14 (step 24) = {158} (only remaining combination) -> R2C2 = 5, R2C5 = 9 (step 22), R23C3 = {18}, locked for N1, clean-up: 3 in R1C23, no 7 in R3C12
55a. Naked pair {47} in R1C23, locked for R1
55b. Naked pair {39} in R3C12, locked for R3
55c. Naked triple {138} in R1C456, locked for N2

56. 18(4) cage in N12 (step 54) = 1{278/458}, no 6
56a. 5 of {1458} must be in R3C4 -> no 4 in R3C4

57. 45 rule on R789 4 innies R7C3 + R8C258 = 16, R8C25 = {14} = 5 -> R7C3 + R8C8 = 11 = [29/56]
57a. If R7C3 = 2 => R6C3 = 6 -> no 6 in R8C3
57b. If R7C3 = 5 => R8C8 = 6 -> no 6 in R8C3
57c. -> no 6 in R8C3

58. Naked pair {25} in R78C3, locked for N7
58a. Naked pair {25} in R8C37, locked for R8
58b. 6 in N7 locked in R9C23, locked for R9

59. 6 in C7 locked in R56C7, locked for N6
59a. 17(3) cage in N6 (step 31) = {278/458}, no 3, 8 locked for N6
59b. R4C789 = {128/245}, 2 locked for R4 and N6, clean-up: no 7 in R4C56, no 9 in R5C6

60. 45 rule on N36 2 innies R25C8 = 2 outies R56C6 + 2
60a. Min R25C8 = 10 -> min R56C6 = 8, no 1 in R6C6

61. 7 in R4 locked in R4C1234
61a. 25(4) cage (step 39a) = 9{178/367/457} (cannot be {3589} which doesn’t contain 7)
61b. 8 of {1789} must be in R4C4
61c. 3,6 of {3679} must be in R4C4 (R4C123 cannot contain both 3,6 which would clash with R6C3)
61d. 4,5 of {4579} must be in R4C4 (R4C123 cannot contain both 4,5 which would clash with 14(3) cage in N4)
61e. -> no 7,9 in R4C4

62. R4C3 = 9 (hidden single in R4), clean-up: no 4 in R5C4
62a. 7 in R4 locked in R4C12, locked for N4, clean-up: no 6 in R5C4

63. R1C3 = 7 (hidden single in C3), R1C2 = 4, R8C25 = [14], R5C1 = 1 (step 18a), R6C3 = 3 (step 18), R7C3 = 5, R7C6 = 2, R8C37 = [25], R56C2 = [28], R6C1 = 5, clean-up: no 5 in R4C6, no 6 in R5C6, no 9 in R5C7
63a. R5C67 = [56], R5C3 = 4, R4C4 = 9, R4C12 = [76], R4C4 = 3 (step 61a), R78C1 = [98], R7C2 = 7, R8C4 = 7, R3C12 = [39], R9C123 = [436], R8C8 = 6 (step 57), R8C9 = 9, R2C4 = 4, R3C4 = 5 (step 54), clean-up: no 1 in R6C5

64. R9C5 = 5 (hidden single in R9), clean-up: no 4 in R4C6

65. Naked pair {18} in R4C56, locked for R4 and N5 -> R4C7 = 2, R5C5 = 7, R5C89 = [38], R1C7 = 9, R2C8 = 8 (step 17a), R3C7 = 7, R3C56 = [26], R2C6 = 7, R23C3 = [18], R2C9 = 6, R6C7 = 1, R9C7 = 8, R6C45 = [26], R7C45 = [68], R4C56 = [18], R1C456 = [831], R9C46 = [19], R6C6 = 4, R7C89 = [13], R3C89 = [41], R4C89 = [54], R1C89 = [25], R9C89 = [72], R6C89 = [97]

3x3::k:8448:3329:1794:2563:4868:1285:1542:7687:7687:3329:8448:1794:2563:1285:4868:4868:1542:7687:8448:1794:8448:8448:8448:3095:3095:4868:7687:5915:1820:5915:3614:2335:2335:4897:2850:2850:1820:5915:3614:3879:3879:4897:4897:2850:4140:5915:3886:5915:3879:4657:4657:1843:1843:4140:3894:3886:4664:4664:4154:4154:4657:4925:4140:3894:4928:4664:4928:4154:5188:4925:4925:4925:4928:3894:4928:2379:2379:4154:5188:5188:4925:

Very nice, Ed. Easy start with a lot of eliminations for N123 but then you have to take your time until it falls.

Here is a walkthrough


Walkthrough SampuZ5

1. N1 : 7(3) = {124} -> 1,2,4 locked for N1
2. N3 : 30(4) = {6789} -> 6,7,8,9 locked for N3
3. N23 : 12(2) = [75|84|93]
4. N2 : 5(2) = {14|23}
5. N3 : 6(2) = {15|24}
6. N2 : 10(2) = {19|28|37|46}
7. N4 : 7(2) = {16|25|34}
8. N45 : 14(2) = {59|68}
9. N47 : 15(2) = {69|78}
10. N5 : 9(2) = {18|27|36|45}
11. N8 : 9(2) = {18|27|36|45}
12. N6 : 7(2) = {16|25|34}
13. N6 : 11(3) : no 9
14. N89 : 20(3) : no 1,2
15. N9 : 1 locked in 19(5) for N9
16. N56 : 19(3) : no 1
17. 45 on N1 : r3c45 = 8(2) = {17|26|35}
18. N12 : 33(6) = 89{1267|1357|2356} -> 8,9 locked for N1 in 33(6)
19. N1 : 13(2) = {67} -> 6,7 locked for N1
20. N12 : 33(6) : {3589} in N1 -> no 3,5 in r3c45
21. N2 : 1,2 locked in 5(2) and r3c45 for N2
22. N2 : 10(2) = {37|46} -> 6,7 locked in 10(2) and r3c45 for N2
23. N23 : 12(2) = [84|93]
24. N2 : 5 locked in 19(4) for N2 -> no 5 in r2c7 r3c8
25. N3 : 5 locked in 6(2) = {15} -> 1,5 locked for N3
26. 5 locked in r3c13 for N1 and r3
27. N23 : 19(4) = 25{39|48} -> no 3,4 in r1c5 r2c6
(other way: N2 : 3,4 locked in 10(2) and 5(2) for N2)
28. c9 : 16(3) has at most one of {6789} because of 30(4) in N3 -> 16(3) = {259|349|358|457} -> no 1,6 -> 6 locked in r123c9 for N3 and c9 -> no 6,7,8,9 in r489c9 (Para's suggestion, much clearer: Killer Quad {6789} in R123C9 + 16(3) which eliminates 6,7,8,9 from r489c9 -> 6 locked in r123C9 for N3)
29. 45 on N6 : r5c6 + r7c9 = 8(2) = [62]|{35|44}
30. N56 : 19(3) : max r5c6 = 6 -> min r45c7 = 13 -> no 2,3 in r45c7
31. N6 : 11(3) = {128|137|146|245} ({236} blocked by 7(2))
32. 45 on N4 : r6c2 + r5c3 = 15(2) = [78]|{69} -> no 7 in r7c2, no 9 in r4c4
33. c2 : 6,7 locked in r167c2 for c2 -> no 1 in r5c1
34. N4 : combination check for 23(5) : 23(5) = {12569|12578|13469|13478|23459} , others blocked
35. 45 on N7 : r6c2 + r78c4 = 22(3) = [679|7{69}|778|9{49|58}|976] -> no 1,2,3 in r78c4, no 7 in r8c4
36. 45 on N78 : r7c2 + r8c6 = 13(2) = [67|85|94] -> r8c6 = {457}
37. N89 : 20(3) = {479|569|578} -> no 3
38. N8 : 16(4) must have two of {123} (three not possible) -> {1456} not possible for 16(4) -> 9(2) must have one of {123} -> {45} not possible for 9(2) -> no 4,5 in 9(2)
Edit: Explanation by sudokuEd:
16(4) cannot have {123} since = 6 -> only other place for {123} in n8 is 9(2) -> no {45} in 9(2). Also, 9(2) can only have 1 of {123} -> 16(4) must have 2 of -> no {1456} in 16(4).
39. 45 on N789 : r7c279 = 19(3) -> r7c7 = {6789}
Edit: Explanation by sudokuEd:
Max r7c29 = [85/94] = 13 -> min r7c7 = 6
40. N6 : combination check: 2 is in 11(2) or 7(2) -> no 2 in r56c9
41. r3 : 1,2,3,4 locked in r3c24578 -> no 3 in r3c13
42. r3 : naked triple {589} -> no 8,9 i r3c9
43. 45 on N47 : r478c4 = 21(3) = [579|8{49] ({678} blocked by 10(2) at r1c4) -> r4c4 = {58}, r7c4 = {479}, r8c4 = {49} ->9 locked in r78c4 for c4 and N8, 4,7 locked in r1278c4 for c4
44. clean-up: no 2 in r9c5, no 1 in r3c5, no 8 in r5c3
45. from step 32: r6c2 + r5c3 = 15(2) = {69} -> 6,9 locked for N4
46. hidden single : r4c7 = 9
47. N47 : 15(2) = {69} -> 6,9 locked for c2
48. r1c2 = 7, r2c1 = 6
49. clean-up: no 1 in r4c2, no 5 in r5c67, no 8 in r5c7
50. 7 locked in r23c9 for c9
51. c9 : 16(3) = {358} -> 3,5,8 locked for c9, 8 locked for N6
52. r1c8 = 8
53. from step 39: r7c279 = 19(3) = [973|685] -> r7c7 = {78}
54. from step 36: r8c6 = {47} -> 4,7,9 locked in r78c4 r8c6 for N8 -> no 2 i r9c4
55. N89 : 20(3) = 7{49|58} -> 7 locked for 20(3) -> no 7 in r8c78, no 5 in r9c7
56. 3 locked in r3c78 for r3 and N3
57. N59 : 18(3) = {189|279|378|468|567}
58. N9 : 6 locked in 19(5) = {12367} -> 1,2,3,6,7 locked for N9
The rest is clean-up
59. r7c79 = [85], r9c78 =[49], r8c6 = 7, r67c2 = [96], r5c3 = 6, r4c4 = 8, r5c67 = [37], r3c67 = [93], 19(4) at r1c5 = [5824], ...

I know there are still a lot of cells to fill but it can be done with simple checks.

As always feel free to comment/correct.

Peter

Hi

I think mine covers a lot more steps but follows basically the same route.
I didn't include the primary eliminations and started my walkthrough from the mark-up pic.

Walk-through SampuZ5

1. {124} locked in 7(3) in R1C3 for N1.
1a. Clean up: no 9 in 13(2) in R1C2
1b. 3 and 9 locked in N1 for 33(6) in R1C1. No 3 or 9 in R3C45.
1c. 33(6) = {135789/234789/235689} -->> 8 obligated in 33(6)

2. {6789} locked in 30(4) for N3.
2a. Clean up: no 3, 4 or 5 in R3C6

3. 45 on N1: 2 outies R3C45 = 8 = {17/26}
3a. 8 locked in 33(6) for N1.
3b. 13(2) in R1C2 = {67} -->> locked in N1
3c. 33(6) in R1C1 = {3589}+{17/26}

4. Killer Pair {67} in R1C2 + 15(2) in R6C2 -->> no 6 or 7 anywhere else in C2
4a. Clean up: No 1 in R5C1

5. 45 on N2: 3 innies: R1C5 + R23C6 = 22 = {589/679}-->> no 1,2,3,4 + 9 locked for N2
5a. 5 locked in R1C5 + R2C6 -->> R1C5 + R23C6 = {589} -->> no 6,7 + 8 locked for N2 + 5 locked in 19(4) in R1C5 -->> no 5 in R2C7 and R3C8
5b. Clean up: no 1 or 2 in R12C4
5c. Clean up: no 5 in R3C7
5d. 5 locked in 6(2) in R1C7: 6(2) = {15} -->> locked for N3

6. Hidden Killer Pair {67} in R3C45 + R3C9 -->> R3C9 = {67} : R3C45 contains either a 6 or a 7. Only other option for a 6 or a 7 in R3 is in R3C9

7. 45 on N4: 2 innies R5C3 + R6C2 = 15 = {69}/[87]
7a. Clean up: R4C4: no 9 ; R7C2: no 7

8. 45 on N5: 4 outies: R5C3 + R457C7 = 30 = 9{489}; 9{579}; 9{678}; 8{589};8{679}; 6{789} -->> no 1, 2 or 3 in R457C7

9. 45 on N6: 2 outies: R5C6 + R7C9 = 8 = [71]/{26/35/44} -->> no 8 or 9 in both cells and no 7 in R7C9
9a. 45 on N6: 4 innies: R45C7 + R56C9 = 27 = {9873/9864/9765} -->> no 1,2

10. 45 on N7: 3 outies: R6C2 + R78C4 = 22 = {994/985/976/877} -->> no 1,2,3

11. 19(5) in R7C8 = {12349/12358/12367/12457/13456} -->> 1 locked in 19(5) for N9
11a. Clean up: no 7 in R5C6 (Step 9)

12. 45 on N569: 2 outies R5C3 + R8C6 = 13 = [67/85/94] -->> R8C6 = {457}
12a. 20(3) in R8C6 = {479/569/578} -->> no 3 in R9C78

13. 45 on N47: 3 outies R478C4 = 21 = {489/579/678} -->> no 5 in R78C4: {579} no possible that way.


14. 5 locked in R3C13 for N1, nowhere else in N1
14a. Hidden Killer Pair {89} in R3C13 + R3C6 -->> R3C13 has to contain 8/9
14b. R3C13 = {589} -->> no 3 in R3C13
14c. 3 locked in R3C78 for N3 -->> no 3 in R2C7

15. 45 on C1234: 3 outies: R359C5 = 18 = {189/279/369/378/468/567} -->> R5C5: no 1,2 ; R9C5: no1 : {189} or {279} not possible with these placements
15a. Clean up: R9C4: no 8

16 Building on step 13: R478C4 = 22
16a. No {678} -->> clashes with 10(2) in R1C4
16b. 9 locked in R78C4 for C4 and N8
16c. No 6 in R478C4
16d. Clean up: R5C3: no 8; R6C2: no 7 (step7); R7C2: no 8
16e. Naked Pair {69} in R5C3 + R6C2 for N4
16f. Naked Pair{69} in R67C2 for C2
16g. R1C2 =7; R2C1 = 6
16h. Clean up: R1C4: no 4; R2C4: no 3; R4C2: no 1

17. Clean up on step 16
17a. R478C4 = {489/579} :R4C4 = {58} -->> R78C4 = {479}

18. 45 on N8: 3 innies R78C4 + R8C6 = 20 = {479} locked for N8 -->> no 5 in R8C6
18a. 20(3) in R8C6 = {479/578} -->> no 6 in R9C78

19. 7 locked in N3 for C9 -->> nowhere else in C9

20. 16(3) in R5C9 = {259/349/358}: {268} clashes with 30(4) : no 6
20a. 30(4) can’t have both 8 and 9 in C9: would clash with 16(3) -->> R1C8 has to contain 8 or 9 -->> R1C8 = {89}
20b. Killer Pair {89} in R123C9 + 16(3) for C9 -->> no {89} anywhere else in C9
20c. 6 locked in N3 for C9 -->> no 6 anywhere else in C9

21. Naked Quad {3689} in R1C1489 -->> not anywhere else in R1
21a. R1C5 = 5 ; R1C7 = 1; R2C8 = 5
21b. Clean up: R2C5 = {13}
21c. Naked Pair {89} in R23C6 for C6

22. Hidden single 9 in R4C7
22a. 9 locked in N9 for C8 -->> R1C8 = 8

23. 16(3) = {358} -->> locked for C9
23a. 8 locked in C9 for N6 -->> no 8 in R5C7
23b. Clean up: R5C6 = {35} (step 9)
23c. 19 (3) in R4C7 = {379} -->> R5C67 = [37]
23d. R7C9 = 5; R5C9 = 8; R6C9 =3

24. 11(3) in R4C8 = {146} -->> locked for N6
24a. 7(2) in R6C7 = [52]
24b. 6 locked in N6 for C8
24c. 2 locked in C9 for N9
24d. Hidden single 2 in R2C7
24e. Naked Pair {34} in R3C78 for R3
24f. 4 locked in N1 for C3

25. Clean up on step 18
25a. 9(2) in R9C4 = [18]/{36}
25b. Hidden single 5 in R9C6

26. 20(3) in R8C6 = {479) -->> R9C7 = 4; R8C6 = 7; R9C8 = 9
26a. R3C78 = [34]
26b. R23C6 = [89]

27. 19(5) in R7C8 = {12367} -->> R8C7: no 8 -->> R8C7 = 6; R7C7 = 8
27a. Naked Pair {12} in R89C9 -->> locked for N9 and C9

28. Bunch of singles
28a. R2C2 = 3; R2C5 = 1; R2C3 = 4; R2C4 = 7; R2C9 = 9
28b. R1C1 = 9; R1C3 = 2; R1C4 = 3; R1C6 = 4; R1C9 = 6
28c. R3C2 = 1; R3C9 = 7
28d. R4C9 = 4; R8C8 = 3; R8C7 = 7

29. 8 locked in N8 for C5
29a. 4 locked in N8 for C4

30. Clean up
30a. R9C5: no 6
30b. R6C56 = [46]/[19]
30c. 7(2) in R4C2 = {25} -->> locked for N4
30d. Naked Quad {2368} in R3789C5 -->> locked for C5

31. Loads of singles
31a. R5C2 = 4; R5C5 = 9; R5C3 = 6; R5C8 = 1; R4C8 = 6
31b. R4C4 = 8; R4C5 = 7; R4C6 = 2; R4C2 = 5; R5C1 = 2; R5C4 = 5
31c. R67C2 = [96]; R6C456 = [146]
31d. R7C6 = 1; R9C45 = [63] ; R3C45 = [26]; R78C5 = [28]
31e. R89C2 = [28]; R89C9 = [12]
32. 19(4) in R8C2 = {1279}
32a. R8C4 = 9

And I trust you all can finish it form here on.

Any corrections/comments are appreciated

greetings

Para

I only tried this puzzle recently, after completing SampuZ4V1. This was a lot easier, about the level of a moderate Assassin whereas SampuZ4 was at Assassin V2 level. I can strongly recommend SampuZ5 for anyone who wants to try a puzzle with crossover cages. Great puzzle Ed!

I was impressed by the fact that the first two walkthroughs were both posted within about half a day of the puzzle being posted and both had a neat elimination in C9 that I never spotted.

Here is my walkthrough.

1. 13(2) cage in N1 = {49/58/67}, no 1,2,3

2. R12C4 = {19/28/37/46}, no 5

3. 5(2) cage in N2 = {14/23}

4. 6(2) cage in N3 = {15/24}

5. R3C67 = {39/48/57}, no 1,2,6

6. 7(2) cage in N4 = {16/25/34}, no 7,8,9

7. 14(2) cage in N45 = {59/68}

8. R4C56 = {18/27/36/45}, no 9

9. R67C2 = {69/78}

10. R6C78 = {16/25/34}, no 7,8,9

11. R9C45 = {18/27/36/45}, no 9

12. 7(3) cage in N1 = {124}, locked for N1, clean-up: no 9 in 13(2) cage

13. 19(3) cage in N56 = {289/379/469/478/568}, no 1

14. 11(3) cage in N6 = {128/137/146/236/245}, no 9

15. 20(3) cage in N89 = {389/479/569/578}, no 1,2

16. 30(4) cage in N3 = {6789}, locked for N3, clean-up: no 3,4,5 in R3C6

17. 19(5) cage in N9 = 1{2349/2358/2367/2457/3456}, 1 locked for N9

18. 45 rule on N1 2 outies R3C45 = 8 = {17/26/35}, no 4,8,9
18a. Remaining part of 33(6) cage in N1 = 25(4) = {3589/3679} = 39{58/67} -> no 3 in R3C45, clean-up: no 5 in R3C45
18b. Remaining part of 33(6) cage in N1 {3589} (cannot be {3679} which clashes with R3C45), locked for N1 -> 13(2) cage = {67}

19. Killer pair 1/2 in 5(2) cage and R3C45 for N2, clean-up: no 8,9 in R12C4

20. Killer pair 3/4 in R12C4 and 5(2) cage for N2

21. Killer pair 6/7 in R12C4 and R3C45 for N2, clean-up: no 5 in R3C7

22. 5 in N2 locked in 19(4) cage -> no 5 in R2C7 + R3C8
22a. 19(4) cage = 5{149/239/248}

23. Killer quad 1/2/3/4 in R3C2, R3C45, R3C7 and R3C8 -> no 3 in R3C12
23a. Naked triple {589} in R3C136, locked for R3
23b. 5 in R3 locked in R3C13, locked for N1

24. 6(2) cage in N3 = {15} (only remaining 5s in N3), locked for N3
24a. 2 in N3 locked in 19(4) cage = 25{39/48}

25. 45 rule on N6 2 outies R5C6 + R7C9 = 8 = {26/35/44} (double possible), no 7,8,9

26. 45 rule on N4 2 innies R5C3 + R6C2 = 15 -> R5C3 = {689}, R6C2 = {679}, clean-up: no 9 in R4C4, no 7 in R7C3

27. 45 rule on N78 2 innies R7C2 + R8C6 = 13 -> R8C6 = {457}

28. 45 rule on N8 3 innies R78C4 + R8C6 = 20 = {479/569/578} (cannot be {389} because no 3,8,9 in R8C6), no 1,2,3
28a. No {45} in R9C45 (clashes with R78C4 + R8C6)

29. 20(3) cage in N89 = {479/569/578} (cannot be 3,8,9 because no 3,8,9 in R8C6), no 3

30. 45 rule on N9 2 innies R7C79 – 6 = 1 outie R8C6, min R8C6 = 4 -> min R7C79 = 10, no 2,3 in R7C7

31. 45 rule on N6 2 innies R45C7 – 11 = 1 outie R7C9, min R7C9 = 2 -> min R45C7 = 13, no 2,3
[Alternatively this comes from combinations for 19(3) cage in N56]

32. 45 rule on C123 5 outies R3C45 + R478C4 = 29, R3C34 = 8 (step 18) -> R478C4 = 21 = {489/579} (cannot be {678} which clashes with R12C4) = 9{48/57}, no 6, 9 locked in R78C4 for C4 and N8, clean-up: no 8 in R5C3, no 7 in R6C2, no 8 in R7C2, no 5 in R8C6 (step 27)
32a. R4C4 = {58} -> no 5,8 in R78C4

33. Naked triple {479} in R78C4 + R8C6, locked for N8, clean-up: no 2 in R9C45

34. Killer pair 4,7 in R12C4 and R78C4 for C4, clean-up: no 1 in R3C5

35. 20(3) cage in N89 (step 29) = {479/578} (cannot be {569} because no 5,6,9 in R8C6) = 7{49/58}, no 6

36. Naked pair {69} in 15(2) cage in N4, locked for N4, clean-up: no 1 in 7(2) cage

37. Naked pair {69} in R67C2, locked for C2 -> R1C2 = 7, R2C1 = 6, clean-up: no 4 in R1C4, no 3 in R2C4
37a. 7 in N3 locked in R23C9, locked for C9

38. 45 rule on R789 3 innies R7C279 = 19 = [685/694/946/964/973/982], no 5 in R7C7
38a. If R7C2 = 9, R8C6 = 4 (step 27) => R9C78 = {79} -> R7C279 cannot be [973]
-> no 7 in R7C7, no 3 in R7C9; clean-up: no 5 in R5C6 (step 25)
38b. 19(5) cage (step 17) = 13{258/267/456} (cannot be {12349} which would make R9C78 = {58} (step 35) and then there is no 7 in N9), no 9

39. 45 rule on C1234 3 outies R359C5 = 18 = {369/378/468/567} (cannot be {189} because no 1,8,9 in R3C5, cannot be {279} because no 2,7,9 in R9C5, cannot be {459} because 4,5,9 only in R5C5), no 1,2, clean-up: no 6 in R3C4, no 8 in R9C4
39a. 4,5,9 only in R5C5 -> no 6 in R5C5
39b. 5 only in R5C5 and {378} must have 7 in R3C5 -> no 7 in R5C5

40. 5,8 in C4 locked in R456C4, locked for N5, clean-up: no 1,4 in R4C56

41. 15(3) cage in N5 = {159/348/456} (cannot be {168} because no 1,6,8 in R5C5, cannot be {249} because 4,9 only in R5C5, cannot be {258} which clashes with R4C4), no 2
41a. No 4 in R56C4 -> no 3 in R5C5

42. R3C4 = 2 (hidden single in C4) -> R3C5 = 6 (step 18), R3C9 = 7, clean-up: no 3 in 5(2) cage in N2, no 3 in R4C6, no 3 in R9C4

43. R2C7 = 2 (hidden single in N3), clean-up: no 5 in R6C8

44. R1C3 = 2 (hidden single in N1)

45. R1C6 = 4 (hidden single in R1), R2C5 = 1 (these are, of course, a naked pair but I was looking for hidden singles and R1C6 was one), R2C4 = 7, R1C4 = 3, R2C3 = 4, R3C2 = 1, R2C8 = 5, R1C7 = 1, R8C6 = 7, clean-up: no 2 in R4C5, no 6 in R6C8, R7C2 = 6 (step 27), R6C2 = 9, R5C3 = 6, R4C4 = 8, no 2 in R5C6 (step 25) -> R5C6 = 3 -> R7C9 = 5 (step 25), R4C5 = 7, R4C6 = 2 (cage sum), R6C5 = 4, R5C5 = 9, clean-up: no 4 in R4C2, no 5 in R5C1, no 3 in R6C78
45a. R8C6 = 7 -> R9C78 = 13 = {49}, locked for R7 and N9-> R7C7 = 8, R7C6 = 1, R6C6 = 6, R6C78 = [52], R56C4 = [51], R9C45 = [63], R7C5 = 2
45b. R7C9 = 5 -> R56C9 = 11 = [83], R2C9 = 9, R1C9 = 6, R1C8 = 8, R1C1 = 9, R2C6 = 8, R1C5 = 5, R2C2 = 3, R3C6 = 9, R3C78 = [34], R9C8 = 9, R9C7 = 4, R5C7 = 7, R5C8 = 1, R4C89 = [64], R4C7 = 9, R8C7 = 6, R4C2 = 5, R5C1 = 2 (cage sum), R5C2 = 4, R8C5 = 8, R89C2 = [28], R89C9 = [12], R8C8 = 3, R7C8 = 7, R9C6 = 5

46. R8C2 = 2, R9C13 = {17} -> R8C4 = 9 (cage sum)

and the rest is naked singles

3x3::k:4864:4864:2818:2818:2820:4613:4613:1799:2056:2569:4864:4864:3340:2820:4622:4613:1799:2056:2569:2835:2835:3340:3340:4622:4622:4121:4121:2075:2075:2835:3870:3870:2080:2080:4121:4131:4644:4644:4390:3870:5160:5160:3882:3882:4131:4644:2862:4390:4390:5160:5160:2355:3882:4131:4662:2862:4664:4409:4409:4923:2355:2355:3646:4662:4664:4664:4409:1091:4923:4923:4923:3646:4662:3657:3657:3657:1091:5709:5709:5709:5709:

Nice puzzle for your first one Peter! Not too difficult but the combination of killer with noncon requires one to think a bit differently to make proper use of the noncon feature.


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. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up. Non-consecutive (noncon) eliminations have been given as separate sub-steps for clarity.

1. R1C34 = {29/38/47} (cannot be {56} which are consecutive), no 1,5,6

2. R12C5 = {29/38/47} (cannot be {56} which are consecutive), no 1,5,6

3. R12C8 = {16/25} (cannot be {34} which are consecutive), no 3,4,7,8,9

4. R12C9 = {17/26/35}, no 4,8,9

5. R23C1 = 10(2), no 5

6. R4C12 = {17/26/35}, no 4,8,9

7. R4C67 = {17/26/35}, no 4,8,9

8. R67C2 = {29/38/47} (cannot be {56} which are consecutive), no 1,5,6

9. R78C9 = {59/68}

10. R89C5 = {13}, locked for C5 and N8, clean-up: no 8 in R12C5
10a. Noncon, no 2 in R7C5, R8C46 and R9C46

11. 11(3) cage in N14, no 9

12. 9(3) cage in N69 = {126/135} (cannot be {234} which are consecutive) = 1{26/35}

13. 2 in N8 locked in R7C46, locked for R7, clean-up: no 9 in R6C2
13a. Only 2 remaining 2 in 9(3) cage in N69 in R6C7 -> no 6 in R6C7

14. 45 rule on R9 2 innies R9C15 = 9 -> R9C1 = {68}
14a. Noncon, no 7 in R8C1 and R9C2

15. 45 rule on C9 2 innies R39C9 = 7 = {16/25/34}, no 7,8,9

16. 45 rule on R123 2 outies R4C38 = 7 = {16/25/34}, no 7,8,9

17. Killer triple 1/2/3 in R4C123678 for R4

18. 16(3) cage in N36, valid combinations with R3C9 + R4C8 = {123456} are {169/259/268/349/358/457} (cannot be {367} because 6,7 would be consecutive) -> R3C8 = {789}
18a. Noncon, no 8 in R3C7 [Thanks Peter. Missed that one]
19. 45 rule on R789 2 outies R6C27 = 11 = [83] (only possible combination), R7C2 = 3, R7C78 = {15}, locked for R7 and N9, clean-up: no 5 in R4C1, no 4 in R4C3, no 5 in R4C6, no 9 in R78C9 = {68}, locked for C9 and N9, clean-up: no 2 in R12C9
19a. Noncon, no 7,9 in R5C2 and R6C13, no 2,4 in R5C7 and R6C68, no 4 in R7C13, no 2,4 in R8C2, no 7 in R6C9 and R8C8
19b. No 1 in R4C8 (from combinations in step 18), clean-up: no 6 in R4C3

20. 45 rule on C123 3 outies R169C4 = 20, no 1,2, clean-up: no 9 in R1C3

21. 45 rule on C1234 3 outies R347C5 = 20, no 2

22. 45 rule on C6789 2 innies R56C6 = 10, no 5, no 2,6,7,8 in R5C6
22a. R56C5 = 10, no 5,7,9, no 2 in R5C5

23. Combinations for R789C1 with R9C1 = {68} are {189/468} (cannot be {567} because 5,6 would be consecutive) -> R7C1 = {689}, R8C1 = {14}
[8 locked in R79C1. Thanks Peter. Missed that one too. I haven’t changed the walkthrough for that because it gets locked in the next step.]

24. 18(3) cage in N4 max R5C2 + R6C1 = 11 -> min R5C1 = 7, only valid combination {459} (cannot be {567} because 6,7 would be consecutive) -> R5C1 = 9, R5C2 + R6C1 = {45}, locked for N4, clean-up: no 1 in R23C1, no 3 in R4C1, no 2 in R4C8, no 1 in R6C6
24a. R79C1 = {68}, locked for C1 and N7 -> R8C1 = 4, R6C1 = 5, R5C2 = 4, clean-up: no 2 in R23C1, no 2 in R4C2, no 6 in R6C56
24b. Noncon, no 3 in R5C3, no 5 in R8C2
24c. R23C1 = {37}, locked for C1 and N1, clean-up: no 4,8 in R1C4, no 1 in R4C2

25. R4C3 = 3 (hidden single in N4) -> R3C23 = 8 = [26] (2 cannot be next to 3)
[I’d worked that out, then typed R3C23 = [62] and put that in my diagram! Thanks Peter for correcting me on this.], clean-up: no 9 in R1C4, no 5 in R4C7, R4C8 = 4
25a. Noncon, no 1 in R2C2, no 5 in R2C3, no 5,7 in R3C4, no 3 in R3C1, no 5 in R4C9 and R5C8, no 2 in R5C3

26. R1C1 = 1 (naked single) -> R4C12 = [26], clean-up: no 6 in R2C8, no 7 in R2C9
26a. R3C1 = 7 (naked single), R2C1 = 3, clean-up: no 5 in R1C9
26b. R4C67 = {17}, locked for R4 -> R4C9 = 9, R56C9 = [52] (only valid combination)
26c. R4C45 = {58}, locked for N5 -> R5C4 = 2, R56C5 = [64], clean-up: no 7 in R12C5 = {29}, locked for C5 and N2, R34C5 = {58}, locked for C5 -> R7C5 = 7
26d. Noncon, no 6,8 in R7C46
26e. R7C3 = 9, R7C4 = 4, R7C6 = 2 (naked singles), R8C4 = 6
26f. Noncon, no 5 in R8C3, no 1 in R7C7 -> R7C78 = [51]

27. 45 rule on N1 2 outies R1C4 + R4C3 = 6 -> R1C4 = 3, R1C3 = 8, R2C3 = 4
27a. Noncon, no 9 in R1C2, no 5 in R2C2 -> R12C2 = [59]

and the rest is naked singles, naked pairs, cage sums, simple and noncon elimination

3x3:d:k:4864:4864:5634:5634:1540:5381:5381:4871:4871:4864:3594:3594:5634:1540:5381:2319:2319:4871:4370:4370:3594:5634:22:5381:2319:5657:5657:4370:28:29:30:31:32:33:34:5657:2596:37:38:39:40:41:42:43:3372:2596:2596:47:48:49:50:51:3372:3372:4150:4150:4150:57:58:59:4668:4668:4668:5439:4150:2369:2369:6211:5444:5444:4668:2631:5439:5439:2369:6211:6211:6211:5444:2631:2631:

Sorry, can't help you with the album, I'm not a music freak. But here is the walkthrough for the Album Killer. That was a real tough nut, Ruud.

Walkthrough Album-Killer

0. Preliminary steps
0a. 19(3) at r1c1 and r1c8 : no 1
0b. 6(2) at r1c5 : no 3,6,7,8,9
0c. 22(3) at r3c8 : no 1,2,3,4
0d. 10(3) at r5c1 and r8c9 : no 8,9
0e. 9(3) at r2c7 and r8c3 : no 7,8,9
0f 21(3) at r8c1 : no 1,2,3

1. 45 on N789 : r7c456 = 16(3)

2. 45 on N7 : r8c4 = 1 -> no 4 in r89c3

3. 45 on N9 : r8c6 = 4 -> r89c7 = {89} -> 8,9 locked for c7 and N9

4. 45 on r89 : r8c28 = 5(2) = {23} -> 2,3 locked for r8 -> no 2,3 in r5c5 and r2c28 (from D/ and D\)
4a. no 5,6 in r9c3

5. 2,3 locked in r9c3 and r9c456 for r9 -> 10(3) = {145} -> r8c9 = 5 -> r9c89 = {14} -> 1,4 locked for r9 and N9

6. r89c3 = [62], r8c28 = [32]

7. 3,6,7 locked in r7c789 for r7 and N9

8. N7 : 21(3) = {579} -> 5,7,9 locked for N9, 5 locked for r9

9. N8 : 24(4) = {3678} -> 3,6,7,8 locked for N8

10. 45 on N1 : r1c3 + 5 = r4c1 -> r3c1 = {134}, r4c1 = {689}

11. 45 on N3 : r1c7 + 5 = r4c9 -> r3c7 = {1234}, r4c9 = {6789}

12. N36 : 22(3) = 9{58|67} -> 9 locked in 22(3) -> no 9 in r12c9

13. N3 : 19(3) : no 3,6 in r1c8

14. 45 on N2 : r3c5 + 4 = r1c37 -> r3c5 = {123}, r1c37 = 5(2), 6(2) or 7(2)
r3c5 = 1 -> r12c5 = {24}, r1c37 = {14}|[32]
r3c5 = 2 -> r12c5 = {15}, r1c37 = [42]
r3c5 = 3 -> r12c5 = {15}|[24], r1c37 = {34}

15. N3 : 9(3) : no 4 in r23c7

16. 4 is either in r3c123 or in r3c4 for r3 -> no 4 in r1c3
16a. no 9 in r4c1, no 1 in r1c7, no 6 in r4c9

17. N3 : 1 locked for N9 in 9(3) = 1{26|35} -> no 4 in r2c8, no 5 in r2c7

18. from step 14 : no 2 in r3c5, no 3 in r1c7, no 8 in r4c9

19. N3 : 5,6 locked in 9(3) and r3c89 for N3

20. no 4 in r1c12489
20a. r1c7 = 4 -> no 4 in r1c12489
20b. 4 in 19(3) at r1c8 -> (step 14) r1c37 = [32], r12c5 = [42] -> no 4 in r1c12489

21. N23 : 21(4) : must have one of {24} -> 21(4) = {1479|2379|2469|2478|2568|3459|3468} -> one of {89} in r123c6

22. N12 : 22(4) : must have one of {13}, can't have both of {14} (step 14), must have one of {89} -> 22(4) = {1579|1678|3469|3478|3568} -> no 2,3 in r123c4 -> one of {89} in r123c4

23. 3 locked in r2c79 for r2 and N3

24. N36 : 22(3) : no 8 in r3c8

25. N14 : 17(3) : must have one of {68} -> 17(3) = {269|278|368|458|467} -> no 1

26. no 4 in r2c1
26a. 19(3) = {469} : r2c1 = 4 -> r1c7 = 4 -> r4c9 = 9 -> r1c8 = 9 -> r1c12 = {69} not possible
26b. 19(3) = {478} -> 14(3) = {356} with 3 in r3c3 -> r3c5 = 1 -> r1c37 = [14] -> r1c89 = [92] -> r12c5 = [24] not possible

27. N1 : 19(3) : no 7 in r1c1

28. N3 : 19(3) : no 7 in r2c9

29. N1 : no 2 in 19(3)
29a. r2c1 = 2 -> r1c12 = {89} -> conflict with 19(3) in N3
29b. r1c2 = 2 -> r12c1 = [89] -> r4c1 = 6 -> r1c3 = 1 -> r3c5 = 1 -> r12c5 = [24] not possible

30. 2 locked in r3c12 for r3, N1 and 17(3) -> r3c12 = {279}

31. N1 : 19(3) = {568} ({379} blocked by 17(3) -> 5,6,8 locked for N1

32. N1 : 14(3) : no 7 in r3c3

33. 3 locked in r13c3 for c3 and N1

34. N4 : 10(3) = 3{16|25} -> 3 locked in r56c1 for c1 and N4 -> no 4,7 in 10(3)

35. single: r7c1 = 4

36. 4 locked in r6c4 and r5c5 for D/ and N5

37. 5 locked in r456c3 for c3 and N4

38. N4 : 10(3) = {136} -> 1,3,6 locked for N4

39. r4c1 = 8, r1c23 = [83], r7c23 = [18], r6c2 = 6, r3c12 = [27], r9c2 = 5, r6c6 = 8, r5c8 = 8

40. 1 locked in r1c56 for r1 and N2

41. r3c5 = 3, r1c78 = [49], r12c9 = [73], r347c9 = [896], r3c8 = 5, r2c7 = 2, r89c1 = [79], r89c7 = [98], r8c5 = 8, r2c4 = 8, r13c4 = [56], r12c1 = [65], r12c5 = [24], r123c6 = [179], r2c238 = [916], r3c37 = [41], r9c89 = [41], r5c5 = 5, r4c6 = 2, r7c456 = [295], r45c2 = [42], r56c9 = [42], r6c458 = [417], r56c1 = [13], r6c37 = [95], r7c78 = [73], r45c3 = [57], r4c4578 = [3761], r5c467 = [963], r9c456 = [763]

OK finally got around to finishing it - quite a challenging one.

Posting my walkthrough as there are a few minor differences from [edit] Peter's (oops! - should take a closer look before I post!) - mainly the same through the opening game but a bit different in the middle-game.

1. 45 on n7 - r8c4=1

2. 45 on n9 r8c6=4

3. 45 on r89 r8c28 total 5 = {23}
3a. naked {23} in r8c28 for r8
3b. no 2/3 at r3c2 r3c8 r5c5

4. 9(3) n78 can only be [531]/[621] r9c3={23}
4a naked {23} at r8c8 r9c3 for n7

5. 21(3) n89 can only be 4{89}
5a. {89} locked in 21(3) for n9 and c7

6. 9(3) in n3 - only combo with 4 is {234} - no 4 in r23c7

7.1 only found in r7 in n7
7a 1 locked in 16(4) n7
7b remove from 18(4)n9 in the same row
7c 1 locked in 10(3)n9

8. 10(3) n9. only allowed combo 5{14}/6{13}/7{12} - no 5/6/7 in r9c89

9 must use 7 in 21(3) n7= - 7 locked for n7

10. only combos for 16(4)n7=2{149}/3{148} - no 5/6 in 16(4), must use 4
10a reasoning - can't use {1258}/{1456}/{2356} because it would break the 9(3)n78

11. 4 locked in 16(4)n7 for r7
11a. 4 locked in 10(3)n9 = 5{14} -> r8c3=6 -> r9c4=2 -> r8c2=3 -> r8c8 =2

12. 16(4)n7 = 3{148} - {148} locked for r7

13. 21(3)n7 = {759} with 5 locked for r9

14. 18(4)n9 = 2{367} - {367} locked for r7

15. 24(4)n8={3678}

16. unmarked 16(3)n8 = {259}

17 must use 9 in 22(3)n36 - no 9 in r4c8 r12c9

18 45 on r123 - r4c19 minus r3c5 = 14
18a max r4c19 is 17, so max r3c5=3 ={123}
18b min r3c5 is 1, so max r4c19 is 15 = {69}
18c. r4c19 both limited to {6789}

19 45 on n2 - r1c37 minus r3c5 = 4
19a. 1/2/3, so r1c37 total 5/6/7
19b. no 7/8/9 in r1c3, no 7 in r1c7

20. 19(3)n3 - no possible combo with 3/6 in r1c8

21. 45 n1 r4c1 minus r1c3 equals 5
21a. r4c1 = {689}, r1c3 = {134}

22. 45 n3 r4c9 minus r1c7 equals 5
22a. r1c7 = {1234}

23. 22(3) n36 - no combo with 8 in r3c8

24. 17(3) n14 - no combo with 8 in r3c1

25. 19(3)+9(3) in n3 - no valid combination with 5 in r2c7

26. 19(3)+14(3) in n1 - must use 8 in one or other - no 8 in r3c2

27. 4 locked for r3 in c1-4, all cells can see r1c3,so no 4 in r1c3 -> no 9 in r4c1
27a. (from 18) r4c19=[69]/[87]/[89] - no 6/8 in r4c9 - no 2 in r3c5
27b. (from 22) no 3 in r1c7
27c. (from 19) r1c37=[14]/[32]/[34] - no 1 in r1c7
27d. no 1 in 17(3)n12

28. 3 locked in n3 for r2

29. must use 1 in 9(3)n3={126}/{135}
29a 19(3)n3 combo {568} would conflict with 9(3)n3 - no 5 in r1c8

30. 22(4)n12 - no 3 in r123c4

31. 19(3)&14(3)n1 - {13} in r1c3 restricts combination with 3 at r1c1

32. 14(3)n1 must contain 1 when r1c3=3 and must contain 3 when r1c3=1 - {13} locked in 14(3)+r1c3
32a. 3 locked in n1 for c3
32b. 3 locked in 10(3)n4 = {136}/{235}
32c. 17(3)n12=6{29}/8{27}/8{45}/6{47} - no 6 in r3c12

33. 9(3)&22(3) n3 must use 1,5,6,9 - no 6 in 19(3)n3

34. Placing 1 at r2c5, r2c6 or r3c6 eliminates all possibles for 1 in row 1 (r2c5=1 -> r3c5=3 -> r3c3=1 -> r1c3=3)
34a no 5 in r1c5

35. 4 in r1 can only be at c5 or c789
35a. n3 - 4 in r1 or 4 in r2c9 -> r1c7=2->r1c5=4

36. 19(3)n1 - no 7 in r2c1, no 5 in r1c2

37. cannot place 7 at r4c9
37a r4c9=7 -> r1c7=2->r1c3=3 -> r3c5=1 -> no valid combo in 6(2)n2
37b. r4c9=9 - 22(3)n35=9{67}/[58] -> unmarked cage=23(5)n6
37c. only value 9 at r1c8 n3
37d. 19(3)n3= 9{28}/[73]
37e. only value 4 at r1c7n3
37f. 6(2)n2=[15]/[24]

38. 8 locked in n3 for c9

39. 19(3)&14(3)n1. Only valid combos remove 2 from r2c1, 5,8 from r2c2 8 from r2c3 5,8 from r3c3
39a 8 locked in 19(3)n1

40. (from 19) r1c3=r3c5
40a. r1c3=r3c5=1 -> 6(2)n2=[24]
40b. r1c3=r3c5=3 -> 6(2)n2=[24] since 1 must be used in the 21(4)
40c. 6(2)n2=[24]

41. 19(3)n3=9[82]/[73]

42. 19(3)n1={568} - locked for n1
42a. 14(3)n1=4{19} - {19} locked for r2,n1
42b. 17(3)n14=8{27} - {27} locked for r3
42c. r1c3=3 -> r3c5=3
42d. 22(3)n3=[589]
42e. 19(3)n3=[973]

43. 22(4)n12=3{568} {568} locked for r4 and n2
43a. 21(4)n23=[4179]
43b. 9(3)n3=[261]

44. 23(5)n6=8{1356} r5c8=8

. . . and it all unravels fairly quick from there with singles and cage sums

I started this puzzle in 2007 when it was first posted. I’m now having another look at it in 2012 and have rewritten some steps in the way I now write them. I’d originally got as far as step 11 (in the renumbered steps), plus a few notes.

Prelims

a) R12C5 = {15/24}
b) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
c) 19(3) cage at R1C8 = {289/379/469/478/568}, no 1
d) 9(3) cage at R2C7 = {126/135/234}, no 7,8,9
e) 22(3) cage at R3C8 = {589/679}
f) 10(3) cage at R5C1 = {127/136/145/235} no 8,9
g) 21(3) cage at R8C1 = {489/579/678}, no 1,2,3
h) 9(3) cage at R8C3 = {126/135/234}, no 7,8,9
i) 21(3) cage at R8C6 = {489/579/678}, no 1,2,3
j) 10(3) cage at R8C9 = {127/136/145/235} no 8,9

1. 45 rule on N7 1 outie R8C4 = 1 -> R89C3 = 8 = {26/35}, no 4

2. 45 rule on N9 1 outie R8C6 = 4 -> R89C7 = 17 = {89}, locked for C7 and N9

3. 45 rule on R123 2 outies R4C19 = 1 innie R3C5 + 14
3a. Min R4C19 = 15, no 1,2,3,4,5 in R4C19
3b. Max R4C19 = 17 -> max R3C5 = 3

4. 45 rule on N1 1 outie R4C1 = 1 innie R1C3 + 5 -> R1C3 = {1234}

5. 45 rule on N3 1 outie R4C9 = 1 innie R1C7 + 5 -> R1C7 = {1234}

6. 45 rule on R89 2 innies R8C28 = 5 = {23}, locked for R8, clean-up: no 5,6 in R9C3 (step 1)
6a. Killer pair 2,3 in R8C2 + R9C3 for N7
[If I’d been doing this step now, I’d also have added
Naked pair {23} in R8C28, CPE no 2,3 in R2C28 + R5C5 using the diagonals. These eliminations are made in step 8b.]

7. 1 in R9 only in R9C89, locked for N9
7a. 10(3) cage at R8C9 = {127/136/145}
7b. R8C9 = {567} -> no 5,6,7 in R9C89

[Several of my original steps only listed valid remaining combinations for cages without making any candidate eliminations; I’ve now omitted those steps until they are required.]

[I originally wrote the next step using hidden killer pairs 2,3 for N8, then for R7 to limit 18(3) cage at R7C7 to one combination. However it’s much simpler as …]
8. 24(4) cage at R7C4 = {2589/2679/3579/3678}
8a. Killer pair 2,3 in R9C3 and 24(4) cage, locked for R9
[Alternatively there’s the fun step
Naked pair {23} in R8C2 + R9C3, naked pair {23} in R8C28 -> naked pair R8C8 + R9C3, CPE no 2,3 in R9C89]
8b. Naked pair {14} in R9C89, locked for R9 and N9, R8C9 = 5 (step 7a), R8C3 = 6, R9C3 = 2 (step 1), R8C2 = 3, placed for D/, R8C8 = 2, placed for D\, clean-up: no 7 in R4C1 (step 4)
8c. Naked triple {367} in R7C789, locked for R7

9. 21(3) cage at R8C1 = {579} (only remaining combination), locked for N7
9a. Naked triple {148} in R7C123, locked for R7
9b. Naked triple {259} in R7C456, locked for N8

10. 22(3) cage at R3C8 = {589/679}
10a. 5 of {589} must be in R3C8 -> no 8 in R3C8

11. 9(3) cage at R2C7 = {126/135/234}
11a. 3 of {135/234} must be in R2C7 -> no 4,5 in R2C7
[Not sure why I missed 4 of {234} must be in R2C7 -> no 4 in R3C7. That would have allowed Nasenbaer’s step 16 4 in R3 only in R3C1234, CPE no 4 in R1C3, clean-up: no 9 in R4C1 (step 4). In a way I’m glad I missed this, since I wouldn’t then have had the opportunity for my step 24. ]

[Remaining original steps omitted. They were based on some incorrect analysis of cage interactions.]

[The next step is one which I would now do immediately after the Prelims. However back in 2007 the term CPE probably hadn't been introduced and I often didn't immediately spot this type of step.]
12. 22(3) cage at R3C8 = {589/679}, CPE no 9 in R12C9

13. 19(3) cage at R1C8 = {289/379/469/478} (cannot be {568} = 5{68} which clashes with 22(3) cage at R3C8, killer ALS block), no 5
13a. 9 of {379/469} must be in R1C8 -> no 3,6 in R1C8

14. 9(3) cage at R2C7 = {126/135} (cannot be {234} which clashes with 19(3) cage at R1C8), no 4, 1 locked for N3, clean-up: no 6 in R4C9 (step 5)
14a. Killer pair 5,6 in 9(3) cage and 22(3) cage at R3C8, locked for N3

15. 19(3) cage at R1C8 (step 13) = {289/379/478}
15a. 4 of {478} must be in R12C9 (R12C9 cannot be {78} which clashes with 22(3) cage at R3C8, killer ALS block), no 4 in R1C8
[It can also be seen, from the interactions between 19(3) cage at R1C8 and 22(3) cage at R3C8, that R1C8 and R4C9 are “clones”. However I can’t see any way to use this.]

16. 14(3) cage at R2C2 = {149/158/167/347/356}
16a. 9 of {149} must be in R2C2 + R3C3 (R2C2 + R3C3 cannot be {14} which clashes with R9C9 using D\), no 9 in R2C3

17. 45 rule on N1 3 innies R1C3 + R3C12 = 12 = {129/138/147/237/246/345} (cannot be {156} because 17(3) cage at R3C1 cannot be {56}6)
17a. 3 of {138} must be in R3C1 (R3C12 cannot be {18} because 17(3) cage at R3C1 cannot be {18}8) -> no 8 in R3C1

18. 17(3) cage at R3C1 = {179/269/278/359/368/458/467}
18a. 1 of {179} must be in R3C1 (R34C1 cannot be [79] which clashes with R9C1), no 1 in R3C2

19. 45 rule on N2 2 outies R1C37 = 1 innie R3C5 + 4
19a. Consider values for R3C5
R3C5 = 1 => R1C37 = 5 = [14/32] and if [32] then R12C5 = [42]
or R3C5 = {23} => R1C37 = 6,7 = [42]/{34}
-> R1C357 must contain 4, locked for R1

20. 19(3) cage at R1C8 (step 13) = {289/379/478}
20a. 3,4 of {379/478} must be in R2C9 -> no 7 in R2C9
20b. Consider the combinations for 19(3) cage
19(3) cage = {289/379} => R1C8 = 9
or 19(3) cage = {478} = {78}4, 7,8 locked for R1
-> R1C12 cannot be {79/89}= 16,17 -> no 2,3 in R2C1

21. Consider values for R1C3
R1C3 = 1 => R1C7 = 4 (because min R1C37 = 5, step 19), R1C37 = 5 => R3C5 = 1 (step 19), R12C5 = [24] => no 2 in R1C2
or R1C3 = {34} => R4C1 = {89} (step 4) => R12C1 cannot be {89} = 17 => no 2 in R1C2
-> no 2 in R1C2

22. 2 in N1 only in R3C12, locked for R3
22a. 17(3) cage at R3C1 = {269/278}, no 1,3,4,5
22b. R1C3 + R3C12 (step 17) contains 2 = {129/237/246}, no 8

23. 9(3) cage at R2C7 (step 14) = {126/135}
23a. 2,3 only in R2C7 -> R2C7 = {23}
23b. R2C8 + R3C7 = {15/16}, 1 locked for D/

24. Variable Caged X-Wing 17(3) cage at R3C1 (step 22a) = {269/278}, 21(3) cage at R8C1 = {579} for C12 -> 19(3) cage at R1C1 cannot contain both of 7,9 -> 19(3) cage = {469/478/568} (cannot be {379} which contains both of 7,9), no 3
24a. 4 of {469/478} must be in R2C1 -> no 7,9 in R2C1

25. 3 in C1 only in R56C1, locked for N4
25a. 10(3) cage at R5C1 = {136/235}, no 4,7

26. R1C3 + R3C12 (step 22b) = {129/237/246}
26a. 14(3) cage at R2C2 = {149/158/347/356} (cannot be {167} which clashes with R1C3 + R3C12
26b. Hidden killer pair 1,3 in R1C3 and 14(3) cage for N1, 14(3) cage contains one of 1,3 -> R1C3 = {13}, clean-up: no 9 in R4C1 (step 4)
26c. 17(3) cage at R3C1 (step 22a) = {269/278}
26d. R4C1 = {68} -> no 6 in R3C12

27. R4C19 = R3C5 + 14 (step 3)
27a. R3C5 = {13} -> R4C19 = 15,17 = [69/87/89], no 8 in R4C9, clean-up: no 3 in R1C7 (step 5)
27b. 3 in N3 only in R2C79, locked for R2

28. 19(3) cage at R1C8 (step 20) = {289/379/478}
28a. 9 of {289} must be in R1C8, 7 of {478} must be in R1C8 (R1C89 + R2C9 cannot be [874] which clashes with 22(3) cage at R3C8, killer ALS block) -> no 8 in R1C8
28b. 8 in N3 only in R123C9, locked for C9

29. 14(3) cage at R2C2 (step 26a) = {149/158/347/356}
29a. 3 of {347} must be in R3C3 -> no 7 in R3C3

30. 22(4) cage at R1C3 = {1489/1579/1678/2389/3469/3478/3568} (cannot be {2479/2569/2578/4567} because R1C3 only contains 1,3)
30a. R1C3 = {13} -> no 3 in R13C4
30b. Hidden killer pair 2,4 in R12C5 and rest of cage for N2, R12C5 must contain both of 2,4 or R123C4 must contain 4 if it contains 2 (because no 4 in R123C6) -> 22(4) cage = {1489/1579/1678/3469/3478/3568} (cannot be {2389} = 3{289} which contains 2 but not 4), no 2 in R12C4

31. 14(3) cage at R2C2 (step 26a) = {149/158/347/356}
31a. R1C37 = R3C5 + 4 (step 19)
31b. R3C5 = {13} -> R1C37 = 5,7 = [14/32/34]
31c. Consider the permutations for R1C37
R1C37 = [14] = 5 => R3C5 = 1, R12C5 = [24] => no 4 in R2C123 => 19(3) cage at R1C1 (step 24) = {568}, locked for N1 => 14(3) cage = {149} (cannot be {347} = {47}3)
or R1C37 = [32/34] => 14(3) cage = {149/158}
-> 14(3) cage = {149/158}, no 3,6,7

32. R1C3 = 3 (hidden single in N1) => R3C12 = {27} (step 26), locked for R3 and N1, R4C1 = 8 (step 4)

33. 7 in N3 only in R1C89, locked for R1
33a. 19(3) cage at R1C8 (step 20) contains 7 = {379/478}, no 2
33b. 3,4 only in R2C9 -> R2C9 = {34}
33c. 2 in N3 only in R12C7, locked for C7

34. 22(4) cage at R1C3 (step 30b) = {3469/3478/3568}
34a. Killer pair 4,5 in 22(4) cage and R12C5, locked for N2

35. 19(3) cage at R1C8 (step 33a) = {379/478} = [784/973]
35a. 19(3) cage at R1C1 (step 24) = {568} (only remaining combination, cannot be {469} = {69}4 which clashes with 19(3) cage at R1C8) -> R1C2 = 8, R12C1 = {56}, locked for C1 and N1, R1C9 = 7, placed for D/, R1C8 = 9, R2C9 = 3, R2C7 = 2, R1C7 = 4,R9C1 = 9, placed for D/, R8C1 = 7, R9C2 = 5, R7C9 = 6
35b. R1C37 = [34] = 7 -> R3C5 = 3 (step 19)
35c. R7C3 = 8 (hidden single in R7), placed for D/

36. R4C9 = 9, R3C89 = 13 = [58]
36a. Naked pair {16} in R2C8 + R3C7, locked for D/
36b. Naked triple {245} in R4C6 + R5C5 + R6C4, locked for N5

37. R1C56 = {12} (hidden pair in R1), locked for N2

38. 21(4) cage at R1C6 = {1479/2469} (cannot be {2478} because 7,8 only in R2C6), no 8, 9 locked for C6 and N2

39. R2C4 = 8 (hidden single in N2) -> 22(4) cage at R1C3 (step 34) = {3568} (only remaining combination) -> R13C4 = [56], R2C5 = 4, R1C5 = 2

40. R2C23 = [91], R3C3 = 4, placed for D\

41. R3C12 = [27], R56C1 = {13}, locked for C1 and N4, R6C2 = 6 (cage sum)

42. R6C6 = 8 (hidden single on D\)

43. R9C9 = 1, R56C9 = {24}, locked for N6, R6C8 = 7 (cage sum), R7C8 = 3, R7C7 = 7, placed for D\

and the rest is naked singles.

I'll rate my walkthrough for Name That Album killer at 1.75. I used several fairly short forcing chains, several killer ALS blocks and some variants on standard steps, including a variable caged X-Wing.

3x3::k:6656:6656:2050:2050:3332:4613:4613:4615:4615:6656:2826:2826:2050:3332:4613:3087:3087:4615:6656:2826:6932:6932:6932:6935:6935:3087:4615:2331:2331:6932:4382:4382:6935:6935:1570:1570:3108:3108:6932:4382:4382:1833:1833:3627:3627:3373:3373:5167:5167:2353:4146:4146:2612:2612:3894:3383:5167:5167:2353:4146:4146:4157:5950:3894:3383:3383:5698:2627:3396:4157:4157:5950:3894:3894:5698:5698:2627:3396:3396:5950:5950:

A really tough nut, Ruud. IMHO it was harder than your album killer. But after I realized what the key was it got easier. So here is the walkthrough...

Edit: Additions/corrections in blue, thanks to Ed

Walkthrough Assassin 38

0. Preliminary steps
0a. 26(4) at r1c1 : no 1
0b. 11(3) at r2c2 : no 9
0c. 8(3) at r1c3 : no 6,7,8,9
0d. 13(2) at r1c5 and r6c1 : no 1,2,3
0e. 27(4) at r3c6 : no 1,2
0f. 9(2) at r4c1 and r6c5 : no 9
0g. 6(2) at r4c8 : no 3,6,7,8,9
0h. 12(2) at r5c1 : no 1,2,6
0i. 7(2) at r5c6 : no 7,8,9
0j. 14(2) at r5c8 : no 1,2,3,4,7
0k. 10(2) at r6c8 and r8c5 : no 5
0l. 22(3) at r8c4 : no 1,2,3,4

1. 45 on N1 : r13c3 = 8(2) = [17|26]|{35}

2. 45 on N7 : r79c3 = 17(2) = {89] -> 8,9 locked for c3 and N7

3. 45 on N3 : r13c7 = 15(2) = {69|78}

4. 45 on N9 : r79c7 = 6(2) = {15|24}

5. 45 on c12 : r28c3 = 9(2) = {27|36|45}

6. 45 on c89 : r28c7 = 9(2) = {18|27|36} ({45} blocked by 6(29 from step 4)

7. 45 on N4 : r456c3 = 11(3) = {137|146|236|245}

8. 45 on N6 : r456c7 = 15(3) = {159|249|357} ({258|456} blocked by 14(2), {267|348} blocked by 6(2) and 14(2), {168} blocked by step 3) -> no 6,8 -> no1 in r5c6

9. 45 on r5 : r5c345 = 12(3) = {129|147|237} ({156} blocked by 14(2), {345} blocked by 12(2), {138} blocked by 12(2) and 14(2), {246} blocked by 7(2)) -> no 5,6,8

10. 45 on c5 : r345c5 = 13(3)
10a. 45 on c5 : r3c5 + 4 = r45c4 -> r3c5 : 1..9 , r45c4 : 5..13

11. r4 : 9(2) = {18|27|36} ({4,5} blocked by 6(2))

12. N12 : 8(3) = 1{25|34} -> 1 locked for 8(3) -> no 1 in r1c6

13. N78 : 22(3) = 9{58|67} -> 9 locked for 22(3) -> no 9 in r9c56 -> no 1 in r8c5

14. N6 : 10(2) = {19|37} ({28|46} blocked by 6(2) and 14(2))

15. N1 : 9 locked in 26(4) = 9{278|368|458|467}

16. N3 : 18(4) = {1269|1278|1359|1458|2349|2358|2457|3456} ({1368|1467|2367} blocked by 15(2) from step 3)

17. 20(4) at r6c3 : must have at least one of {89} -> {2567|3467} not possible

18. 45 on N8 : r7c456 = r9c37 = 9..14

19. 45 on N2 : r1c37 + 6 = r3c456 -> r1c37 : 7..14 , r3c456 : 13..20

20. 45 on N123 : r45c3 + 2 = r3c67
using step 7:
20a. min r3c67 : 9 -> min r45c3 : 7
20b. max r45c3 : 10 -> max r3c67 : 12
20c. -> r6c3 = {1234}, r3c6 = {345}

21. 27(4) at r3c6 : no 3,4,5 in r4c67

22. from step 7 : 11(3) : no 2,4 at r4c3 possible

23. from step 8 : 15(3) : no 7,9 at r6c7 possible -> 7,9 locked in r4c7 and r6c89 for N6
other way to put it: Killer pair {79} in r4c7 and r6c89 -> 7,9 locked for N6 (thanks Ed)

24. N6 : 14(2) = {68} -> 6,8 locked for r5 and N6

25. no 4 in r5c12, no 1 in r5c7

26. 7,9 locked in r134c7 for c7 -> no 2 in r28c7

27. from step 9 : 1 locked in 12(3) = 1{29|47} -> no 3

28. N9 : 16(3) : {259|457} blocked by 6(2)

29. N9 : 23(4) : {2579|4568 blocked by 6(2)

30. r6c12 : {49} not possible
30a. r6c12 = {49} -> r5c12 = {57}, r456c3 = {236} (from step 7), r4c12 = {18}, r4c89 = {24}, r6c89 = {37} -> conflict, can't place r5c67 = {34}

31. N4 : 12(2) = {39} -> 3,9 locked for r5 and N4

32. r5 : 7(2) = {25} -> 2,5 locked for r5

33. no 6 in r4c12

34. from step 8 : no 2,5 in r6c7

35. from step 7 : 11(3) = 4{16|25} -> no 7 in 11(3), no 1 at r4c3

36. -> 4 locked in r56c3 for c3 and N4 -> no 5 in r28c3

37. 7 locked in r5c45 for r5, N5 and 17(4) -> no 2 at r7c5

38. 3 locked in r4c45 for r4, N5 and 17(4)

39. N5 : 17(4) = {1736} -> 1,3,6,7 locked for N5

40. r456c3 = [542], r13c3 = [17]

41. N4 : 9(2) = {18} -> 1,8 locked for r4 and N4

42. r4c67 = [97], r13c7 = [96], r35c6 = [52], r56c7 = [53]

43. r79c7 = {24} -> 2,4 locked for c7 and N9

44. N12 : 8(3) = {134} -> r12c4 = {34} -> 3,4 locked for c4 and N2

45. r4c45 = [63], r12c6 = [81], r28c7 = [81], r6c6 = 4, r7c67 = [72], r9c7 = 4

46. N2 : 13(2) = {67} -> 6,7 locked for c5 and N2

47. r5c45 = [71]

48. N78 : 22(3) = {589} -> 5 locked in r89c4 for c4 and N8

49. r6c45 = [85], r7c345 = [914], r9c3 = 8, r89c5 = [82], r3c45 = [29], r2c1 = 9, r5c12 = [39]

50. N1 : 11(3) = {236}

51. r2c23 = [26], r3c12 = [83], r4c12 = [18], r12c5 = [67], r8c3 = 3, r78c2 = [64], r1c12 = [45], r12c4 = [34], r6c12 = [67], r789c1 = [527], r9c2 = 1, r89c6 = [63], r78c8 = [87], r1c89 = [27], r4c89 = [42], r5c89 = [68], r23c8 = [31], r23c9 = [54], r6c89 = [91], r78c9 = [39], r9c89 = [56], r89c4 = [59]

Comments appreciated.

Peter

Looks like we've both used contradiction moves Peter. Found this to be the only way to break open this very tricky puzzle.

My 'squeeze' move comes much earlier than Peter's. Please let me know if anything can be more accurate or clearer.

First, one column
1. "45"n3 -> r13c7 = 15 = h15(2) = {69/78}

2. 6(2)n6 = {15/24}

3. 14(2)n6 = {59/68}

4. "45" n6 -> r456c7 = 15 = h15(3) = {159/249/357} (no 6, 8) = [2/5,4/5,5/9..]
4a. {168} blocked by h15(2)step 1
4b. {258} blocked by 14(2)n6
4c. {267} blocked by h15(2)step 1
4d. {348} -> 14(2)n6 = {59} but clashes with 6(2)n6
4e. {456} blocked by 6(2)n6
4f. no 1 r5c6

5.from h15(3)n6(step 4)-> 14(2)n6 {59} blocked = {68} only: locked for n6, r5

6. 10(2)n6 = {19/37} (no 2, 4) [edit: thanks Andrew]

7. "45" n9 -> r79c7 = 6 = h6(2) = {15/24} = [2/5,4/5..]

8. Killer pairs {25} and {45} in h6(2)n9 and h15(3)n6: (steps 4,7)
8a. 2,4,5 locked for c7

9. "45" c89 -> r28c7 = 9 = h9(2) = {18/36} (no 7.9)

10. 7(2)r5 = {25/34} (no 1) = [3/5..]

11. 12(2)n4 = {39/57} (no 4) = [3/5..]

12. -> Killer pair {35}: locked for r5

Now the second column

13. "45" n7 -> r79c3 = 17 = h17(2) = {89}:locked for n7, c3

14. "45" n4 -> r456c3 = 11 = h11(3) = {146/236/245} (no 7) = [2/6..] ({137} blocked by 12(2)n4)

15."45"n1 -> r13c3 = 8 = h8(2) = [17/]/{35} ([26] blocked by h11(3)n4)
15a. r1c3 = {135}, r3c3 = {357}

16. 1 locked in h8(2) or h11(3), r13456 for c3 (steps 14, 15)

17. "45"c12 -> r28c3 = 9 = h9(2)c3 = {27/36/45}

Now the squeeze

18. "45" r5 -> r5c345 = 12 = h12(3) = {129/147} = [7/9..] with [7/9] only in n5 in 17(4)
18a.can't have both 7 and 9 in a 17(4) (since 7+9=16) -> no 7,8 or 9 r4c45

19. don't know what happened to this one.

20. "45" r5 -> r5c3 + 5 = r4c45

21. Putting steps 18 and 20 together
1.r5c3 -> {r5c45}(step 18)
2.r5c3 + 5 -> {r4c45}(step 20)

21a.
1.r5c3 = 1 -> r5c45 = {29}
2.r5c3:1 + 5 = 6 -> r4c45 = {15} ({24} blocked by 21a.1)

21b.
1.r5c3 = 1 -> r5c45 = {47}
2.r5c3:1 + 5 = 6 -> r4c45 = {15} ({24} blocked by 21b.1)

21c.
1.r5c3 = 2 -> r5c45 = {19}
2.r5c3:2 + 5 = 7 -> r4c45 = Blocked:
..............................{16} blocked by 21c.1
..............................{25} blocked by 6(2)r4
..............................{34} -> 9(2)n4 = {18} only (remembering 2 in r5c3), but {14} in r4c1245 clashes with 6(2)n6.

21d.
1.r5c3 = 4 -> r5c45 = {17}
2.r5c3:4 + 5 = 9 -> r4c45 = {36} only
..................................{18/27} blocked by 21d.1
..................................{45} blocked by 6(2)r4

22. In summary:
22a. r5c3 = {14}
22b. r4c45 = {15/36} (no 2,4)
22c. 17(4)n5 = {1259/1457/1367} = 1{259/367/457}: 1 locked for n5
22d. no 8 r7c5

23. (step 14) h11(3)n4 = {146/245} (no 3) ({236} blocked by r5c3)
23a. = 4{16/25}: 4 locked for n4, c3

24. 13(2)n4 = {58/67} (no 9)

25. 9 n4 only in 12(2) = {39}:locked n4, r5

26. 7(2)r5 = {25}:locked r5

27. 7 for r5 only in n5: locked for n5
27a. no 2 r7c5

28. 9(2)n4 = {18/27} (no 5,6) = [1/2..]
28a. Killer pair {12} with 6(2)n6: locked for r4

29. 17(4)n5 = 17{36/45}: 1 locked for r5
29a. r5c3 = 4

30. r5c45 = {17} -> r4c45 = {36} locked for r4,n5
30a. no 3,6 r7c5

31. r4c3 = 5 -> r6c3 = 2 (h11(3))

32. 6(2) n6 = {24}:locked r4,n6

33. 7(2) r5 = [25]

34. 9(2)n4 = {18}:locked n4,r4

35. r4c67 = [97]

36. r6c7 = 3 (h15(3))

37. r13c7 = [96] (h15(2))

38. r13c3 = [17] (h8(2))

39. 9 n1 in c1: 9 locked c1 -> r5c12 = [39]

40. r3c6 = 5, r12c4 = {34}:locked n2,c4
40a. r4c45 = [63]

41. r12c6 = [81]

42. r28c7 = [81] (h9(2))

43. r12c5 = {67}:locked c5

the rest goes on

I'm with you Para, I didn't see anything particularly tricky in this one.
After the obvious 45 rule moves, it fell fairly quickly to cage combinations.

It was really the cage combinations in r4r5 r6 and n4 n6 that did it for me.

Here's a complete run-through of the order I did it.

[Edit] - tried to recreate this and realised I made an error at step 9 - couldn't see how I'd managed to get that elimination (although it was correct!!) I've gone through it again this evening, and had to solve around it a different way.

[Edit]Thanks to Andrew for some constructive comments on this one

1. 45 on n1: r13c3=8=[17]/[26]/{35}
Remember this one for later on - thanks Andrew

2. 45 on n3: r13c7=15={69}/{78}
... and remember this one for later on

3. 45 on n7: r79c3=17={89} locked for n7 and c3

4. 45 on n9: r79c7=6={15}/{24}

5. 45 on c12 r28c3=9={27}/{36}/{45} - no 1

6. 45 on c89 r28c7=9={18}/{27}/{36}/{45} - no 9
Missed an extra elimination here
6a. No {45} in r28c7 because it would clash with r79c7 from step 4 - thanks Andrew

7. 9(2)n4 - no 4,5 because of conflict with 6(2)n6

8. 9(2), 12(2) and 13(2) in n4 must use 79 - nowhere else in n4
8a. 12(2)n4 can only be {39}/{75} - {48} would force 9(2)={72} or {63} leaving no possible for 13(2)
Andrew comments I should have seen that this also eliminates {36} from 9(2) - I didn't find that until step 17

9b. 14(2)n6 {59} blocked by 12(2)n4 from 8a - can only be {68} locked for n6 - 10(2)n6={19}/{37}
9. 14(2) and 7(2) in r5 must use 35 - nowhere else in r5
9a. 7(2)n56 can only be {25}/{34}
I've reversed steps here, as it is more logical

10. 45 on r123 - r45c3+r4c67=25
10a. max r4c67=17, min r45c3=8, r45c3=[54]/[62]/[64]
10b. r45c3=8,9,10 -> r4c67=17,16,15
10c. r4c67=17 -> [89]
10d. r4c67=16 -> {79}
10e. r4c67=15 -> [87]
10f. r4c3={56}, r5c3={24}, r4c6={789}, r4c7={79}

11 killer pair {79} in n6 - r4c7 & 10(2) - no 7,9 in r6c7

12. 27(4)n2356 no 6,7,8,9 in r3c6

13. 45 n4 r456c3=11 =[641]/[623]/5{42} - no 5,6 in r6c3

14. must use 1 in 8(3)n12 - no 1 in r1c6

15. must use 9 in 22(3)n78 - no 9 in r9c56, no 1 in r8c5

16. 45 on r5 r5c345=12=2{19}/4{17}
16a. 1 locked in 17(4)n5, r5
16b. 17(4)n5 no 1,7,8,9 r4
16c. 9(2)n58 no 8 in r7c5

No longer need this step
17. 45 on r4 r4c34567=30 - must use 6 - no 6 in 9(2)n4 and no 3

18. 3 locked in 17(4)n5 for r4
18a 17(4)n5={1349}/{1367}
18b. 9(2)n58 - no 6 in r7c5
18c. 7(2)n56 - no 4 in r5c7

19. 9 locked in 27(4)n2356 for r4
19a. from 2, no 6 in r1c7

Major rework on this step after some pointers from Andrew
20. 45 rule n2. r1345c3+r4c6+r134c7=48
Remembering that from step 1 r13c3 total 8 and from step 2 r13c7 total 15
i) Possibilities for r13c3=8=[17]/[26]/{35}
ii) Possibilities for r13c7=15={78}/[96]
iii) so Possibilities for r134c7+r4c6 are...
20a. [9678]=30 - r1345c3=18 - r45c3=10 - r1345c3=[1764]/{35}[64]
20b. [9679]=31 - r1345c3=17 - r45c3=9 - r1345c3=[1754]/[2654]
20c. [7897]=31 - r1345c3=17 - r45c3=9 - r1345c3=[1754]/[2654]
20d. [8798]=32 - r1345c3=16 - r45c3=8 - r1345c3={35}[62] - but this would remove 8&2 from 9(2)n4 so not possible
20e. therefore r5c3=4

21 13(2)n4={58}/{76}

22 7(2)n56={25} - locked for r5
22a 12(2)n4={39} - locked for n4 and r5

23 17(4)n5={3617} - locked for n5

24 4 locked in 6(2)n6 for r4 ={42} locked for n6
24a 7(2)n56=[25]

25 1 locked in 9(2)n4 for r4 = {18} locked for n4, r4
25a 13(2)n4={67} locked for n4,r6
25b r4c3=5
25c 10(2)n6={19} locked for n6,r6
25d. r6c7=3
25e r4c7=7
25f r4c6=9

26 Naked single 2 at r6c3

27 hidden single 1 at r1c3 for c3

28 27(4)n2356 = [5697]/[3897]

29 killer pair {35} in n2 8(3) & r3c6
29a 13(2)={49}/{67}
29b. 18(3)=8{64}/[918]

30 9(2)n58={45}/[81]

31 45 on c5 r345c5=13=[931]
31a 17(4)n5=[6371]
31b. 13(2)n2={67} locked for n2, c5
31c. 9(2)n58={45} locked for c5
31d 10(2)n8={28} locked for n8
31e. 18(3)n23=[891]

32. 27(5)n124=[72954]

33 22(3)n78=8{59} - {59} locked for n8, c6
33a 10(2)n8=[82]
33b. 8(3)n12=1{34} - {34} locked for n2,c4
33c. 20(4)n4578 = [2891]
33d 27(4)n2356=[5697]
33e 16(4)n5689 = [4372]
33f 9(2)n58=[54]

34 13(3)n89={36}4
34a r8c7=1, r2c7=8

35 naked {36} at r8c36 for r8

36 12(3)n3=[831]
36a 10(2)n6=[91]
36b r3c9=4
36c 6(2)n6=[42]
36d 8(3)n12=[134]
36e r2c3=6, r8c3=3, r8c6=6, r9c6=3

37. 11(2)n1=[263]
37a r3c1=8
37b 9(2)n4=[18]
37c 12(2)n4=[39]

38 {45} locked in r1 of 26(4) n1
38a r2c1=9
38b 18(4)n3=[2754]
38c 13(2)n2=[67]
38d r8c9=9, r9c9=6
38e 14(2)n6=[68]
38f 23(4)n9=[3965]
38g 22(3)n78=[598]
38h 16(3)n9=[817]
38i 13(3)n7=[643]
38j 15(4)n7=[5271]
38k 26(4)n1=[4598]
38l 13(2)n4=[67]

Another one that I only did fairly recently. It does have one contradiction move but a much simpler one than in the first two walkthroughs for this puzzle.

Richard's walkthrough did some excellent combination work that avoided the need for any contradiction moves. He always seems to get more out of large groups of innies or outies than the rest of us. I was particularly impressed by step 20 which broke the puzzle open.

Here is my walkthrough. There were times when I was struggling to find the way forward ](*,) so a few steps are just statements of combinations that I found while searching for useful steps. I've left them in because they show what I was looking at. Then I found the eliminations in N6 (step 28) and the key 45 in R5 (step 32). After that a short contradiction move in step 33 breaks it open.

Clean-up is used in various steps, using the combinations in steps 1 to 10 for further eliminations from these two cell cages and for split cages formed by the use of the 45 rule.

Thanks Ed for pointing out the flaw in step 28b.

1. R12C5 = {49/58/67}, no 1,2,3

2. R4C12 = {18/27/36/45}, no 9

3. R4C89 = {15/24}

4. R5C12 = {39/48/57}, no 1,2,6

5. R5C67 = {16/25/34}, no 7,8,9

6. R5C89 = {59/68}

7. R6C12 = {49/58/67}, no 1,2,3

8. R67C5 = {18/27/36/45}, no 9

9. R6C89 = {19/28/37/46}, no 5

10. R89C5 = {19/28/37/46}, no 5

11. 8(3) cage in N12 = 1{25/34}, no 1 in R1C6

12. 11(3) cage in N1 = {128/137/146/236/245}, no 9

13. 22(3) cage in N78 = 9{58/67}, no 9 in R9C56, clean-up: no 1 in R8C5

14. 26(4) cage in N1 = {2789/3689/4589/4679/5678}, no 1

15. 27(4) cage in N2356 = 9{378/468/567}, no 1,2

16. 45 rule on N1 2 innies R13C3 = 8 = [17/26/35/53]

17. 9 in N1 locked in 26(4) cage = 9{278/368/458/467}

18. 45 rule on N3 2 innies R13C7 = 15 = {69/78}

19. 45 rule on N7 2 innies R79C3 = 17 = {89}, locked for C3 and N7

20. 45 rule on N9 2 innies R79C7 = 6 = {15/24}

21. 45 rule on C12 2 outies R28C3 = 9 = {27/36/45}, no 1

22. 45 rule on C89 2 outies R28C7 = 9 = {18/27/36} (cannot be {45} which would clash with R79C7), no 4,5,9

23. 13(3) cage in N7 = {157/247/256/346}

24. 15(4) cage in N7 = {1257/1347/1356/2346}

25. 45 rule on N4 3 innies R456C3 = 11 = {146/236/245} (cannot be {137} which clashes with all possible combinations for R56C12), no 7
25a. R456C3 must contain 1 or 2 -> R4C12 must contain 1 or 2 = {18/27}, no 3,4,5,6
[Edit. Step number corrected.]
25b. If R4C12 = {18} => R5C12 cannot be {48}
If R4C12 = {27} => R456C3 => {146} => R5C12 cannot be {48}
-> no 4,8 in R5C12
[Edit. Step 25b added for use with the corrected step 28b. Thanks Richard for giving me the idea for step 25b which I hadn’t spotted before I saw your walkthrough.]

26. Killer pair 1/2 in R4C12 and R4C89 for R4

27. 9 in C2 locked in R156C2
27a. 45 rule on C1 5 outies R14569C2 = 30 = 9{1578/2478/2568/3468/3567}, must contain two of 6,7,8
27b. 45 rule on C2 4 innies R2378C2 = 15 = {1248/1257/1347/1356/2346}

28. 45 rule on N6 3 innies R456C7 = 15 = {159/168/249/357} (cannot be {258/456} which clash with R5C89, cannot be {267/348} which clash with the combination of R45C89)
28a. For the valid combinations for R456C7, after working the interactions with the other three cages in N6, R6C89 can only be {19/37}, no 2,4,6,8
[Edit. An alternative way, suggested by Ed, that eliminates 2,4,6,8 from R6C89.]
If R4C89 = {24} => no 2,4,6,8 in R6C89
If R4C89 = {15} => R5C89 = {68} => no 2,4,6,8 in R6C89
-> R6C89 = {19/37}

The original step 28b was flawed. Thanks Ed for pointing that out. Here is a replacement step.
28b. R5C12 = {39/57} -> R5C89 = {68} (cannot be {59} which clashes with R5C12)
28c. R5C89 = {68} (hidden pair in N6), locked for R5, clean-up: no 1 in R5C67
[Edit. Clean-up edited after adding step 25b and changing step 28b.]
28d. Killer pair 3/5 in R5C12 and R5C67 for R5
28e. 45 rule on R5 3 innies R5C345 = 12 = 1{29/47}
28f. R456C7 = {159/249/357} [1/2/3, 4/5, 7/9]

29. Killer pair 7/9 in R13C7 and R456C7 for C7, clean-up: no 2 in R28C7

30. 45 rule on R4 5 innies R4C34567 = 30 and must contain 3,6,9 = 369{48/57}

31. 45 rule on R6 5 innies R6C34567 = 22 and must contain 2 = 2{1469/1568/3458/3467} (cannot be 2{1379/1478} which clash with R6C89)

32. 45 rule on R5 2 outies R4C45 – 5 = 1 innie R5C3, min R4C45 = 7 -> min R5C3 = 2, max R5C3 = 4 -> max R4C45 = 9, no 7,8,9
32a. R5C345 = 1{29/47}, R5C3 = {24}, no 2,4 in R5C45
32b. 1 in R5 locked in R5C45, locked for N5, clean-up: no 8 in R7C5

33. If R5C3 = 2, R5C67 = {34}, R4C45 = 7 (step 32) = {34} -> R5C3 <>2
33a. R5C3 = 4, clean-up: no 5 in R28C3, no 3 in R5C67 = {25}, locked for R5, R5C45 = {17} (step 28e), locked for R5 and N5, R5C12 = {39}, locked for N4, R4C45 = 9 = {36} (cannot be {45} which would clash with R4C89), locked for R4 and N5, clean-up: no 2,3,6 in R7C5

34. R4C3 = 5 (naked single), clean-up: no 3 in R13C3, no 1 in R4C89 = {24}, locked for R4 and N6, no 8 in R6C12 = {67}, locked for R6 and N4, no 3 in R6C89 = {19}, locked for R6 and N6

35. R5C67 = [25], R4C7 = 7, R6C3 = 2, R6C7 = 3, R1C3 = 1 (naked singles), clean-up: R3C3 = 7, no 7 in R7C5, no 8 in R13C7 = {69}, locked for C7 and N3, no 1 in R79C7 = {24}, locked for N9

36. R6C456 = {458}, locked for N5

37. R4C6 = 9, R3C7 = 6, R1C7 = 9 (naked singles), R3C6 = 5 (cage sum), clean-up: no 2 in R12C4, no 8 in R12C5, no 4 in R2C5

38. R12C4 = {34}, locked for C4 and N2, clean-up: no 9 in R2C5
38a. R12C5 = {67}, locked for C5 and N2, clean-up: no 3,4 in R89C5

39. R4C45 = [63], R5C45 = [71], R12C6 = [81], R28C7 = [81], R6C6 = 4, R79C7 = [24] (naked singles), R7C6 = 7 (cage sum), clean-up: no 9 in R8C5

40. R3C4 = 2 (hidden single in C4) -> R3C5 = 9

41. 22(3) cage in N78 = {589} -> no 8 in R9C5, R89C5 = [82], R89C4 = {59}, locked for C4, N8 and 22(3) cage

42. R6C45 = [85], R7C45 = [14], R79C3 = [98] (naked singles)

43. R2C1 = 9 (hidden single in N1) -> R5C12 = [39]

44. R2C7 = 8 -> R23C8 = 4 = [31] (only remaining combination)

and the rest is naked singles

3x3:d:k:6144:1793:1793:3331:3331:3331:2566:2566:6144:3337:5386:3586:3586:3853:3076:3076:5386:3089:3337:3859:6676:3853:3853:3853:6676:3609:3089:3337:3859:3859:11550:11550:11550:3848:3609:3089:5156:5156:5156:11550:11550:11550:3848:3848:3848:3373:1070:5156:11550:11550:11550:5171:5171:5685:3373:1070:6676:3129:3129:3129:6676:5171:5685:3373:5386:2881:2881:3129:2882:2882:5386:5685:6144:1541:1541:5383:5383:5383:2373:2373:6144:

Hey Ed

Nice Killer. Had to result to one ugly move. Must be a nicer way through it.
So here is my walk-through. Any comments or corrections are appreciated.

Walk-through Black Hole Killer

1. R1C23 = {16/25/34}
2. R1C78 = {19/28/37/46}
3. R2C34 and R34C8 = {59/68}
4. R2C67 = {39/48/57}
5. R37C37 = 26(4): no 1
6. 20(3) in R6C7 = {389/479/569/578}
7. 22(3) in R678C9 = {589/679} -->> 9 locked for C9
8. R8C34 and R8C67 = {29/38/47/56}
9. R9C23 = {15/24}
10. R9C456 = {489/579/678}
11. R9C78 = {18/27/36} : {45} clashes with 6(2) in R9C2
12. 12(4) in R7C4 = {1236/1245} -->> 1 and 2 locked for N8
12a. Clean up : R8C37: no 9
13. R67C2 = {13} locked for C2
13a. Clean up : R1C3: no 4,6; R9C3: no 5
14. Sum 3 remote cages = 71 -->> 2*R5C5 + R46C46 = 19: min. R46C46 = 10; max. R5C5 = 4: no 5,6,7,8,9
14a. Max R46C46 + R5C5 = 18: no 9
14b. 45 on N5: Min. R46C5 + R5C46 = 27 : no 1 or 2
15. 45 on R1: 2 innies: R1C19 = 15 = {78}/[96]
15a. 45 on R1: 2 outies: R9C19 = 9 = {18/27/36}: {45} clashes with 6(2) in R9C2
16. 45 on N2: 2 innies: R2C46 = 17 = {89}: locked for N2 and R2
16a. Clean up : R2C3 = {56}; R2C7 = {34}
17. 9 locked in R9 and 21(3) in R9C4 for N8
17a. 21(3) in R9C4 = {489/579}: no 6
18. 45 on N8: 2 innies: R8C46 = 12 = {48/57}
18a. Clean up: R8C37 : no 2,5,8
19. 6 locked in N8 for 12(4) in R7C4: 12(4) = {1236}
19a. 1 and 3 can’t both be in R7C456: clashed with R7C2: R8C5 = {13}
19b. 2 and 6 locked in N8 for R7
19c. Killer Pair {13} in R7C2 + R7C456
20. Killer Pair {67} in R1C19 and 13(3) in R1C4 for R1
20a. Clean up: R1C3: no 1; R1C78: no 3,4
21. 45 on C9: 3 innies: R159C9 = 11 = [8]{12}/[7]{13}/[6]{23}/[6][41]
21a. R5C9 = {1234}; R9C9 = {123}; R9C1 = {678}
21b. R59C9 always has 1 or 3(or both): 12(3) in R2C9 can’t be {138}: no 8 in R34C9
22. 45 on C1: 3 innies: R159C1 = 19: [928]/[937]/{8[4]7}/[856] ([946] clashes with R1C19 = [96] using D/)-->> R5C1 = {2345}
22a. R159C1 = [937] -->> R159C9 = [632] : 2 3’s in R5: R159C5 can’t be [937]: R5C1: no 3
23. 24(4) in R1C1 = {1689/2679/3678}: 6 locked in 24(4) for D/
24. 1 locked in N1 for C1 and 13(3) in R2C1
24a. 13(3) in R2C1 = {139/148/157}: no 2,6
24b. 1’s in R8: R8C58: crossover: no 1 in R5C5
24c. Max. R46C46 + R5C5 = 17: no 8 (step 14)
24d. Min R5C46+R46C5 = 28: no 3 (step 14)
25. 13(3) in R6C1 = {238/247/256/346}: no 9
25a. 9 locked in N7 for D/: R3C7: no 9
26. 3 locked in R9 for N9
26a. Clean up: R8C6: no 8; R8C4: no 4; R8C3: no 7
27. Hidden killer Triple {123} in N9 in R8C8 + 9(2) in R9C7 +R9C9 -->> R8C8 = {12}
28. R8C3: no 6
28a. R8C3 = 6 -->>R8C46 = [57] -->> R8C7 = 4
28b. R8C3 = 6 -->>R2C3 = 5 -->> R2C46 = 98 -->> R2C7 = 4 : 2 4’s in C7
28c. Clean up: R8C4: no 5; R8C6: no 7; R8C7: no 4
29. Killer Triple {134}for N7 in R7C2 + R8C3 + 6(2) in R9C2
30. 13(3) in R6C1 = [382]/[472]/{2[5]6} (needs 5, 7 or 8 in R7C1)
30a. R6C1 = {2346}; R8C1 = {26}
30b. 2 locked in 13(3) in R6C1 for C1: R5C1: no 2
31. R159C1 = {8[4]7}/[856] (step 22): R1C1: no 9
31a. R1C19 = {78} -->> locked for R1 and 24(4) in R1C1 (step 15)
31b. Naked single 6 in R1C9 -->> R9C9 = 3 (step 15a)
31c. R8C1 = 2; R9C23 = [51]; R67C2 = [13]
31d. R8C34 = [47]; R8C67 = [56]; R8C58 = [31];
32. 9 locked in 13(3) in R2C1 -->> 13(3) = {139} locked for C1
32a. R67C1 = [47]; R5C1 = 5; R1C1 = 8; R1C9 = 7; R1C78 = [19]
32b. R4C6 = 1; R5C9 = 1; R6C4 = 3; R2C7 = 3; R5C8 = 3 (hidden singles)
32c. R2C6 = 9; R2C34 = [68]; R2C1 = 1
33. Naked Pair {89} in R7C3 + R8C2 for D/
33a. Hidden single 8 in R3C8; R4C8 = 6; Hidden single 6 in R3C9 and R5C2
33b. Hidden single 6 in R6C6 for D\; R7C6 = 2
34. 9’s on D\ in R3C3 + R7C7 locked for 26(4) -->> R7C3: no 9
34a. R7C3 = 8; R8C2 = 9; R8C9 = 8; R4C2 = 8 (hidden)
34b. 15(3) in R3C2 = [483] -->> R3C2 = 4; R4C3 = 3
And the rest is singles or cage sums.

There must be a nicer way past step 28.

Ok here is the cleaned up and nicer walk-through. Without the ugly moves.

Walk-through Black Hole Killer

1. R1C23 = {16/25/34}
2. R1C78 = {19/28/37/46}
3. R2C34 and R34C8 = {59/68}
4. R2C67 = {39/48/57}
5. R37C37 = 26(4): no 1
6. 20(3) in R6C7 = {389/479/569/578}
7. 22(3) in R678C9 = {589/679} -->> 9 locked for C9
8. R8C34 and R8C67 = {29/38/47/56}
9. R9C23 = {15/24}
10. R9C456 = {489/579/678}
11. R9C78 = {18/27/36} : {45} clashes with 6(2) in R9C2
12. 12(4) in R7C4 = {1236/1245} -->> 1 and 2 locked for N8
12a. Clean up : R8C37: no 9
13. R67C2 = {13} locked for C2
13a. Clean up : R1C3: no 4,6; R9C3: no 5
14. Sum 3 remote cages = 71 -->> 2*R5C5 + R46C46 = 19: min. R46C46 = 10; max. R5C5 = 4: no 5,6,7,8,9
14a. Max R46C46 + R5C5 = 18: no 9
14b. 45 on N5: Min. R46C5 + R5C46 = 27 : no 1 or 2
15. 45 on R1: 2 innies: R1C19 = 15 = {78}/[96]
15a. 45 on R1: 2 outies: R9C19 = 9 = {18/27/36}: {45} clashes with 6(2) in R9C2
16. 45 on N2: 2 innies: R2C46 = 17 = {89}: locked for N2 and R2
16a. Clean up : R2C3 = {56}; R2C7 = {34}
17. 9 locked in R9 and 21(3) in R9C4 for N8
17a. 21(3) in R9C4 = {489/579}: no 6
18. 45 on N8: 2 innies: R8C46 = 12 = {48/57}
18a. Clean up: R8C37 : no 2,5,8
19. 6 locked in N8 for 12(4) in R7C4: 12(4) = {1236}
19a. 1 and 3 can’t both be in R7C456: clashed with R7C2: R8C5 = {13}
19b. 2 and 6 locked in N8 for R7
19c. Killer Pair {13} in R7C2 + R7C456
20. Killer Pair {67} in R1C19 and 13(3) in R1C4 for R1
20a. Clean up: R1C3: no 1; R1C78: no 3,4
21. 45 on C9: 3 innies: R159C9 = 11 = [8]{12}/[7]{13}/[6]{23}/[6][41]
21a. R5C9 = {1234}; R9C9 = {123}; R9C1 = {678}
21b. R59C9 always has 1 or 3(or both): 12(3) in R2C9 can’t be {138}: no 8 in R34C9
22. 45 on C1: 3 innies: R159C1 = 19: [928]/[937]/{8[4]7}/[856] ([946] clashes with R1C19 = [96] using D/)-->> R5C1 = {2345}
23. 45 on C19: 6 innies = 30: R5C19 = 30 – 24(4) in R19C19 = 6 = {24}/[51]
23a. R159C1 : not [937]
24. 24(4) = {1689/2679/3678}: 6 locked in 24(4) for D/
25. Combining step 15 and 21-24
25a. 24(4) = [9681/8763]: [9672] clashes with step 23a.; [7863] clashes with step 21
25b. R1C1 = {89}; R1C9 = {67}; R5C1 = {25}; R5C9 = {14}; R9C1 = {68}; R9C9 = {13}
25c. 22(3) in R6C9 = {589}-->> locked for C9: {679} clashes with R1C9
25d. 2 locked in 12(3) in R2C9-->> 12(3) = {237/246}: no 1
26. 1 locked in N1 for C1 and 13(3) in R2C1
26a. 13(3) = {139/148/157} : no 2 or 6
26b. 13(3) in R6C1 = {238/247/256/346}: no 9
26c. 9 locked in N7 for D/: R3C7: no 9
27. 3 locked in R9 for N9
27a. Clean up: R8C6: no 8; R8C4: no 4; R8C3: no 7
27b. Hidden Killer Triple {123}in N9 in R8C8 + R9C9 + 9(2) in R9C7 -->> R8C8 = {12}
28. {4,7} locked in R8C3467 for R8, like this...
28a. R8C67 = {47}/[56]; R8C67 = [56] -->> R8C34 = [47] (step 18) (alternatively, Hidden killer pair 4,7 in N8)
28b. 7 locked in N7 for R7
28c. Naked quad {4589} in R7C789 + R8C9 for N9
28d. Clean up: R9C78: no 1; R8C6: no 7; R8C4: no 5; R8C3: no 6
29. Killer Triple {134}for N7 in R7C2 + R8C3 + 6(2) in R9C2
30. 13(3) in R6C1 = [382]/[472]/{2[5]6} (needs 5, 7 or 8 in R7C1)
30a. R6C1 = {2346}; R8C1 = {26}
30b. 2 locked in 13(3) in R6C1 for C1: R5C1: no 2
31. R5C19 = [51]; R19C9 = [73] (step 21); R19C1 = [86] (step 22)
31a. R678C1 = [472]; R9C23 = [51]; R67C2 = [13]
31b. R8C34 = [47]; R8C67 = [56]; R8C8 = 1; R8C5 = 3
31c. R1C78 = [19]; R4C6 = 1(hidden)
32. Crossover 9’s on D\ in R3C3 and R7C7 -->> R7C3: no 9
32a. R7C3 = 8; R8C29 = [98]
32b. R3C8 = 8(hidden); R4C8 = 6
33. 20(4) = 5{267} -->> locked for N4
33a. R4C2 = 8
33b. 15(3) in R3C2 = 8{34} -->> R3C2 = 4; R4C3 = 3
33c. R3C1 = 9; R1C23 = [25]; R2C34 = [68]; R3C67 = [93]
33d. R23C1 = [13]; R2C2 = 7; R3C3 = 9; R5C2 = 6; R2C8 = 4 (cage sum)
And the rest is singles and cage sums.

greetings

Para

Thanks Ed for another nice puzzle.

Para's walkthroughs made interesting use of the sum of those three cages by using combined 45 rule on the two diagonals (step 14). Step 23 in Para's second walkthrough was a nice improvement on step 22a in his first one.

This puzzle wasn't too difficult, even though I didn't spot Para's combined 45 rule on the two diagonals which probably gave a shorter solving path.

Here is my walkthrough for Black-Hole remote killer-X.

Prelims

a) R1C23 = {16/25/34}, no 7,8,9
b) R1C78 = {19/28/37/46}, no 5
c) R2C34 = {59/68}
d) R2C67 = {39/48/57}, no 1,2,6
e) R34C8 = {59/68}
f) R67C2 = {13}
g) R8C34 = {29/38/47/56}, no 1
h) R8C67 = {29/38/47/56}, no 1
i) R9C23 = {15/24}
j) R9C78 = {18/27/36/45}, no 9
k) 20(3) cage at R6C7 = {389/479/569/578}, no 1,2
l) 22(3) cage at R6C9 = {589/679}
m) 21(3) cage in N8 = {489/579/678}, no 1,2,3
n) 12(4) cage in N8 = {1236/1245}, no 7,8,9
o) 26(4) disjoint cage at R37C37 = {2789/3689/4589/4679/5678}, no 1
p) And, of course, 45(9) cage in N5 = {123456789}

Steps resulting from Prelims
1a. Naked pair {13} in R67C2, locked for C2, clean-up: no 4,6 in R1C3, no 5 in R9C3
1b. 22(3) cage at R6C9 = {589/679}, 9 locked for C9
1c. 12(4) cage in N8 = {1236/1245}, 1,2 locked for N8, clean-up: no 9 in R8C37
1d. R9C78 = {18/27/36} (cannot be {45} which clashes with R9C23), no 4,5

2. 45 rule on R1 2 innies R1C19 = 15 = [78/87/96], no 6 in R1C1

3. 45 rule on N2 2 innies R2C46 = 17 = {89}, locked for R2 and N2, clean-up: no 5,7 in R2C7

4. 45 rule on R9 2 innies R9C19 = 9 = {18/27/36} (cannot be {45} which clashes with R9C23), no 4,5,9

5. 45 rule on N8 2 innies R8C46 = 12 = {39/48/57}, no 6, clean-up: no 5 in R8C37

6. 9 in R9 only in 21(3) cage, locked for N8, clean-up: no 2 in R8C37
6a. 21(3) cage = {489/579} (cannot be {678} which doesn’t contain 9), no 6

7. 6 in N8 only in 12(4) cage = {1236} (only remaining combination), locked for N8, clean-up: no 8 in R8C37
7a. 2,6 of {1236} must be in R7C456 (R7C456 cannot be {123/136} which clash with R7C2), locked for R7 and N8
7b. Killer pair 1,3 in R7C2 and R7C456, locked for R7

8. Hidden killer pair 8,9 in R1C19 and R1C78 for R1, R1C19 contains one of 8,9 -> R1C78 must contain one of 8,9 -> R1C78 = {19/28}, no 3,4,6,7

9. 45 rule on C1 3 innies R159C1 = 19 = {289/379/469/478/568}, no 1, clean-up: no 8 in R9C9 (step 4)

10. 45 rule on C9 3 innies R159C9 = 11 = {128/137/146/236} (cannot be {245} because R1C9 only contains 6,7,8), no 5
10a. R1C9 = {678} -> no 6,7,8 in R59C9, clean-up: no 2,3 in R9C1 (step 4)
10b. 3 in R9 only in R9C789, locked for N9, clean-up: no 8 in R8C6, no 4 in R8C4 (step 5), no 7 in R8C3

11. 24(4) disjoint cage at R19C19 = {1689/2679/3678}, 6 only in R1C9 + R9C1, locked for D/

12. R159C1 (step 9) = {289/379/568} (cannot be {469/478} which clash with R1C19, CCC using D/), no 4
12a. 2,3,5 only in R5C1 -> R5C1 = {235}
12b. 9 of {379} must be in R1C1 -> no 7 in R1C1, clean-up: no 8 in R1C9 (step 2)
[Ed told me that SudokuSolver found my CCC and also a similar one on the other side
R159C9 = {128/137/146/236} cannot be [623/821] which clash with R9C19, CCC using D/ -> no 2 in R5C9]

13. 22(3) cage at R6C9 = {589} (only remaining combination, cannot be {679} which clashes with R1C9), locked for C9

[At this stage I looked at the permutations for 24(4) disjoint cage at R19C19 = {1689/2679/3678} (step 11), taking into account steps 2 and 4, which gave [9681/9672/8763] and spotted that [9672] is blocked because R159C1 = [937] clashes with R159C9 = [632] => 8 locked for C1, R5C1 = {25}, R5C9 = {14}, etc.
This looks rather heavy so I’ll continue looking for simpler steps.]

14. Hidden killer pair 4,5 in R1C23 and 13(3) cage for R1, R1C23 and 13(3) cage can each only contain one of 4,5 -> R1C23 = [25/43/52], no 1,6, 13(3) cage = {157/256/346} (cannot be {247} which clashes with R1C23)

15. 1 in N1 only in R23C1, locked for C1
15a. 13(3) cage at R2C1 = {139/148/157}, no 2,6
15b. 13(3) cage at R6C1 = {247/256/346} (cannot be {238} which clashes with R159C1), no 8,9
15c. 5 of {256} must be in R7C1 -> no 5 in R68C1
15d. 1 in R8 only in R8C58, CPE no 1 in R5C5 using D\

16. 8,9 in N7 only in R7C3 + R8C2 + R9C1, locked for D/

17. Hidden killer pair 8,9 in R1C78 and R3C8 for N3, R1C78 can only contain one of 8,9 -> R3C8 = {89}, clean-up: no 8,9 in R4C8
17a. 6 in N3 only in R123C9, locked for C9

18. 5 in N3 only in R2C8 + R3C7, locked for D/
[Note. This means that one or both of the disjoint cages R28C28 and R37C37 must contain 5. Also from step 16 at least one of these cages must contain 8 or 9. Don’t know how this can be used.]

19. R8C46 (step 5) = [57/75/84]
19a. 45 rule on N8 2 outies R8C37 = 10 = [37/46/64]
19b. Killer pair 4,7 in R8C37 and R8C46, locked for R8

20. 26(4) disjoint cage at R37C37 = {2789/3689/4589/4679/5678}
20a. 6 of {3689} must be in R3C3 -> no 3 in R3C3

21. 3 in N1 only in R1C23 = [43] or in 13(3) cage at R2C1 (step 15a) = {13}9 -> no 4 in R23C1 (locking-out cages)
21a. 13(3) cage at R2C1 (step 15a) = {139/148/157}
21b. 4 of {148} must be in R4C1 -> no 8 in R4C1

22. Hidden killer triple 1,2,3 in R8C8, R9C78 and R9C9 for N9, R9C78 must contain one of 1,2,3, R9C9 = {123} -> R8C8 = {12}
22a. 45 rule on N2 2 outies R2C37 = 9 = [54/63]
22b. 21(4) disjoint cage R28C28 = {1479/1578/2379/2478} (cannot be {1389} because 8,9 only in R8C2, cannot be {1569/2568} which clash with R2C3, cannot be {2469} which clashes with R2C37, cannot be {3459/3468/3567} which don’t contain 1 or 2), no 6, 7 locked for R2, CPE no 7 in R5C5 using diagonals
22c. R8C8 = {12} -> no 1,2 in R2C28 + R8C2

23. 1 on D/ only in R4C6 + R6C4, locked for N5
23a. 1 on D\ only in R8C8 + R9C9, locked for R9C78, clean-up: no 8 in R9C78

[Now for a bit of nibbling at cages, until I find something more interesting.]
24. 15(3) cage at R3C2 = {159/168/249/258/267/348/357/456}
24a. 1 of {159} must be in R4C3, 9 of {249} must be in R34C2 (R34C2 cannot be {24} which clashes with R19C2, ALS block) -> no 9 in R4C3
24b. 1,3 of {168/348} must be in R4C3, 8 of {258} must be in R34C2 (R34C2 cannot be {25} which clashes with R19C2, ALS block) -> no 8 in R4C3

25. 20(3) cage at R6C7 = {389/479/578} (cannot be {569} which clashes with R4C8 because all cells of 20(3) cage “see” R4C8), no 6
25a. 3 of {389} must be in R6C8 (R67C8 cannot be {89} which clashes with R3C8) -> no 3 in R6C7

26. R2C37 = [54/63] (step 22a), R2C28 = {3457} -> variable combined cage R2C2378 (permutations written as R2C37 + R2C28) = [54][73] / [63]{47} / [63]{57}, 3 must be in R2C78, locked for R2 and N3

[Another nibble, this time using bigger ALS blocks. With hindsight this step wasn’t necessary, because of steps 28 and 29, but I’ve left it in because of the interesting ALS blocks.]
27. 15(3) cage at R3C2 = {159/168/249/258/267/348/456} (cannot be {357} which clashes with R129C2, ALS block)
27a. 1 of {168} must be in R4C3, 6 of {456} must be in R34C2 (R34C2 cannot be {45} which clashes with R19C2, ALS block), 6 of {267} must be in R34C2 (R34C2 cannot be {27} which clashes with R129C2, ALS block) -> no 6 in R4C3

28. 15(4) cage in N6 = {1239/1248/1257/1347/2346} (cannot be {1356} which clashes with R4C8)
28a. Consider combinations for R1C78
R1C78 = {19}, locked for N3 => R3C8 = 8, R4C8 = 6 => 15(4) cage not {2346}
R1C78 = {28} => 1 in N3 only in R23C9, locked for C9 => 1 in N6 only in 15(4) cage
-> 15(4) cage in N6 = {1239/1248/1257/1347}, no 6, 1 locked for N6

29. R4C8 = 6 (hidden single in N6), R3C8 = 8, clean-up: no 2 in R1C78, no 3 in R9C7

30. Naked pair {19} in R1C78, locked for R1 and N3 -> R1C1 = 8, placed for D\, R1C9 = 7 (step 2), placed for D/, R9C1 = 6, R9C9 = 3 (step 4), placed for D\, R5C1 = 5 (step 9), R5C9 = 1 (step 10), clean-up: no 5 in R8C4

31. Naked pair {27} in R9C78, locked for R9 and N9 -> R8C8 = 1, R1C78 = [19], R8C5 = 3, R8C1 = 2, R8C3 = 4, R8C4 = 7, R8C67 = [56], R9C23 = [51], R67C2 = [13], R7C1 = 7, R6C1 = 4 (step 15b), clean-up: no 2 in R1C3

32. R2C2 = 7 (hidden single in R2), placed for D\
32a. 21(4) disjoint cage R28C28 (step 22b) = {1479/1578}, no 3

33. R4C6 = 1, R6C4 = 3 (hidden singles on D/)

34. R2C7 = 3 (hidden single in N3), R2C6 = 9, R2C4 = 8, R2C3 = 6, R2C1 = 1
34a. R3C9 = 6 (hidden single in C9)

35. R6C6 = 6 (hidden single on D\), R7C6 = 2

36. 26(4) disjoint cage at R37C37 = {4589} (only remaining combination), no 2 -> R7C3 = 8, R8C2 = 9, R9C8 = 8

37. Naked pair {24} in R13C2, locked for C2 -> R45C2 = [86]

38. R5C12 = [56] = 11 -> R56C3 = 9 = {27}, locked for C3 and N4 -> R4C3 = 3, R34C1 = [39], R1C3 = 5, R1C2 = 2, R3C2 = 4, R3C3 = 9, placed for D\, R3C7 = 5, R2C8 = 4, placed for D/, R5C5 = 2

39. R7C8 = 5, R6C8 = 7 -> R6C7 = 8 (step 25)

and the rest is naked singles, without using the diagonals.


Rating Comment. I'll rate my walkthrough for Black-Hole remote killer-X at Easy 1.5; I used a short forcing chain.

I missed Andrew's really neat step 12. Great work!! It didn't end up making a big difference to my alternate ending in terms of step count but would have made my path a bit simpler. Looks like my step 18a is the key difference to Para and Andrew. No chains or combined cages.

Andrew's end step 11 here

.-------------------------------.-------------------------------.-------------------------------.
| 789       2456      1235      | 1234567   1234567   1234567   | 1289      1289      678       |
| 1234567   24567     56        | 89        1234567   89        | 34        123457    1234567   |
| 123456789 2456789   23456789  | 1234567   1234567   1234567   | 2345789   5689      12345678  |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 2456789   123456789 | 123456789 123456789 12345789  | 123456789 5689      12345678  |
| 23456789  2456789   123456789 | 123456789 12345789  123456789 | 123456789 123456789 1234      |
| 123456789 13        123456789 | 12345789  123456789 123456789 | 3456789   3456789   56789     |
:-------------------------------+-------------------------------+-------------------------------:
| 45789     13        45789     | 1236      1236      1236      | 45789     45789     5789      |
| 123456789 245789    346       | 578       13        457       | 467       12456789  56789     |
| 678       245       124       | 45789     45789     45789     | 123678    123678    123       |
'-------------------------------.-------------------------------.-------------------------------'

12. Hidden killer pair 4,5 in r1: 13(3)n2 can only have one of 4/5 -> 7(2)n1 must have 4/5 = {25/34}(no 1,6)

13. 1 in n1 only in c1: locked for c1
13a. -> 13(3)r2c1 must have 1 = {139/148/157}(no 2,6)

14. Hidden killer triple 1,2,3 in n9: r9c789 has two of 1,2,3 -> r8c8 = (12)
14a. 21(4)r2c2 remote cage: can't have more than one of 1,2 -> no 1,2 in r2c28, r8c2

15. 1 in D/ only in n5: locked for n5

16. 1 in r8 only in r8c58: -> no 1 in r5c5 (CPE using D\)

17. 1 on D\ only in n9: locked for n9
17a. no 8 in 9(2)n9

18. "45" on n2: 2 outies r2c37 = 9 = [54/63] = [4/6..]
18a. "45" on n8: 2 outies r8c37 = 10: but [64] clashes with h9(2)r2c37 (step 18)
18b. -> r8c37 = 10 = [37/46]
18c. no 5 in r8c4, no 7 in r8c6

19. 9(2)n9 = {27/36} ->Killer pair 6,7 with r8c6: both locked for n9

20. 22(3)r6c9 must have two of 5,8,9 for r78c9 = {589} only: 5,8 locked for c9
20a. no 7 in r1c1 (h15(2)r1c19)

21. 4 in n9 only in r7: locked for r7

22. 13(3)r6c1 = {238/247/256}(no 9)({346} doesn't have any candidates in r7c1)
22a. must have 2 -> 2 locked for c1

23. there is a h19(3)r159c1 and h11(3)r159c9 -> since 4 cells overlap completely with the 24(4)r19c19 -> the other two cells at r5c19 = the difference (19+11-24=6)
23a. -> h6(2)r5c19 = [42/51]

24. 12(3)r2c9 must have 4 for c9 = {147/246}(no 3)

25. r9c9 = 3 (hsingle c9): Placed for D\
25a. r9c1 = 6 (h9(2)r9c19): Placed for D/
25b. r1c9 = 7: placed for D/
25c. r1c1 = 8 (h15(2)r1c19)

cracked:

3x3::k:7936:7936:7936:5123:5123:3077:3077:3077:6152:3593:1034:7936:7936:5123:5646:2575:2575:6152:3593:1034:6676:6676:5123:5646:5646:6152:6152:3593:6676:6676:4126:4126:4126:5646:6152:2851:2852:2597:2597:1319:4126:6185:3370:3370:2851:2852:7726:5423:1319:6185:6185:4915:4915:3125:7726:7726:5423:5423:6202:4915:4915:829:3125:7726:3136:3136:5423:6202:6468:6468:829:3125:7726:1609:1609:1609:6202:6202:6468:6468:6468:

All that practice on this cage pattern helped a lot. All the openings were exactly the same. The big difference was that they actually opened the puzzle in this one. Those other 2 are cruel.
Got a few numbers placed in those other 2 but that doesn't seem to help me on much yet.

Well here is the walk-through for this assasin.

Walk-Through Assasin 39

1. R2C78 and R5C23 = {19/28/37/46}
2. R45C9, R56C1 = {29/38/47/56}
3. R5C78 = {49/58/67}
4. R8C23 = {39/48/57}
5. R56C4 = {14/23}
6. 26(4) in R3C3 : no 1
7. R9C234 = {123} -->> locked for R9
8. 24(3) in R5C6 = {789} -->> locked for N5
9. R78C8 = {12} -->> locked for C8 and N9
9a. Clean up: R2C7: no 8 or 9
10. R23C2 = {13} -->> locked for C2 and N1
10a. R9C2 = 2; Clean up : R5C3: no 7,8,9; R8C3: no 9
10b. Killer pair {13} in 5(2) in R5C4 + R9C4 -->> locked for C4
11. 45 on C789: 5 outies: R12378C6 = 16 = {12346} -->> locked for C6
11a. R4C6 = 5
11b. Clean up: R5C9: no 6
12. 45 on C1234: 2 innies: R14C4 = 10 = {46}/[82]
12a. Killer Pair {24} in R14C4 + 5(2) in R5C4 -->> locked for C4
13. 45 on R1234: 2 outies: R5C59 = 10 = [19/28/37/64] -->> R5C5 = {1236}; R5C9 = {4789}
13a. Clean up: R4C9: no 6,8,9
14. 45 on R6789: 4 innies: R6C1456 = 28 = {4789} -->> locked for R6
14a. R56C4 = [14]; R9C34 = [13]; R14C4 = [82] (step 12)
14b. Clean up: R5C2: no 9; R5C9: no 9; R5C1 = {234}
14c. Naked Pair {36} in R45C5 -->> locked for C5
14d. Clean up: R5C9: no 8 (step 13); R4C9: no 3
15. Naked Pair {47} in R45C9 -->> locked for C9 and N6
15a. 13(2) in R5C7 = {58} -->> locked for R5 and N6
15b. Clean up : R5C3: no 2
15c. Killer Pair {47} in 10(2) in R5C2 + R5C9 -->> locked for R5
15d. R5C6 = 9; R6C1 = 9(hidden); R5C1 = 2
16. Naked triple {356} in R6C238 -->> locked for R6
16a. Naked Pair {12} in R6C79 -->> locked for N6
16b. R4C1 =1 (hidden)
16c. Naked triple {369} in R4C578 -->> locked for R4
17. 21(4) in R6C3 = {3459/3468/3567} -->> 3 locked in 21(4) for C3
17a. R6C3 = 3 (hidden in N4); R6C8 = 6; R6C2 = 5
17b. 10(2) in R5C2 = {46} -->> locked for R5 and N4
17c. R45C5 = [63]; R45C9 = [47]
18. 45 on N7: 1 innie: R7C3 = 5
18a. 21(4) in R6C3 = 35{67} -->> R78C4 = {67} -->> locked for C4 and N8
18b. R9C6 = 8; R6C56 = [87]
18c. 24(4) in R7C5 = {259}8 -->> {259} locked for C5 and N8
18d. Naked Pair {14} in R78C6 -->> locked for C6
19. 31(5) in R1C1 = {25789/45679} -->> 7 locked in 31(5) for N1
19a.14(3) in R2C1 = {58}1 -->> {58} locked in C1 and N1
19b. 31(5) in R1C1 = {45679}
19c. R2C4 = 5 (only 5 in 31(5)); R3C4 = 9; R23C1 = [85]; R3C3 = 2 (Hidden)
20. 3 locked in C1 in 30(5) in R6C2 -->> 30(5) = 5{3679} -->> {3679} locked for N7
20a. R7C2 = 9 (hidden); R7C5 = 2; R78C8 = [12]; R78C6 = [41]; R1C1 = 4 (hidden)
20b. R8C23 = {48} locked for R8
21. 12(3) in R1C6 = {129/156/237}
21a. Killer Pair {17} in R1C5 + 12(3) in R1C6 -->> locked for R1
21b. R1C2 = 6; R12C3 = [97]
21c. 12(3) in R1C6 = {237} -->> locked for R1
21d. R1C9 = 5; R123C5 = [147]; R2C78 = [19]; R4C78 = [93]
21e. R1C8 = 7; R6C79 = [21]
And the rest is singles and cage sums.

Awaiting comments or corrections.

greetings

Para

I only did this Assassin a few days ago, having got sidetracked by other interesting puzzles in this forum.

Here is my walkthrough. I missed Para's very powerful step 11 so followed a different path with some interesting moves.

Clean-up is used in various steps, using the combinations in steps 1 to 8 for further eliminations from these two cell cages and for split cages from applying the 45 rule. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R23C2 = {13}, locked for C2 and N1

2. R2C78 = {19/28/37/46}, no 5

3. R45C9 = {29/38/47/56}, no 1

4. R5C23 = {28}/{46}/[73]/[91], no 5, no 7,9 in R4C3

5. R56C4 = {14/23}

6. R5C78 = {49/58/67}, no 1,2,3

7. R78C8 = {12}, locked for C8 and N9, clean-up: no 8,9 in R2C7

8. R8C23 = {48}/{57}/[93], no 1,2,6, no 9 in R8C3

9. R9C234 = {123} -> R9C2 = 2, R9C34 = {13}, locked for R9, clean-up: no 8 in R5C3

10. R678C9 = {138/147/156/237/246/345} (cannot be {129} because no 1,2 in R78C9), no 9
10a. Min R78C9 = 7 -> max R6C9 = 5

11. 24(3) cage in N5 = {789}, locked for N5

12. 5,6 in N5 locked in 16(4) cage = 56{14/23}

13. 26(4) cage in N124 = {2789/3689/4589/4679/5678}, no 1

14. 45 rule on C1 2 outies R67C2 – 10 = 1 innie R1C1, max R67C2 = 17 -> max R1C1 = 7

15. 45 rule on C9 2 outies R34C8 – 2 = 1 innie R9C9 -> max R34C8 = 11, no 9

16. 45 rule on N9 1 innie R7C7 – 5 = 2 outies R6C9 + R8C6 (doubles possible) -> min R7C7 = 7
16a. Max R6C9 + R8C6 = 4 = {11/12/13/22}, R6C9 = {123}, R8C6 = {123}

17. 45 rule on N14 2 outies R23C4 – 6 = 2 innies R6C23, min R6C23 = 5 -> min R23C4 = 11, no 1
17a. Max R23C4 = 17 -> max R6C23 = 11, no 8,9 in R6C3

18. 45 rule on C1234 2 innies R14C4 = 10 = {46}/[73]/[82]/[91], no 5, no 1,2,3 in R1C4
[I had missed the 1/3 killer pair in C4, which was in Para’s walkthrough. It would have simplified this step further and led to his 2/4 killer pair in C4 with further eliminations in that column.]

19. 45 rule on C6789 2 innies R49C6 – 5 = 1 outie R6C5, min R6C5 = 7 -> min R49C6 = 12, no 1,2 in R4C6, no 4,5,6 in R9C6 (R49C6 cannot be [66])
19a. Naked triple {789} in R569C6 for C6
[Step 19 was useful but not as powerful as Para’s 5 outies on C789 which I missed.]

20. 45 rule on R6789 4 innies R6C1456 = 28, max R6C4 = 4 -> R6C4 = 4, R6C156 = {789}, locked for R6, clean-up: R5C1 = {234}, R5C4 = 1, no 9 in R5C2
20a. R9C4 = 3, R9C3 = 1
20b. Clean-up: no 6 in R4C4 = 2, R1C4 = 8, clean-up: no 9 in R5C9

21. R4C1 = 1 (hidden single in N4), R23C1 = 13 = {49/58/67}, no 2

22. 8 in N8 locked in 24(4) cage = 8{169/259/457}[5/6, 7/9]
22a. R78C4 contains 7/9 -> must contain 5/6
22b. Killer pair 5/6 in R78C4 and 24(4) cage for N8
22c. 5/6 and 7/9 in R78C4 -> 5/6 and 7/9 in R23C4
22d. Max R23C4 = 15 -> max R6C23 = 9 (step 17) -> R6C3 = {23}, R6C2 = {56}
22e. Min R6C23 = 7 -> min R23C4 = 13, cannot be {57} -> R78C4 cannot be {69}

23. 21(4) cage in N478, R6C3 = {23}, R78C4 must contain 5/6 and 7/9 (step 22a), valid combinations {2469/3459/3567}, no 3,8,9 in R7C3
23a. Cannot be {2469} because R78C4 cannot be {69} (step 22e), 21(4) cage = {3459/3567} -> R6C3 = 3, clean-up: no 7 in R5C2, no 8 in R6C1, no 9 in R8C2
23b. 8 in R6 locked in R6C56, locked for N5
23c. R6C28 = {56}, locked for R6
23d. R6C79 = {12}, locked for N6, clean-up: no 9 in R4C9

24. R23C4 – 6 = R6C23 (step 17), R6C23 = 3{5/6} -> R23C4 = 9{5/6}, no 7, 9 locked for C4 and N2
24a. 7 in N2 locked in 20(4) cage = 78{14/23}, no 5,6, 7 locked in R123C5 for C5
24b. 7 in C4 locked in R78C4, locked for N8 and 21(4) cage

25. 21(4) cage in N478 = {3567} (only valid combination), no 4

26. 8,9 in N8 locked in 24(4) cage = 89{16/25}, no 4

27. R7C6 = 4 (hidden single in N8)

28. 4 in C5 locked in R123C5 for N2, 20(4) cage = {1478}, no 2,3, 1 locked in R123C5 for C5 and N2

29. R8C6 = 1 (hidden single in N8) -> R78C8 = [12], R7C5 = 2 (hidden single in N8)
29a. 24(4) cage in N8 = {2589}, no 6, 5 locked in R89C5 for C5 and N8

30. R78C4 = {67}, locked for C4 -> R7C3 = 5, clean-up: no 7 in R8C23 = {48}, locked for R8 and N7
30a. 8 in N8 locked in R9C56, locked for R9

31. 4 in N9 locked in R9C789, 25(5) cage = {14569} (only valid combination), no 3,7, 5,6,9 locked for N9

32. 3 in N9 locked in R78C9, locked for C9, clean-up: no 8 in R45C9

33. 30(5) cage in N47 = {35679}, R6C2 = 5 (hidden single in 30(5) cage) -> R6C8 = 6, clean-up: no 4 in R2C7, no 7 in R5C78, no 5 in R45C9 = {47}, locked for C9 and N6, clean-up: no 9 in R5C78 = {58}, locked for R5 and N6, clean-up: no 2 in R5C3, R5C23 = {46}, locked for R5 and N4

34. R45C9 = [47], R56C1 = [29], R6C56 = [87], R5C6 = 9, R9C6 = 8, R4C8 = 3, R4C7 = 9, R4C5 = 6, R4C6 = 5, R5C5 = 3, R8C9 = 3, R7C9 = 8, R7C7 = 7, R78C4 = [67], R7C12 = [39], R89C1 = [67], R8C7 = 5, R89C5 = [95], R5C78 = [85] (naked singles with cage sum for R6C1), R6C7 = 2, R6C9 = 1 (cage sums), clean-up: no 4 in R23C1, no 8 in R2C8

35. R3C8 = 8 (hidden single in C8) -> R3C1 = 5, R12C1 = [48], R23C4 = [59] (naked singles)

36. R4C23 = {78}, R3C4 = 9 -> R3C3 = 2 (cage sum)

and the rest is naked singles and a cage sum

3x3::k:7680:7680:7680:4099:4099:3077:3077:3077:4616:4617:1290:7680:7680:4099:6670:3087:3087:4616:4617:1290:4628:4628:4099:6670:6670:4616:4616:4617:4628:4628:5662:5662:5662:6670:4616:3619:804:2853:2853:3111:5662:2857:2858:2858:3619:804:6702:4399:3111:2857:2857:5939:5939:3637:6702:6702:4399:4399:5690:5939:5939:1853:3637:6702:3648:3648:4399:5690:6212:6212:1853:3637:6702:4937:4937:4937:5690:5690:6212:6212:6212:

Finally got a chance to look at Richard's alternate way - a very fine way it is too. Makes a very satisfying way of unlocking the puzzle. But rather than wait until step 50, why not step 16?

So, here's an alternative solution to Assassin 39V2 using Richard's key moves as quickly as possible. It's not really a condensed walk-through - perhaps "sweetened condensed" . I add many new clean-up-at-the-end steps to keep the solution simple. It makes you wonder why we thought it was difficult .

Assassin 39V2

note: L = "locked for"
1. 3(2)n4 = {12}:L c1, n4
1a. 11(2)n4: no 9

2. 14(2)n6 = {59/68} = [5/6..]
2a. {56} blocked from 11(2)n6 = {29/38/47}

3. "45"r1234 -> r5c59 = 11 = [29/38/56/65]
3a. r5c5 = {2356}

4. "45" c1234 -> r14c4 = 7 (no 789)

5. 22(4) n5 - no 1 in r4c56 - here's how
5a. 2 combos with 1 - {1579} & {1678} - ({1489} blocked by r5c5)
5b. {1579} - 7,9 must be in r4c56
5c. {1678} - 7, 8 must be in r4c56
5d. -> no 1 r4c56

6. 1 in n7 only in r7: L r7
6a. no 6 r8c8

7. "45" c9: r34c8 - r9c9 = 1
7a. -> min r9c9 = 2

8. no 1 in r8c4. Here's how
8a. r8c4 = 1 -> r4c8 = 1 -> r1c7 = 1 -> no place for 1 in n9
8b. -> no 1 in r8c4

9. no 1 in r8c6
9a. r8c6 = 1 -> no place for 1 in n9
9b. -> no 1 in r8c6

10. n8 : 1 locked in 22(4) -> 22(4) = 1{489|579|678} -> no 2,3

11. no 1 in r1c4: here's how.
11a. r1c4 = 1 -> r4c8 = 1 -> no place for 1 in n3
11b. -> no 6 in r4c4 (step 4)

12. 16(4)n2: any combination with 1 also needs a 6: here's why
12a. 16(4) has 1 -->> R123C5 = 1 -->> R9C6 = 1 -->> R4C4 = 1 -->> R1C4 = 6 (step 4)

13a. 16(4) = {1267/1456/2347/2356}(no 8 or 9)

14. "45" n6789 -> r6c23 - 11 = r4c78
14a. -> min r6c23 = 14 -> no 3,4 r6c23
14b. max r6c23 = 17 -> max r4c78 = 6
14c. max r4c7 = 5
14d. min r4c7 = 2 -->> max r4c8 = 4

15. "45" n3 -> 2 outies + 3 = 1 innie
15a. max r3c7 = 9 -> max r1c6 + r4c8 = 6
15b. max r1c6 = 5

16. Can't have 8 at r3c7 as it removes all possible 8s in row 1. Here's why
16a. r3c7=8 -> r1c789<>8
16b. r3c7=8 -> r3c46<>8, r2c6<>8 -> r2c4=8 -> r1c123<>8
16c. -> no 8 r3c7

17. Can't have 9 at r3c7 as it removes all possible 9s in row 1. Here's why
17a. r3c7 = 9 -> r1c78<>9
17b. r3c7=9 -> r3c46<>9, r2c6<>9 -> r2c4=9 -> r1c123<>9
17c. -> no 9 r3c7

18. "45" n3 -> 2 outies + 3 = 1 innie
18a. min r1c6 + r4c8 = {12} = 3 (can't have {11}:leaves no 1 for n3)
18b. min r3c7 = 6
18c. max r3c7 = 7 -> max. 2 outies = 4
-> max value r1c6 & r4c8 = 3

19. 26(4)n236: must have 6/7 = {2789/3689/4679/5678}
19a. 2,3,4,5 are all required in r4c7 -> no 2,3,4,5 r23c6

20. "45"n2 : 5 innies = 29 = 29(5)
20a. 16(4)n2 blocks {34679}, {25679}, {35678}
20b. only combinations for 29(5) with a 1 are {14789} and {15689} with the 1 only in r1c6
20c. ->no 1 in r23c4

21. r4c4 = 1 (h single c4)
21a. 22(4)n5 now 21(3) = {579/678}(no 234) ({489} blocked by r5c5)
21b. = 7{59/68}: 7 L n5, r4
21c. -> r5c5 = {56}
21d. r4c56 = {789}

22. 11(3)n5= 2{36/45}(no 8)

23. r5c1 = 1 (hsingle r5)
23a r6c1 = 2

24. r5c6 = 2 (hsingle n5)

25. r5c9 = {56} (step 3)
25a. {56} pair r5c59: L r5
25b. r4c9 = {89}
25c. Killer Pair {89} with r4c56:L r4

26. 11(2)n4 & 6={38/74}(no 9):all L r5

27. r5c4 = 9 (single r5), r6c4 = 3

28. r6c56 = {45}: L n5, r6
28a. r5c59 = [65], r4c9 = 9
28b. r4c56 = {78}:L r4

29. r1c4 = 6 (step 4)
29a. r123c5 = {127/235}(no 4) ({145} blocked by r6c5)
29b. = 2{17/35}: 2 L n2

30. 6 r4 in n4: L n4

31. 9 n2 in c6: L c6

32. 9 n8 in 22(4) = 19{48/57} (no 6)

33. "45"n3 -> r1c6 + r4c8 + 3 = r3c7 = 6/7
33a. ->r1c6 + r4c8 = 3/4 = [12/13]
33b. -> r1c6 = 1

34. 16(4)n2 now 10(3) = {235}: L n2,c5
34a. r6c56 = [45]

35. 4 n2 in c4: L c4

36. 22(4)n8 = {1489}
36a. r9c6 = 4
36b. r789c5 = {189} L n8, c5
36c. r4c56 = [78]

37. r23c6 = {79}:L n2,c6, 26(4)
37a. -> r34c7 = [64]

38. 11(2)n6 = {38}: L n6, r5
38a. r4c8 = 2, r1c78 = [29] (hsingle 2)

39. 11(2)n4 = {47} L n4

40. 23(4)n6 must have 8/9 -> r7c7 = {89}

41. "45"n9 -> r6c9 + r8c6 = r7c7 = 8/9
41a. -> r6c9 + r8c6 = [63], r7c7 = 9, r7c6 = 6, r9c8 = 6 (hsingle n9), 7(2)n9 = [34], r78c9 = [71]
etc
[edit step 20 for clarity]

As with other recent puzzles added to the archive, I've repeated earlier steps so that analysis can be understood without looking back to see the relationships used. This makes my walkthrough a bit longer.

Prelims

a. R23C2 = {14/23}
b. R2C78 = {39/48/57}, no 1,2,6
c. R45C9 = {59/68}
d. R56C1 = {12}
e. R5C23 = {29/38/47/56}, no 1
f. R56C4 = {39/48/57}, no 1,2,6
g. R5C78 = {29/38/47/56}, no 1
h. R78C8 = {16/25/34}, no 7,8,9
i. R8C23 = {59/68}
j. 11(3) cage at R5C6 = {128/137/146/236/245}, no 9
k. 19(3) cage at R9C2 = {289/379/469/478/568}, no 1
l. 26(4) cage at R2C6 = {2789/3689/4589/4679/5678}, no 1
m. 18(5) cage at R1C9 = {12348/12357/12456}, no 9

Steps resulting from Prelims
1a. Naked pair {12} in R56C1, locked for C1 and N4, clean-up: no 9 in R5C23
1b. 18(5) cage at R1C9 = {12348/12357/12456}, CPE no 1,2 in R1C8
1c. 1 in N7 only in R7C23, locked for R7, clean-up: no 6 in R8C8
1d. 1 in R9 only in R9C56789, CPE no 1 in R8C6

2. 45 rule on R6789 4 innies R6C1456 = 14 = {1238/1247/1256/1346/2345}, no 9, clean-up: no 3 in R5C4

3. 45 rule on R1234 1 innie R4C9 = 1 outie R5C5 + 3, R4C9 = {5689} -> R5C5 = {2356}

4. 1 in R5 only in R5C16
4a. 45 rule on R6789 3 outies R5C146 = 12 = {129/138/147/156}
4b. 3,6 of {138/156} must be in R5C6 -> no 5,8 in R5C6

5. 45 rule on C1234 2 innies R14C4 = 7 = {16/25/34}, no 7,8,9

6. 26(5) cage at R6C2 either contains one of 1,2 which must be in R7C2 or is {34568}, CPE, no 3,4,5,6,8 in R89C2 => R8C2 = 9, R8C3 = 5 => 5 of {34568} must be in R6C2 -> R7C2 = {123468}, no 5,7,9
[This interesting step wouldn’t have been necessary if I’d spotted step 8 earlier; fortunately I saw this step first.]

7. 45 rule on N3 1 innie R3C7 = 2(1+1) outies R1C6 + R4C8 + 3
7a. Min R1C6 + R4C8 = 3 (cannot be [11] because R1C6 + R4C8 “see” all places for 1 in N3) -> min R3C7 = 6
7b. Max R1C6 + R4C8 = 6 -> no 6,7,8,9 in R1C6 + R4C8
[Note. It also follows that R1C6 and R4C8 cannot both be 2.
Also that R1C6 and R4C8 cannot both be 3 which would make R3C7 = 9, no 3,9 in R2C78 and R1C6 + R4C8 would “see” all remaining places for 3 in N3.]

8. 45 rule on N7 2(1+1) outies R6C2 + R9C4 = 1 innie R7C3 + 14
8a. Max R6C2 + R9C4 = 18 -> max R7C3 = 4
8b. Min R6C2 + R9C4 = 15, no 2,3,4,5 in R6C2 + R9C4

9. 26(5) cage at R6C2 (step 6) cannot be {34568} (which clashes with R8C23) -> 26(5) cage must contain one of 1,2 -> R7C2 = {12}
9a. Killer pair 1,2 in R23C2 and R7C2, locked for C2

10. 45 rule on N9 2(1+1) outies R6C9 + R8C4 = 1 innie R7C7
10a. Min R6C9 + R8C4 = 3 -> min R7C7 = 3
10b. Max R6C9 + R8C4 = 9 -> no 8,9 in R6C9, no 9 in R8C4

11. 45 rule on C1 2 outies R67C2 = 1 innie R1C1 + 2, min R67C2 = 7 -> min R1C1 = 5

12. 45 rule on C9 2 outies R34C8 = 1 innie R9C9 + 1, min R34C8 = 3 -> min R9C9 = 2

13. 45 rule on C6789 2 innies R49C6 = 1 outie R6C5 + 8, IOU no 8 in R9C6

14. 45 rule on N6 5 innies R4C78 + R6C789 = 20 = {12359/12368/12467/13457} (cannot be {12458/23456} which clash with R45C9)
14a. R5C78 = {29/38/47} (cannot be {56} which clashes with R4C78 + R6C789), no 5,6 in R5C78

15. 19(3) cage at R9C2 = {289/379/469/478} (cannot be {568} which clashes with R8C23), no 5 in R9C23
15a. Hidden killer quad 1,2,3,4 in R7C3, 26(5) cage at R6C2 and 19(3) cage for N7, R7C3 contain one of 1,2,3,4, 19(3) cage contains one of 2,3,4 in N7 -> 26(5) cage must contain one of 1,2 in R7C2 and one of 3,4 (but not both) in R789C1
15b. Hidden killer pair 3,4 in 18(3) cage at R2C1 and 26(5) cage for C1, 26(5) cage contains one of 3,4 in C1 -> 18(3) cage must contain one of 3,4 -> 18(3) cage = {369/378/459/468} (cannot be {567} which doesn’t contain 3 or 4)
15c. 26(5) cage at R6C2 = {13589/13679/14579/14678/23579/23678/24569/24578} (cannot be {12689} because 1,2 only in R7C2, cannot be {23489} which contains both of 3,4, {34568} has already been eliminated in step 9)

16. 45 rule on N6789 2 outies R6C23 = 2 innies R4C78 + 11
16a. Min R4C78 = 3 -> min R6C23 = 14, no 3,4 in R6C3
16b. Max R6C23 = 17 -> max R4C78 = 6, no 6,7,8,9 in R4C7, no 5 in R4C8

17. 45 rule on N69 2 outies R78C6 = 2 innies R4C78 + 3
17a. Max R4C78 = 6 (step 16b) -> max R78C6 = 9, no 8,9 in R78C6

18. 26(4) cage at R2C6 = {2789/3689/4589/4679/5678}
18a. 2,3 of {2789/3689} must be in R3C7 -> no 2,3 in R23C6

19. 22(4) cage at R4C4 = {1579/1678/2389/2569/2578/3469/4567} (cannot be {1489} because R5C5 only contains 2,3,5,6, cannot be {2479/3478/3568} which clash with R56C4)
19a. R4C9 = R5C5 + 3 (step 3) -> R4C9 + R5C5 = [52/63/85/96]
19b. 22(4) cage = {1579/1678/2389/2569/4567} (cannot be {2578} = {278}5/{578}2 which clash with R4C9 + R5C5 = [52/85], IOD clash, cannot be {3469} = {349}6/{469}3 which clash with R4C9 + R5C5 = [63/96], IOD clash)
19c. 7,8,9 of {1579/1678/2389} can only be in R4C56 -> no 1,3 in R4C56
19d. R6C1456 (step 2) = {1238/1247/1256/1346/2345}
19e. Consider placement for 1 in N6
R4C8 = 1 => 22(4) cage = {2389/2569/4567}
or 1 in R6C789, locked for R6 => R6C1456 = {2345}, R6C456 = {345} locked for N5 => 22(4) cage = {1678}
-> 22(4) cage = {1678/2389/2569/4567}
19f. 11(3) cage at R5C6 {128/137/146/245} (cannot be {236} which clashes with 22(4) cage)

20. Consider interactions between R14C4 (step 5) = {16/25/34}, R4C9 + R5C5 (step 19a) = [52/63/85/96] and 22(4) cage at R4C4 (step 19e) = {1678/2389/2569/4567}
22(4) cage = {1678}
or 22(4) cage = {2389} => R56C4 = {57}, locked for C4 => no 2 in R14C4 => {2389} = 3{89}2
or 22(4) cage = {2569} = {269}5 (cannot be {259}6 which clashes with R4C9 + R5C5 = [96], IOD clash, cannot be {569}2 which clashes with R4C9 + R5C5 = [52], IOD clash)
or 22(4) cage = {4567} => R56C4 = [93] => no 4 in R14C4 => 5,6 of {4567} must be in R4C4 + R5C5
-> no 4 in R4C4, no 5 in R4C56, no 3 in R5C5, clean-up: no 3 in R1C4 (step 5), no 6 in R4C9 (step 3), no 8 in R5C9

21. R6C1456 (step 2) = {1238/1247/1256/1346/2345}, 11(3) cage at R5C6 (step 19f) = {128/137/146/245}
21a. Consider combinations for 22(4) cage at R4C4 (step 19e) = {1678/2389/2569/4567}
22(4) cage = {1678/2569/4567}, 6 locked for N5 => R6C1456 = {1238/1247/2345}
or 22(4) cage = {2389} => 11(3) cage = {146} => R56C4 = {57} => R6C1456 = {1247/1256} (cannot be {1238/1346} which don’t contain 5 or 7, cannot be {2345} because R6C56 only contain 1,4,6) -> R6C1456 = {1238/1247/1256/2345}, 2 locked for R6

22. R4C9 + R5C5 (step 19a) = [52/63/85/96], R4C78 + R6C789 (step 14) = {12359/12368/12467/13457}
22a. Continuing analysis of 22(4) cage at R4C4 (step 19e) = {1678/2389/2569/4567}
22(4) cage = {1678} = 1{78}6 => R4C9 = 9, R5C9 = 5 => R4C78 + R6C789 = {12368/12467}
or 22(4) cage = {2389} = 3{89}2 => R4C9 = 5 => R4C78 + R6C789 = {12368/12467}
or 22(4) cage = {2569} = {269}5 (step 20), 9 locked for R4, R4C9 = 8, R5C9 = 6 => R6C23 = {69} (hidden pair in N4), locked for R6, 5 in N4 only in R4C123, locked for R4 => R4C78 + R6C789 = [31]{457} (cannot be [41]{357} because R8C9 = 1 (hidden single in C9) => R67C9 = [49], cannot be [58/76] which clash with R45C9)
or 22(4) cage = {4567}, 4 locked for R4, R56C4 = [93], 4 in N4 only in R5C23 = {47}, locked for R5 => 3 in R5 only in R5C78 = {38} , locked for N6 => R4C78 + R6C789 = {12467}
-> R4C78 + R6C789 = {12368/12467}/[31]{457}, no 5 in R4C7, no 9 in R6C78
22b. 9 in R6 only in R6C23, locked for N4

23. 26(4) cage at R2C6 = {2789/3689/4589/4679} (cannot be {5678} because R4C7 only contains 2,3,4)
23a. R4C7 = {234} -> no 4 in R23C6
23b. 26(4) cage at R2C6 = {2789/3689/4589/4679}, CPE no 9 in R3C45

24. Continuing analysis of 22(4) cage at R4C4 (step 19e) = {1678/2389/2569/4567} a bit further
22(4) cage = {1678/2389}, 8 locked for N5
or 22(4) cage = {2569} = {269}5 => R4C9 = 8, R5C9 = 6 => R6C23 = {69} (hidden pair in N4) => 8 in N4 only in R5C23 = {38}, locked for R5
or 22(4) cage = {4567} => R56C4 = [93]
-> no 8 in R5C4, clean-up: no 4 in R6C4
24a. 8 in R5 only in R5C23 = {38} or in R5C78 = {38}, 3 locked for R5 (locking cages)

25. R6C1456 (step 21a) = {1238/1247/1256/2345}
25a. 11(3) cage at R5C6 (step 19f) = {128/137/146/245}
25b. 7 of {137} must be in R5C6 (R6C56 cannot be {37} because R6C1456 cannot contain both of 3,7) -> no 7 in R6C56

26. R4C78 + R6C789 (step 22a) = {12368/12467}/[31]{457}
26a. R6C1456 (step 21a) = {1238/1247/2345} (cannot be {1256} which clashes with R4C78 + R6C789), no 6 in R6C56
26b. 5 of {2345} must be in R6C56 (R6C56 cannot be {34} because 11(3) cage at R5C6 cannot be 4{34}) -> no 5 in R6C4, clean-up: no 7 in R5C4
26c. R5C146 (step 4a) = {129/147/156}
26d. R5C4 = {459} -> no 4 in R5C6

27. Analysing R4C78 + R6C789 (step 22a) = {12368/12467}/[31]{457}
27a. R4C78 + R6C789 = {12368}, R4C78 = [21/31] => R8C9 = 1 (hidden single in C9) => R67C9 = 13, no 3 in R6C9 or R4C78 = {23}, locked for N6
or 3 of R4C78 + R6C789 = [31]{457} is in R4C7
-> no 3 in R6C9
27b. R4C78 + R6C789 = [31]{457}, locked for N6 => R45C9 = [86], R8C9 = 1 (hidden single in C9) => R67C9 = 13 = [49] -> no 5 in R6C9

28. 14(3) cage at R6C9 = {149/167/248/257/347}
28a. R4C78 + R6C789 (step 22a) = {12368/12467}/[31]{457}
28b. Consider placements for R6C9
R6C9 = 1 => R4C78 = {23/24}, R6C78 = {67/68}
or R6C9 = 4 => R4C78 = [21/31], R6C78 = {57/67}
or R6C9 = 6 => R78C9 = [71], R6C78 = {17/18/47} (R6C78 cannot be {38} because no valid combination for 23(4) cage at R6C7 since {3578} clashes with R7C9)
or R6C9 = 7 => R6C78 = {16/45/46}
-> R6C78 = {16/17/18/45/46/47/57/67/68}, no 3

29. 3 in R6 only in R6C456, locked for N5, clean-up: no 4 in R1C4 (step 5)
29a. 22(4) cage at R4C4 (step 19e) = {1678/2569/4567}
29b. R56C4 = [48/93] (cannot be [57] which clashes with 22(4) cage), no 5 in R5C4, no 7 in R6C4
29c. R5C146 (step 26c) = {129/147}, no 6
29d. R6C1456 (step 26a) = {1238/2345}

30. 22(4) cage at R4C4 (step 29a) = {1678/2569/4567}
30a. 22(4) cage = {2569} must be {269}5 (step 20) -> no 2 in R5C5, clean-up: no 5 in R4C9 (step 3), no 9 in R5C9
30b. Naked pair {56} in R5C59, locked for R5

[A small amount of analysis in a different area]
31. R78C6 = R4C78 + 3 (step 17), 11(3) cage at R5C6 (step 19f) = {128/137/245}
31a. Consider combinations for 22(4) cage at R4C4 (step 29a) = {1678/2569/4567}
22(4) cage = {1678}, 8 locked for N5 => R56C4 = [93] => 11(3) cage = 2{45}
or 22(4) cage = {2569}, 2,9 locked for N5 => R56C4 = [48]
or 22(4) cage = {4567}, 4 locked for N5 => R56C4 = [93], R4C78 = [21] (hidden pair in R4) = 3 => R78C6 = 6 = {24}, locked for C6 => 11(3) cage = [128]
-> no 4 in R4C6, no 8 in R6C5, no 2 in R6C6
[Note for simplicity of writing the next step, 22(4) cage = {4567} = [5476] because R4C9 = 9 (hidden single in R4) -> R5C5 = 6, step 3]

[Then I was pleasantly surprised to find that the previous step leads to a combination elimination. I could have continued the previous step to a contradiction but I’ll write it as a forcing chain.]
32. 22(4) cage at R4C4 (step 29a) = {1678/2569/4567}
32a. Consider combinations for 26(4) cage at R2C6 (step 23) = {2789/3689/4589/4679}
26(4) cage = {2789} => R23C6 = {789} => 22(4) cage = {1678} (cannot be {4567} = [5476] because 11(3) cage at R5C6 must then be [128] and R46C6 = [78], clashes with R23C6, ALS block)
or 26(4) cage = {3689/4589/4679} => 1,2 in R4 only in 22(4) cage and R4C8 => 22(4) cage must contain one of 1,2 = {1678/2569}
-> 22(4) cage = {1678/2569}, no 4
32b. 5 of {2569} must be in R5C5 (step 20) -> no 5 in R4C4, clean-up: no 2 in R1C4 (step 5)
32c. 11(3) cage at R5C6 (step 31) = {137/245} (cannot be {128} which clashes with 22(4) cage), no 8 in R6C6
32d. 11(3) cage = {137/245} = 2{45}/7{13} -> R5C6 = {27}

33. R5C1 = 1 (hidden single in R5) -> R6C1 = 2
33a. 5 in R4 only in R4C123, locked for N4
33b. Killer pair 8,9 in 22(4) cage at R4C4 and R4C9, locked for R4

34. 45 rule on N1 2(1+1) outies R2C4 + R4C1 = 1 innie R3C3 + 8
34a. Max R2C4 + R4C1 = 16 -> max R3C3 = 8
34b. Min R2C4 + R4C1 = 9, max R4C1 = 7 -> min R2C4 = 2

35. 45 rule on N78 2 outies R6C23 = 2 innies R78C6 + 8
35a. Max R6C23 = 17 -> max R78C6 = 9 = {36/45} (cannot be {27} which clashes with R5C6), no 7 in R78C6

36. R6C23 = R4C78 + 11 (step 16)
36a. Min R6C23 contains 9 = {69} = 15 -> min R4C78 = 4
36b. R4C78 + R6C789 (step 22a) = {12368/12467}/[31]{457} = {23}{168}/{24}{167}/[31]{457} (R4C78 cannot be [21] because min R6C23 = 4)
[Alternatively R4C78 cannot be [21] which clashes with 22(4) cage at R4C4]
-> R6C789 = {167/168/457}
36c. Max R7C67 = 15 -> min R6C78 = 8 => 7 of {167} must be in R6C78
or R6C789 = {457} must have 4 in R6C9 (step 27b)
-> no 7 in R6C9

[Taking some of this analysis a bit further]
37. R4C78 + R6C789 (step 22a) = {12368/12467}/[31]{457}
37a. Consider combinations for 22(4) cage at R4C4 (step 32a) = {1678/2569}
22(4) cage = {1678} = 1{78}6 => R56C4 = [93]
or 22(4) cage = {2569} = {269}5 => R4C9 = 8, R5C9 = 6, R4C78 + R6C789 = [31]{457} => 14(3) cage at R6C9 = [491] (step 27b), R6C78 = {57} = 12 => R7C67 = 11 = [38]
-> 3 in R6C4 or R7C6, CPE no 3 in R6C6 + R78C4
37b. 11(3) cage at R5C6 (step 31) = {137/245}
37c. 3 of {137} must be in R6C5 -> no 1 in R6C5

38. 11(3) cage at R5C6 (step 31) = {137/245}
38a. R49C6 = R6C5 + 8 (step 13)
38b. Consider combinations for 22(4) cage at R4C4 (step 32a) = {1678/2569}
22(4) cage = {1678} => R4C56 = {78} => 11(3) cage = {245} = 2{45} => R6C5 + R49C6 = 4[84]/5[76]/5[85]
or 22(4) cage = {2569} = {269}5 => 11(3) cage = {137} = [731] => R6C5 + R49C6 = 3[29]/3[65]/3[92]
-> R9C6 = {24569}, no 1,7

39. Consider placements for 1 in R4
R4C4 = 1 => R1C6 = 1 (hidden single in C6)
or R4C8 = 1 => R1C7 = 1 (hidden single in N3)
-> 1 in R1C67, 1 locked for R1, clean-up: no 6 in R4C4 (step 5)
39a. 1 in 12(3) cage at R1C6 = {129/138/147} (cannot be {156} which clashes with R1C4), no 5,6
39b. 22(4) cage at R4C4 (step 32a) = {1678/2569}
39c. R4C4 = {12} -> no 2 in R4C56, clean-up: no 9 in R9C6 (step 38b)

40. 16(4) cage at R1C4 = {1258/1267/1357/1456/2356} (cannot be {1249/1348/2347} because R1C4 only contains 5,6), no 9 in R12C5

[I was surprised to find that the next step finally cracks the puzzle. It only works after 3 has been eliminated from R78C4 and 2 has been eliminated from R4C56.]
41. 22(4) cage at R4C4 (step 32a) = {1678/2569}
41a. R14C4 (step 5) = [52/61], R6C2 + R9C4 = 1 innie R7C3 + 14 (step 8)
41b. Consider combinations for 17(4) cage at R6C3 = {1259/1268/1349/1358/1367/1457/2348/2357/2456} which must contain one or more of 3,5,8
R7C3 = 3 => R6C2 + R9C4 = 17 = {89}, CPE no 8 in R6C4
and/or 5 in R78C4, locked for C4 => R1C4 = 6 => R4C4 = 1 => 22(4) cage = {1678} => R5C4 = 9 (hidden single in N4) => R6C4 = 3
and/or 8 in R6C3 + R78C4, CPE no 8 in R6C4
-> R6C4 = 3, R5C4 = 9, 22(4) cage = {1678} -> R4C4 = 1, R1C4 = 6, R5C5 = 6, R5C9 = 5, R4C9 = 9, R4C56 = {78}, locked for R4 and N5 -> R5C6 = 2
41c. Naked pair {45} in R6C56, locked for R6
41d. 6 in R4 only in R4C123, locked for N4
[Things are a lot easier now.]

42. R1C6 = 1 (hidden single in C6) -> R1C78 = 11 = [29]/{38/47}, no 9 in R1C7

43. 45 rule on N23 2 innies R23C4 = 2 outies R4C78 + 6
43a. 2 in R4 only in R4C78 = {23/24} = 5,6 -> R23C4 = 11,12 = {47/48/57}, no 2 in R23C4

44. 2,6 in N2 only in 16(4) cage at R1C4 = 6{235}, 2,3,5 locked for C5 and N2 -> R6C56 = [45]

45. R78C4 = {25} (hidden pair in C4), locked for N8 and 17(4) cage at R6C3
45a. R78C4 = {25} = 7 -> R67C3 = 10 = [73/91]

46. 1,9 in N8 only in 22(4) cage at R7C5 = {1489} (only possible combination) -> R9C6 = 4, R789C5 = {189}, locked for C5 and N8 -> R4C56 = [78]

47. R9C4 = 7 -> R9C23 = 12 = {39}, locked for R9 and N7 -> R7C23 = [21], clean-up: no 3 in R23C2, no 5 in R8C23
47a. Naked pair {68} in R8C23, locked for R8 and N7 -> R9C1 = 5
47b. R7C3 = 1, R78C4 = [52] -> R6C3 = 9 (cage sum)
47c. Naked pair {47} in R78C1, locked for C1 and 26(5) cage at R6C2 -> R6C2 = 8, clean-up: no 3 in R5C23
47d. Naked pair {14} in R23C2, locked for C2 and N1 -> R5C23 = [74]
47e. Naked pair {38} in R5C78, locked for N6

48. 18(3) cage at R2C1 = {369} (only remaining combination), 9 locked for C1 and N1 -> R1C1 = 8, R2C4 = 4, R23C2 = [14], clean-up: no 3 in R1C78 (step 42), no 8 in R2C78

49. R3C4 = 8 -> R3C3 + R4C23 = 10 = {235} (only possible combination) -> R3C3 = 2, R4C23 = {35}, locked for R4 -> R4C1 = 6
49a. Naked pair {39} in R23C1, locked for N1 -> R1C2 = 5, R12C3 = [76]

50. Naked pair {79} in R23C6, locked for 26(4) cage at R2C6 -> R3C7 = 6, R4C7 = 4 (cage sum), R1C7 = 2 -> R1C8 = 9 (cage sum), clean-up: no 3 in R2C78
50a. Naked pair {57} in R2C78, locked for R2 and N3

51. R9C89 = {26} (hidden pair in R9), locked for N9, clean-up: no 1,5 in R78C8
51a. R8C6 = 3, R9C89 = {26} = 8 -> R89C7 = 13 = [58]

and the rest is naked singles.

3x3::k:5632:5632:5632:5635:5635:4357:4357:4357:4360:3849:3850:5632:5632:5635:4110:2575:2575:4360:3849:3850:5140:5140:5635:4110:4110:4360:4360:3849:5140:5140:5662:5662:5662:4110:4360:3363:2852:2085:2085:2855:5662:3113:1322:1322:3363:2852:6702:4655:2855:3113:3113:6963:6963:3637:6702:6702:4655:4655:6714:6963:6963:2877:3637:6702:3648:3648:4655:6714:6212:6212:2877:3637:6702:2377:2377:2377:6714:6714:6212:6212:6212:

Hmm. Don't know how. So, this is one of Para & Glyn's Unsolvables. Thanks Ruud.
Managed to find some nice tricks, but stuck now. Thought I'd had it.... Pretty sure this far is correct. Anyone want to lend a hand? Over to A44 V1.5 for the interim.

Cheers
Ed

A39 V3
Prelims
i. 17(5)n3: no 8,9
ii. 15(2)n1 = {69/78}
iii. 10(2)n3: no 5
iv. 13(2)n6: no 1,2,3
v. 11(2)n4, 5 & 9: no 1
vi. 8(2)n4: no 4,8,9
vii. 5(2)n6 = {14/23}
viii. 27(4)n6: no 1,2
ix. 26(4)n8: no 1
x. 14(2)n7 = {59/68}
xi. 9(3)n7: no 7,8,9

1. "45" r6789: r5c146 = h23(3) = {689}
1a. all locked for r5
1b. r4c9: no 4,5,7
1c. r6c14 = {235}
1d. no 2 r5c23

2. "45" r1234: r5c59 = h9(2) = [27/45/54](no 1,3)
2a. r5c5 = {245}

3. "45"r1234: r4c4569 = h26(4)

4. "45" r6789: r6c1456 = h11(4) = {1235}: all locked for r6
4a. 1 only in r6c56 in 12(3)
4b. 1 locked for n5 & 12(3) must have 1 = {129/138/156}

5. "45" c6789: r4569c6 = h26(4) = {2789/3689/5678}(no 1, 4) ({4589/4679} are blocked by combo's in 12(3)n5)
5a. = 8{279/369/567}
5b. 8 locked for c6

6. 12(3)n5 = {129/138/156} = 1{..}
6a. r6c5 = 1
6b. r56c6 = [92/83/65]
6c. -> from step 5, r49c6 = {78/69} = {6789}

7. 22(4)n5 must have 4 & 7 for n5 = 47{29/38/56}.
7a. r4c456+r4c9 = h26(4) must have 3 cells overlapping with the 22(4) = {2789/4679/5678}
7b. = {279}[8]/{467}[9]/{479}[6]/{567}[8]
7c. no 3 or 8 r4c456
7d. 22(4) = {2479/4567}
7e. 7 locked for r4
7f. r4c456 = [6/9..]

8. 3 in n5 only in r6: 3 locked for r6
8a. no 8 r5c1

9. deleted

10.from step 5, r4569c6 = h26(4) = {2789/3689/5678} =
10a. = [7928]/{6[83]9}/[7658]
10a. no 7 r9c6

11. "45" c1234: r14c4 = h11(2) = {29/47/56}(no 1,3,8)

12. 11(2)n4 = [92/65]
12a. 15(3)n1 = {168/249/348/357/456} ({159/258/267} all blocked by 11(2))

13. "45" n3:2 outies + 1 = 1 innie r3c7
13a. min r3c7 = 3
13b. max 2 outies = 8 -> no 9 r1c6

14. "45" n236: r23c4 + r6c789 = 35
14a. max r6c789 = {679} = 22 ({789/689} blocked by 13(2)n6)
14b. -> min r23c4 = 13 (no 1..3)

15. r23c4 = 13..17 -> r6c789 = 22..18
15a. r6c789 =
15b. = 18 Blocked: {468} clashes with 13(2)n6
15c. = 19 = {469} ({478} clashes with 13(2)
15d. = 20 = {479}
15e. = 21 = {489/678}
15f. = 22 = {679}
15f. [89] must be in r6c789 or 13(2) for n6: no 89 in r4c7

16. Not sure if this is strictly logical. No 4 in r5c78. Here's how.
16a. from step 15c..f: 4 must be in r6c789
16b. or it is forced into 13(2) by {678}
16c. or it is forced into r5c5 by {679} through h9(2)r5c59 = [45]
16d. -> no 4 r5c78

17. 5(2)n6 = {23}: both locked for n6 & r5

18. 8(2)n4 = {17}: both locked for n4 & r5
18a. no 6 r4c9
18b. 13(2)n6 = [94/85]

19. from step 15d..f. r6c789 must have 7
19a. = 20 = {479}
19b. = 21 = {678}
19c. = 22 = {679}
19d. -> innies n236 = 35 -> r23c4 = 13-15

20. 1 in c4 only in n8: 1 locked for n8

21. from step 19a.r6c789 = 20{479}/21{678}/22{679}
21a. "45" n9: 5 outies = 31
21b. min r6c789 = 20 -> max r78c6 = 11
21c. min r7c6 = 3 -> max r8c6 = 8

22. Now a nice trick to get of 9 from r7c6
22a. 9 in r7c6 -> since max r78c6 = 11 (step 21b) and since min r8c6 = 2 -> r78c6 can only be [92] = 11
22b. from outies n9 = 31, when r78c6 = 11 -> r6c789 = 20 = {479} only (step 22)
22c. since r6c78 is in the same cage as r7c6 -> 9 can only go in r6c9
22d. However, cannot have {47} in r6c78 in a 27(4) cage {4+7+9+7} clash
22e. -> no 9 r7c6

23. Generalized X-wing on 9 in c78: must be in 27(4)n6 or c78 in n3: 9 locked for c78
23a. no 2 r78c8

24. 27(4)n6
24a. {3789} cannot have {89} in r6c78 because of r4c9 -> must have 7 in r6c78 -> 3 must be in r7c6 -> no 3 in r7c7
24b. {4689} same logic -> no 4 r7c7

25. 17(5)n3 = 123{47/56} = [123]
25a. 23 locked for n3
25b. CPE: no 1 r12c8

26. 10(2)n3 = {46}/[19]
26a. since 17(5) = 123{47/56} = [14/16] -> whichever combination is in 10(2), the leftover 1/4/6 has to be in r4c8
26b. -> r2c7 = r4c8 (no 5)
26c. -> r24c8 = [46/64/91]

27. "45" c9: r34c8 + 1 = r9c9
27a. -> min r9c9 = 4
27b. -> r4c8 != r9c9
27c. -> from step 26b, the 14(3)n6 must have [1/4/6] in r78c9
27d. -> 14(3)n6 = {149/158/167/347/356}(no 2) ({248} clashes with 13(2)n6)
27e. 14(3)n6 = [1/3..]

28. 2 in c9 only in n3
28a. no 2 r3c8

29. because r78c9 = [1/3..] (step 27e)-> the 1 & 3 required in 17(5)n3 cannot both be in c9
29a. they also cannot both be in r34c8 since that would leave 1/3 missing from c9 (r78c9 can only have 1 of 1/3)
29b. -> 1/3 must be in r34c8 = [14/16/41/51/61/71](cannot have [36] since sum is max 8);[34] forces 8 into both r9c9 & r4c9
29c. no 3 r3c8
29d. 3 locked for c9 in r123c9
29e. 1 must be r34c8: no 1 r9c8, no 1 r123c9
29f. min r34c8 = {14} -> min r9c9 = 6
29g. r234c8 = [614/416/941/951/961/971]

30. 14(3)n6 = 1{49/58/67}
30a. 1 locked for n9
30b. 8 in {158} must be in r6c9 -> no 8 r78c9

31. 16(4)n2 must have 1 because of 1's in n6. Here's how.
31a. "45" n3: 2 outies + 1 = 1 inn.
31b. 1 in r4c8 -> min r1c6 = 2 -> 1 in n2 in r23c6 in 16(4)
31c. 1 in r4c7 must be in 16(4) = 1{..}
31d. {2347/2356} blocked

32. no 1 in r1c7, r2c6 or r3c13 because of 1s in c8. Here's how.
32a. 1 in r4c8 -> 1 in r2c7 (step 26b) -> 1 in 16(4) in r3c6 -> no 1 in r1c7, r2c6 or r3c13
32b. 1 in r3c8 -> no 1 in r1c7 & r3c13 -> 1 in n6 in r4c7 -> no 1 in r2c6

33. "45" n3: 2 outies + 1 = r3c7 -> no 4 in r4c8. Here's how.
33a. 4 in r4c8 -> 4 in r2c7 (step 26b) -> 6 in r2c8
33b. 4 in r4c8 -> 1 in r4c7 -> 1 in n2 in r1c6 -> 6 in r3c7 (step 33)
33c. but this means 2 6s n3
33d. no 4 r4c8
33e. no 4 r2c7
33f. no 6 r2c8

34.when 6 in r4c8 -> 1 in r4c7 -> 1 in n2 in r1c6 -> max 2 outies n3 = 7
34a. -> max r3c7 = 8

35. {1249} combo blocked from 16(4)n2. Here's how.
35a. "45" n3: 2 outies + 1 = r3c7
35b. {1249} combo must have 4 in r3c7 -> 2 outies = 3 = [21]: but this forces 2 into both r1c6 & r23c6
35c. 16(4) = {1258/1267/1348/1357/1456} (no 9)

36. 9 in c6 only in h26(4)r4569c6
36a. = 9{278/368} (no 5)
36b. -> no 6 r5c6
36c. = [7928/6839/9836]

Prelims

a. R23C2 = {69/78}
b. R2C78 = {19/28/37/46}, no 5
c. R45C9 = {49/58/67}, no 1,2,3
d. R56C1 = {29/38/47/56}, no 1
e. R5C23 = {17/26/35}, no 4,8,9
f. R56C4 = {29/38/47/56}, no 1
g. R5C78 = {14/23}
h. R78C8 = {29/38/47/56}, no 1
i. R8C23 = {59/68}
j. 9(3) cage at R9C2 = {126/135/234}, no 7,8,9
k. 27(4) cage at R6C7 = {3789/4689/5679}, no 1,2
l. 26(4) cage at R7C5 = {2789/3689/4589/4679/5678}, no 1
m. 17(5) cage at R1C9 = {12347/12356}, no 8,9

1. 17(5) cage at R1C9 = {12347/12356}, CPE no 1,2,3 in R12C8, clean-up: no 7,8,9 in R2C7

2. 45 rule on C1234 2 innies R14C4 = 11 = {29/38/47/56}, no 1

3. 45 rule on R6789 3 outies R5C146 = 23 = {689}, locked for R5, clean-up: R4C9 = {689}, R6C1 = {235}, R6C4 = {235}, no 2 in R5C23
3a. 45 rule on R1234 1 innie R4C9 = 1 outie R5C5 + 4, R4C9 = {689} -> R5C5 = {245}

4. 45 rule on R6789 4 innies R6C1456 = 11 = {1235}, locked for R6, 1 also locked for N5

5. 7 in N5 only in R4C456, locked for R4
5a. 4,7 in N5 only in 22(4) cage at R4C4 = {2479/3478/4567} (only combinations containing both of 4,7)
5b. R4C9 = R5C5 + 4 (step 3a) -> R4C9 + R5C5 = [62/84/95]
5c. 22(4) cage = {2479/4567} (cannot be {3478} = {378}4 which clashes with R4C9 + R5C5 = [84], IOD clash), no 3,8 in R4C456, clean-up: no 3,8 in R1C4 (step 2)
5d. 8 in N5 only in R5C46, locked for R5, clean-up: no 3 in R6C1

6. 45 rule on N3 5(3+2) outies R123C6 + R4C78 = 15
6a. Min R123C6 = 6 -> max R4C78 = 9, no 9 in R4C7
6b. Min R4C78 = 5 (R4C78 cannot be {12/13} which clash with R5C78) -> max R123C6 = 10, no 8,9 in R123C6

7. 45 rule on N23 2 innies R23C4 = 2 outies R4C78 + 8
7a. Min R4C78 = 5 (step 6b) -> min R23C4 = 13, no 1,2,3 in R23C4
7b. 1 in C4 only in R789C4, locked for N8

8. 45 rule on C6789 4 innies R4569C6 = 26, no 1 in R6C6
8a. R6C5 = 1 (hidden single in N5)
8b. 45 rule on C6789 2 remaining innies R49C6 = 15 = [69/78/96]
8c. R4569C6 = {2789/3689/5678}, 8 locked for C6

9. 45 rule on C9 4 innies R1239C9 = 18 = {1269/1278/1359/1368/1458/1467/2349/2358/2367} (cannot be {2457/3456} which clash with R45C9)
9a. 8,9 of {1269/1278/1359/1368/1458/2349/2358} must be in R9C9, {1467/2367} must have one of 6,7 in R9C9 (because 17(5) cage at R1C9 cannot contain both of 6,7) -> R9C9 = {6789}

10. 45 rule on C1 2 outies R67C2 = 1 innie R1C1 + 7
10a. Max R67C2 = 15 (cannot be 16 = {79} which clashes with R23C2) -> max R1C1 = 8

11. 45 rule on N6 5 innies R4C78 + R6C789 = 27 = {14589/14679/15678/23679} (cannot be {12789/13689/24579/24678/34569/34578} which clash with R5C78, cannot be {23589} because no 2,3,5 in R6C789)
11a. 1,5 of {14589/15678} must be in R4C78 -> no 8 in R4C7

12. 45 rule on N89 3 outies R6C789 = 3 innies R789C4 + 12
12a. Max R6C789 = 22 (because R6C789 cannot be {689/789}, step 11) -> max R789C4 = 10, no 8,9 in R78C4

13. Consider combinations for R4C78 + R6C789 (step 11) = {14589/14679/15678/23679}
R4C78 + R6C789 = {14589/14679/15678}, 1 locked for N6 => R5C78 = {23}, locked for R5 => R5C23 = {17}, locked for N4
or R4C78 + R6C789 = {23679}, 7 locked for R6
-> no 7 in R6C23
13a. 7 in N4 only in R5C23 = {17}, locked for R5 and N4, clean-up: no 6 in R4C9, no 4 in R5C78
13b. Naked pair {23} in R5C78, locked for R5
13c. R4C78 + R6C789 = {14679/15678} (cannot be {14589} which clashes with R4C9)
13d. 17(5) cage at R1C9 = {12347/12356}, 2,3 locked for N3, clean-up: no 7,8 in R2C8

14. 17(5) cage at R1C9 = {12347/12356}
14a. Consider combinations for R2C78 = [19]/{46}
R2C78 = [19] => R4C8 = 1 (hidden single in R4)
or R2C78 = {46} => R4C8 = {46} (only remaining place for 4 or 6 in 17(5) cage) => R4C7 = 1 (hidden single in R4)
-> 1 in R24C7, locked for C7, R4C8 = {146}, no 5
14b. 1 in C7 only in R24C7, CPE no 1 in R2C6
14c. Taking this forcing chain a little further
R2C78 = [19] => R4C8 = 1 (hidden single in R4)
or R2C78 = {46} => R4C8 = {46} (only remaining place for 4 or 6 in 17(5) cage) => R4C7 = 1 (hidden single in R4) => R1C6 = 1 (hidden single in C6) => 17(3) cage at R1C6 = {179}, 7,9 locked for N3 => 17(5) cage = {12356} => R4C8 = 6
-> no 6 in R2C8, no 9 in R3C7, no 4 in R4C8, clean-up: no 4 in R2C7
[One of these paths could be taken further to reach a contradiction, but I prefer to avoid this if possible so will look at other steps.]

15. 14(3) cage at R6C9 = {149/158/167/239/257/347/356} (cannot be {248} which clashes with R56C9)
15a. 8 of {158} must be in R6C9 -> no 8 in R78C9

16. R4569C6 (step 8c) = {2789/3689/5678}
16a. 8 in N8 only in 26(4) cage at R7C5 = {2789/3689/5678} (cannot be {4589} which clashes with R5C5), no 4 in R789C5
16b. 45 rule on N69 2 outies R78C6 = 2 innies R4C78 + 4
16c. Max R4C78 = {16} = 7 -> max R78C6 = 11
16d. Consider placement for 3 in N8
3 in R789C4, locked for C4 => R6C4 = 3 (hidden single in N5) => R4569C6 = {3689}, locked for C6
or 3 in 26(4) cage at R7C5 = {3689}, locked for N8
or 3 in R78C6, max R78C6 = 11 => no 9 in R78C6
-> no 9 in R78C6
[Alternatively as a contradiction move, max R78C6 = 11 cannot be [92] because R4569C6 = {5678}, 26(4) cage at R7C5 = {5678} when 3 in C5 and C6 only in N2.]
16e. 8,9 in N8 only in 26(4) cage = {2789/3689}, no 5 in R789C5
16f. 9 in C6 only in R4569C6 (step 8c) = {2789/3689}, no 5, clean-up: no 6 in R5C6

[I originally analysed 27(4) cage at R6C7 next, taking account of R4C78 + R6C789, but this heavy combination analysis didn’t achieve much, so I then looked at R56C1.]

17. 45 rule on N9 5(3+2) outies R6C789 + R78C6 = 31
17a. R4C78 + R6C789 (step 13c) = {14679/15678} -> R6C789 = {479/678/679} = 20,21,22 -> R78C6 = 9,10,11
17b. R4569C6 (step 16f) = {2789/3689}
17c. Consider permutations for R56C1 = [65/92]
R56C1 = [65]
or R56C1 = [92] => 12(3) cage at R5C6 = [813] => R56C4 = [65] => R49C6 = [96] (from R4569C6) => 5 in N8 only in R78C6 = {45} = 9 => R6C789 = 22 = {679}, locked for R6)
-> no 6 in R6C23

18. 6 in R6 only in R6C789, locked for N6 -> R4C8 = 1
18a. Naked pair 4,5 in R4C7 + R5C9, locked for N6
18b. 4 in R6 only in R6C23, locked for N4
18c. R2C7 = 1 (hidden single in N3) -> R2C8 = 9, clean-up: no 6 in R3C2, no 2 in R78C8
18d. R1239C9 (step 9) = {2349/2358/2367}, 2,3 locked for C9 and N3
18e. Min R6C9 = 6 -> max R78C9 = 8, no 9 in R78C9
18f. 2 in N9 only in R8C7 + R9C78, locked for 22(5) cage at R8C6, no 2 in R8C6

19. 17(3) cage at R1C6 = {278/368/458/467}, no 1
19a. R3C6 = 1 (hidden single in C6)

20. R6C789 + R78C6 = 31 (step 17)
20a. R6C789 (step 17a) = {678/679} = 21,22 -> R78C6 = 9,10 = {36/45/46} (cannot be {37} which clashes with 26(4) cage at R7C5), no 7 in R78C6

21. R3C6 = 1 -> 16(4) cage at R2C6 = {1258/1348/1357/1456} (cannot be {1267} because R4C7 only contains 4,5)
21a. 3 of {1357} must be in R2C6 -> no 7 in R2C6

22. R123C6 + R4C78 = 15 (step 6)
22a. Max R3C67 = [18] = 9 -> min R2C6 + R4C7 = 7
22b. R3C6 = 1, R4C8 = 1 -> R12C6 + R4C7 = 13, min R2C6 + R4C7 = 7 -> max R1C6 = 6

23. R4C6 = 7 (hidden single in C6) -> R9C6 = 8 (step 8b), R5C6 = 9, R5C1 = 6 -> R6C1 = 5, R5C4 = 8 -> R6C4 = 3, R6C6 = 2

24. R23C4 = R4C78 + 8 (step 7)
24a. R4C78 = [41/51] = 5,6 -> R23C4 = 13,14 = [49]/{67}/[59], no 4,5 in R3C4

25. 45 rule on N7 5(2+3) outies R6C23 + R789C4 = 22
25a. 4 in N4 only in R6C23 = {48/49} = 12,13 -> R789C4 = 9,10 = {127} (only possible combination, cannot be {126} which clashes with 26(4) cage at R7C5, cannot be {145} which clashes with R23C4 + R4C4, killer ALS block), locked for C4 and N8, 7 also locked for 18(4) cage at R6C3, no 7 in R7C3
25b. R789C4 = {127} = 10 -> R6C23 = 12 = {48}, locked for R6 and N4
25c. Naked triple {369} in R789C5, locked for C5 and N8
[Cracked, although some may think it was cracked after step 17, or after step 22.]

26. R4C9 = 8 (hidden single in N6) -> R5C9 = 5, R4C7 = 4, R4C45 = [65], R5C5 = 4
26a. Naked triple {278} in R123C5 -> R1C4 = 5 (cage sum)

27. R3C4 = 9, R4C23 = {23}, locked for R4, R3C3 = 6 (cage sum)
27a. Naked pair {78} in R23C2, locked for C2 and N1

28. R2C4 = 4, R4C1 = 9 -> R23C1 = 6 = [24]
28a. Naked triple {139} in R1C123, locked for R1 and N1

29. R1C6 = 6 -> R1C78 = 11 = [74], R1C9 = 2, R23C9 = {36}, locked for C9 and N3

30. R78C4 = {14} (hidden pair in C9) -> R6C9 = 9 (cage sum), R6C78 = [67] = 13 -> R7C67 = 14 = [59], R9C9 = 7

31. 8 in N9 only in R78C8 = {38}, locked for C8 and N9, R9C8 = 6 (hidden single in C8

32. R9C8 = 6 (hidden single in C8)
32a. 9(3) cage at R9C2 = {234} (only remaining combination, cannot be {135} which clashes with R9C1) -> R9C4 = 2, R9C23 = {34}, locked for R9 and N7
32b. R78C6 = {17} = 8 -> R67C3 = 10 = [82]
32c. R8C2 = 5 (hidden single in C2)

and the rest is naked singles.

I'll rate my walkthrough at 1.75. Once I'd realised that I didn't need to use heavy combination analysis, I found it quite a lot easier that Assassin 42 V2 and Assassin 48-Hevvie, both of which I've done in recent weeks.

3x3::k:5376:4097:4097:4097:5892:3077:3077:3077:3336:5376:5376:3851:5892:5892:5892:5135:3336:3336:2578:3851:3851:4885:4885:4885:5135:5135:4890:2578:2578:6429:6429:4885:3616:3616:4890:4890:2852:6429:6429:3367:3367:3367:3616:3616:2860:2852:2350:1583:1583:5169:4146:4146:1332:2860:2852:2350:4408:5169:5169:5169:3644:1332:2860:4159:4159:4408:3650:3650:3650:3644:3910:3910:4159:3913:3913:3913:3650:4173:4173:4173:3910:

Not sure that I consider myself an Advanced Player but I took up the challenge and it's the first time that I've finished an Assassin within a few hours of it being posted.

A nice smooth one Ruud! It just seemed to flow!

Over to you Ed for a V2. This Assassin needs one.

Here is my walkthrough.

I've had feedback that there was a flaw in this walkthrough at step 32a. Thanks Para. I've re-written the later stages and also taken the opportunity to include the steps that were only comments before. I hope it is now correct.

Clean-up is used in various steps, using the combinations in steps 1 and 6 for further eliminations from these two cell cages; it is also used for the two cell sub-cages that are produced by applying the 45 rule.

1. R6C34 = {15/24}

2. R6C67 = {79}, locked for R6

3. R67C2 = {18/27/36/45}, no 9, no 2 in R7C2

4. R67C8 = {14/23}

5. R78C3 = {89}, locked for C3 and N7, clean-up: no 1 in R6C2

6. R78C7 = {59/68}

5. 21(3) cage in N1 = {489/579/678}, no 1,2,3

6. 20(3) cage in N3 = {389/479/569/578}, no 1,2

7. 10(3) cage in N14 = {127/136/145/235}, no 8,9

8. 19(3) cage in N36 = {289/379/469/478/568}, no 1

9. 11(3) cage in N47 = {128/137/146/236/245}, no 9 [Para pointed out that R5C2 is now a hidden single 9 in N4. I only saw that at step 32!]

10. 11(3) cage in N69 = {128/137/146/236/245}, no 9

11. 14(4) cage in N56 = {1238/1247/1256/1346/2345}, no 9

12. 14(4) cage in N8 = {1238/1247/1256/1346/2345}, no 9

13. 45 rule on N1 1 outie R1C4 – 7 = 1 innies R3C1 -> R1C4 = {89}, R3C1 = {12}

14. 45 rule on R123 1 outie R4C5 – 4 = 2 innies R3C19, min R3C19 = 3 -> min R4C5 = 7, max R4C5 = 9 -> max R3C19 = 5 -> max R3C9 = 4, R3C19 = [12/13/14/23]

15. 45 rule on N3 1 innie R3C9 = 1 outie R1C6 -> R1C6 = {234}

16. 45 rule on N7 1 outie R9C4 – 3 = 2 innies R7C12, min R7C12 = 3 -> min R9C4 = 6, max R9C4 = 9 -> max R7C12 = 6, no 6,7, clean-up: no 2,3 in R6C2

17. 45 rule on N8 2 innies R9C46 – 11 = 1 outie R6C5, max R9C46 = 17 -> max R6C5 = 6, min R9C46 = 12, no 1,2

18. 45 rule on R12 2 innies R2C37 = 5 = [14/23]
18a. Killer pair 1/2 in R2C3 and R3C1 for N1
18b. 45 rule on R89 2 innies R8C37 = 14 = [86/95], clean-up: no 5,6 in R7C7
18c. Killer pair 8/9 in R7C37 for R7

19. 15(3) cage in N1 = {159/168/249/258/267} [7/8/9], no 3 in R3C23, no 4 in R3C2
19a. 21(3) cage in N1 = {489/579/678}, must contain two of 7,8,9 -> killer triple 7/8/9 in 15(3) and 21(3) cages for N1

20. 3 in N1 locked in R1C23, locked for R1 -> 16(3) cage = 3{49/58}, no 6
20a. R3C9 = {24} (step 14)
[Para pointed out that this can be used with the combinations at the end of step 14 to give R3C1 = 1 and then R1C4 = 8 from step 13. Very neat!]

21. 12(3) cage in N23 = {129/147/246}, no 5,8

22. 20(3) cage in N3 = {389/479} = 9{38/47} -> R3C78 = {789}, 9 locked for R3 and N3
22a. 15(3) cage in N1 now {168/258/267}, no 4, no 5 in R3C2

23. 5 in N3 locked in 13(3) cage = 5{17/26}, no 3,4,8

24. R2C7 = 3 (hidden single in N3), R3C78 = {89}, locked for R3

25. 15(3) cage in N1 = {267} -> R2C3 = 2, R3C23 = {67}, locked for R3 and N1, R3C1 = 1; clean-up: no 4 in R6C4
25a. R3C19 = 3 or 5 -> R4C5 = {79}
25b. Killer pair 7/9 in R4C5 + R6C8, locked for N5

26. R1C4 = 8 (step 13) -> R1C23 = {35} (step 2), locked for R1 and N1

27. Naked quad 2/3/4/5 in R1C6 + R3C456, locked for N2

28. 19(3) cage in N36 = {289/469/478} -> R4C89 = {6789}

29. R37C7 = {89}, locked for C7 -> R6C7 = 7, R6C6 = 9, R4C5 = 7, clean-up: no 2 in R4C12
[Edit. Step 29 rewritten after adding steps 18b and 18c. Steps resulting from R4C5 = 7 have been left until later since they weren’t in the original version of this walkthrough.]

30. 11(3) cage in N47 = {236/245} = 2{36/45}, no 7,8, 2 locked for C1

31. R2C1 = 8, R1C1 = 9 (hidden singles in C1) -> R2C2 = 4, clean-up: no 5 in R4C1, no 5 in R67C2

32. R5C2 = 9, R6C2 = 8 (hidden singles in C2), R7C2 = 1, clean-up: no 4 in R6C8
32a. R9C4 – 3 = R7C12 (step 16), R9C4 = {679} -> R7C1 = {235}
[Corrected from {245}, thanks Para. My only excuse for that error is that I was doing this puzzle too late at night! Steps 35 onward are new; the original step 35 no longer works.]

33. 16(3) cage in N7 = {367/457} = 7{36/45}, no 2, 7 locked for N7

34. 15(3) cage in N78 = {249/267/456} (cannot be {357} which would clash with 16(3) cage), no 3, no 6 in R9C2, no 5 in R9C3

35. R4C5 = 7 (step 29) -> R3C456 = 12 = {345}, locked for R3 and N2 -> R1C6 = 2, R3C9 = 2
35a. R1C6 = 2 -> R1C78 = 10 = {46}, locked for R1 and N3, R5C1 = 1
35b. R3C9 = 2 -> R4C89 = 17 = {89}, locked for R4 and N6

36. R9C2 = 2 (hidden single in C2), 15(3) cage in N78 (step 34) = 2{49/67} -> no 6 in R9C4

37. 2 in C1 locked in R56C1 -> R56C1 = {246}, no 3,5

38. 7 in C1 locked in R89C1 -> R3C2 = 7 (hidden single in C2), R3C3 = 6, R9C3 = 4, R9C4 = 9, clean-up: no 2 in R6C4

39. 16(3) cage in N7 (step 33) = {367}, locked for N7 -> R7C1 = 5 -> R56C1 = {24}, locked for N4, clean-up: no 5 in R4C2
39a. R6C34 = {15}, locked for R6, clean-up: no 4 in R7C8
39b. R4C12 = {36}, locked for R4 and N4
39c. R67C8 = {23}, locked for C8

40. R5C3 = 7 (hidden single in N4), R46C3 = {15}, locked for C3 -> R1C23 = [53]

41. R5C23 = [97] -> R4C34 = [54] (cage sum), R6C34 = [15], R4C67 = [12], R67C8 = [32]

42. R4C67 = [12] -> R5C78 = {56} (cage sum), locked for R5 and N6 -> R6C9 = 4, R5C9 = 1, R6C1 = 2, R5C1 = 4 (naked singles), R7C9 = 6

and the rest is naked and hidden singles, simple elimination and cage sums

3x3::k:4864:3585:3585:3585:6404:3589:3589:3589:3336:4864:4864:4363:6404:6404:6404:4623:3336:3336:7442:4363:4363:4885:4885:4885:4623:4623:7442:7442:7442:6429:6429:4885:4128:4128:7442:7442:2852:6429:6429:2855:2855:2855:4128:4128:3372:2852:1838:5679:5679:5681:5679:5679:1332:3372:2852:1838:3896:5681:5681:5681:3644:1332:3372:4671:4671:3896:4162:4162:4162:3644:3398:3398:4671:3913:3913:3913:4162:3661:3661:3661:3398:

After going through Ed's ever growing Killer archive I noticed that A40 V2 didn't have a walkthrough, so here is my take on it. This Assassin is cracked after the Killer quad in step 1 which is needed for important steps like 3a or 3b. Afterwards it's just one big mop-up.

A40 V2 Walkthrough:

1. R123 !
a) Innies R12 = 5(2) = {14/23}
b) Outies R12 = 30(4) = {6789} locked for R3
c) Innies+Outies N1: 5 = R1C4 - R3C1 -> R1C4 = (6789), R3C1 <> 5
d) Innies+Outies N3: R1C6 = R3C9 = (12345)
e) ! Killer quad (6789) locked in 19(3) + R3C23 for N1
f) Innies N1 = 9(3) = 3{15/24} -> 3 locked for N1
g) Innies R12 = 5(2): R2C7 <> 2

2. R456
a) Outies R1234 = 29(4) = {5789} locked for R5
b) 11(3) @ N5 = 6{14/23} -> 6 locked for R5+N5
c) Innies R1234 = 12(4) = 12{36/45} -> 1,2 locked for R4
d) 16(4): R5C78 <> 9 because it must be <= 13 -> R5C78 = 5{7/8}
-> 5 locked for R5+N6+16(4)
e) 16(4) = 15{28/37} -> 1 locked for R4
f) 9 locked in R5C23 for N4

3. C123
a) Naked quad (6789) locked in R3578C3 for C3
b) 25(4) = 59{47/38} -> 5 locked R4
c) Innies N7 = 12(4) = 12{36/45} -> 1,2 locked for N7

4. R456
a) Innies R1234 = 12(4) = {1245} locked for R4
b) 25(4) = {4579} -> 7 locked for R5+N4
c) Naked pair (58) locked in R5C78 for N6

5. C123
a) Killer pair (79) locked in R5C3+15(2) for C3
b) 17(3) must have one of (79) and it's only possible @ R3C2 -> R3C2 = (79)
c) Naked pair (79) locked in R35C2 for C2
d) 17(3) <> 1 because R3C3 = (68)
e) Innies N1 = 9(3) = {135} because R2C3 = (24) blocks {234}
-> 5 locked for R1+N1
f) 5 locked in 14(3) @ N1 = 5{18/36}

6. R123
a) 7,9 locked in 25(4) @ N2 = 79{18/36/45} <> 2
b) Innies R12 = 5(2): R2C7 <> 4
c) 19(4) = 5{149/239/248/347} because (789) only possible @ R4C5; R4C5 <> 3
-> 5 locked for R3+N2
d) 5 locked in 13(3) @ R2 = 5{17/26}
e) 25(4) = 79{18/36}
f) 4 locked in R2C123 @ R2 for N1
g) 4 locked in 14(3) @ R1C6 @ R1 = 4{19/28/37} <> 6
h) 29(6) = 1368{29/47} because R4C12 = (368)
-> 1 locked for R3, 8 locked for R4+N4
i) 19(4) = 35{29/47} -> 3 locked for R3+N2
j) R3C1 = 1
k) 14(3) @ N1 = {356} -> R1C4 = 6, 3 locked for R1

7. R789
a) 8 locked in R789C9 @ C9 for N9
b) 14(2) = {59} locked for C7+N9
c) Innies R89 = 14(2) = {59} -> R8C3 = 9 -> R7C3 = 6, R8C7 = 5
d) 18(3) = {378} locked for N7, 7 locked for C1
e) 7(2): R6C2 <> 1,4

8. N134
a) Hidden Single: R6C3 = 1 @ N4 -> 22(4) @ R6C3 = 1{489/579/678} <> 2,3
b) R5C7 = 8, R5C8 = 5, R3C3 = 8
c) Hidden Single: R1C8 = 8 @ N1 -> R1C67 = 6(2) = {24} locked for R1

9. R12+N5
a) 18(3) = {369} -> R2C7 = 3, R3C7 = 6, R3C8 = 9
b) 22(4) @ R6C3 = 19{48/57} -> 9 locked for R6+N5
c) Hidden Single: R4C9 = 9 @ N6
d) 29(6) = {123689} -> R3C9 = 2
e) R1C7 = 4, R1C6 = 2, R4C6 = 1, R4C7 = 2
f) Hidden Single: R5C9 = 1 @ R5 -> R67C9 = 12(2) = [48] -> R6C9 = 4, R7C9 = 8
g) R6C8 = 3 -> R7C8 = 2
h) Innies R12 = 5(2) = {23} -> R2C3 = 2

10. R789
a) 13(3) @ R8C8 = {346} -> R8C8 = 4, {36} locked for N9
b) 14(3) = {167} because R9C78 = (17) -> R9C6 = 6, {17} locked for R9
c) 15(3) = 2{49/58} because R9C23 = (245) -> R9C2 = 2, R9C4 = (89)

11. Rest is singles.

Rating: 1.25. I used one Killer quad.

Thanks Ed for a nice variant. Solving it in 2011 I didn't find it a particularly hard variant, more like a V1.5 and maybe Afmob felt the same in 2008; it was probably felt harder in early 2007 when you posted it. The two disjoint cages messed up vertical 45s but they weren't a problem for horizontal 45s.

The first parts of Afmob's walkthrough and my one are fairly similar but then we diverged. I found my step 15 very helpful. Afmob's breakthrough seemed to come from analysing the larger cages, including the disjoint 29(6) cage, more than I did; that was possibly a quicker way to finish this puzzle.

Here is my walkthrough for A40 V2.

Prelims

a) R67C2 = {16/25/34}, no 7,8,9
b) R67C8 = {14/23}
c) R78C3 = {69/78}
d) R78C7 = {59/68}
e) 19(3) cage in N1 = {289/379/469/478/568}, no 1
f) 11(3) cage at R5C1 = {128/137/146/236/245}, no 9
g) 11(3) cage in N5 = {128/137/146/236/245}, no 9

1. 45 rule on R12 2 innies R2C37 = 5 = {14/23}

2. 45 rule on R12 4 outies R3C2378 = 30 = {6789}, locked for R3

3. 45 rule on R123 1 outie R4C5 = 2 innies R3C19 + 4
3a. Min R3C19 = 3 -> min R4C5 = 7
3b. Max R3C19 = 5, no 5 in R3C19
3c. 5 in R3 only in R3C456, locked for N2

4. 45 rule on R1234 4 innies R4C3467 = 12 = {1236/1245}, 1,2 locked for R4

5. 45 rule on R1234 4 outies R5C2378 = 29 = {5789}, locked for R5
5a. Min R5C78 = {57} = 12 -> max R4C67 = 4 -> R4C67 = 3,4 = {12/13}, 1 locked for R4
5b. Min R4C67 = 3 -> max R5C78 = 13 -> R5C78 = 12,13 = {57/58}, 5 locked for R5 and N6
5c. 9 in R5 only in R5C23, locked for N4

6. 45 rule on R6789 2 outies R5C19 = 5 = {14/23}
6a. 11(3) cage in N5 = {146/236}, 6 locked for N5

7. Naked quad {6789} in R3578C3, locked for C3

8. R4C3467 (step 4) = {1245} (only remaining combination) -> R4C67 = {12}, R4C34 = {45}, locked for R4
8a. R4C67 = {12} = 3 -> R5C78 = 13 = {58}, locked for R5 and N6
8b. Naked pair {79} in R5C23, locked for N4
8c. Killer pair 7,9 in R5C3 and R78C3, locked for C3
8d. Killer pair 5,8 in R5C7 and R78C7, locked for C7

9. 45 rule on R789 2 outies R7C37 = 15 = [69/78/96], no 8 in R7C3, no 5 in R7C7, clean-up: no 7 in R8C3, no 9 in R8C7

10. 45 rule on N1 3 innies R1C23 + R3C1 = 9 = {126/135/234}, no 7,8,9
10a. 45 rule on N1 1 outie R1C4 = 1 innie R3C1 + 5 -> R1C6 = {6789}

11. 45 rule on N3 1 outie R1C6 = 1 innie R3C9 -> R1C6 = {1234}

12. 17(3) cage in N1 = {269/278/368/467} (cannot be {179} because R3C3 only contains 6,8), no 1, clean-up: no 4 in R2C7 (step 1)
12a. R1C23 + R3C1 (step 10) = {135} (only remaining combination, cannot be {126} which clashes with 17(3) cage, cannot be {234} which clashes with R2C3), locked for N1, 5 also locked for R1, clean-up: no 2 in R2C7 (step 1)
12b. 17(3) cage = {269/278/467}
12c. 7,9 only in R3C2 -> R3C2 = {79}
12d. Naked pair {79} in R35C2, locked for C2

13. 45 rule on N7 1 outie R9C4 = 2 innies R7C12 + 3
13a. Min R7C12 = 3 -> min R9C4 = 6
13b. Max R7C12 = 6 -> no 6,7,8 in R7C12, clean-up: no 1 in R6C2
13c. Max R57C1 = 9 -> min R6C1 = 2

14. 45 rule on N7 4 innies R7C12 + R9C23 = 12 = {1236/1245}, no 8, 1,2 locked for N7

15. 45 rule on C9 2 innies R34C9 = 2 outies R28C8 + 6
15a. Min R28C8 = 5 (cannot be {12/13} which clash with R67C8) -> min R34C9 = 11 -> min R3C9 = 2, min R4C9 = 7, clean-up: no 1 in R1C6 (step 11)
15b. Max R34C9 = 13 -> max R28C8 = 7, no 7,8,9 in R28C8
15c. R28C8 = 5,6,7 = {14/23/15/16/25} (cannot be {24/34} which clash with R67C8)
15d. Killer pair 1,2 in R28C8 and R67C8, locked for C8
[Alternatively step 15a could be expressed at Min R28C8 = 5 because min R2678C8 = 10 ...]

16. 45 rule on R789 4 innies R7C1289 = 1 outie R6C5 + 8
16a. Min R7C1289 = 10 -> min R6C5 = 2

17. 29(6) disjoint cage = {123689/134678} (cannot be {124679} because 1,2,4 only in R3C19) -> R3C1 = 1, R1C4 = 6 (step 10a), clean-up: no 4 in R5C9 (step 6)
17a. Naked pair {35} in R1C23, locked for R1, clean-up: no 3 in R3C9 (step 11)
17b. Naked quad {2345} in R1C6 + R3C456, locked for N2
17c. 8 in 29(6) disjoint cage only in R4C12, locked for R4 and N4
17d. Naked pair {79} in R4C59, locked for R4

18. 11(3) cage at R5C1 = {236/245}, 2 locked for C1
18a. 6 of {236} must be in R6C1 -> no 3 in R6C1
18b. 2 in N1 only in R2C23, locked for R2

19. 14(3) cage at R1C6 = {149/248}, no 7, 4 locked for R1
19a. 1 of {149} must be in R1C7 -> no 9 in R1C7
19b. 8,9 only in R1C8 -> R1C8 = {89}

20. 5 in N3 only in 13(3) cage = {157/256}, no 3,4,8,9

21. R2C7 = 3 (hidden single in N3), R2C3 = 2 (step 1)
21a. 18(3) cage in N3 = {369/378}
21b. 8 of {378} must be in R3C8 -> no 7 in R3C8

22. 19(3) cage in N1 = {469/478}
22a. 9 of {469} must be in R1C1 -> no 9 in R2C1
22b. 9 in R2 only in R2C456, locked for N2

23. R9C8 = 7 (hidden single in C8), R9C67 = 7 = [16/34/52/61], no 8,9, no 2,4 in R9C6

24. 9 in C8 only in R13C8, locked for N3
24a. 18(3) cage in N3 (step 21a) = {369/378}
24b. 8,9 only in R3C8 -> R3C89 = {89}

25. Naked pair {89} in R13C8, locked for C8 -> R5C8 = 5, R5C7 = 8, clean-up: no 6 in R78C7
25a. R78C7 = [95], R7C3 = 6 (step 9), R8C3 = 9, R5C23= [97], R3C23 = [78], R3C78 = [69], R1C8 = 8, R2C8 = 1, clean-up: no 4 in R67C8, no 1 in R9C6 (step 23)
25b. Naked pair {24} in R1C67, locked for R1 -> R1C1 = 9, R12C9 = [75], R1C5 = 1, R4C9 = 9

26. Naked pair {23} in R67C8, locked for C8 -> R4C8 = 6, R8C8 = 4, clean-up: no 3 in R9C6 (step 23)

27. Naked pair {12} in R49C7, locked for C7 -> R1C67 = [24], R3C9 = 2, R6C7 = 7, R4C67 = [12], R9C7 = 1, R9C6 = 6 (step 23), R67C8 = [32], R5C9 = 1, R5C1 = 4 (step 6), R2C12 = [64], R4C34 = [54], R6C1 = 2, R7C1 = 5 (step 18), R6C2 = 6, R7C2 = 1, R6C9 = 4, R7C9 = 8 (cage sum)

28. R7C456 = {347} = 14 -> R6C5 = 8

and the rest is naked singles.


Rating Comment. I'll rate my walkthrough for A40 V2 at 1.25. This is mainly based on my 2 innies, 2 outies in step 15 and the fairly routine analysis in that step. I also used a couple of naked quads. Afmob's rating of 1.25 using a killer quad seems a bit low; that was the same rating that he gave for A58 V1.5 where he used a triple blocker.