SudokuSolver Forum

3x3::k:4608:4608:4608:4355:4355:5381:5381:5381:3848:7177:7177:4355:4355:4877:4877:2319:5381:3848:7177:1811:1811:4877:4877:3095:2319:3865:3848:7177:4636:2845:2845:5919:3095:5409:3865:3865:5668:4636:4636:5919:5919:5919:5409:5409:3865:5668:5668:4636:4400:5919:3378:3378:5409:5429:1846:5668:2872:4400:3898:3898:1852:1852:5429:1846:5184:2872:3898:3898:5444:5444:5429:5429:1846:5184:5184:5184:5444:5444:4430:4430:4430:

Hey, finally managed to get my walkthrough in first!

Thanks to Para and Cathy for not rushing too much!

Have to post the walkthrough in 2 parts, though, due to size problems.

Here's part 1:


Assassin 54 Walkthrough

1. 28/4 at R2C1 = {(47|56)89} (no 1,2,3)
1a. Common Peer Elimination (CPE): no 8,9 in R1C1

2. 9/2 at R2C7: no 9

3. 7/2 at R3C2: no 7,8,9

4. 12/2 at r3C6: no 1,2,6

5. 11/2 at R4C3: no 1

6. 17/2 at R6C4 = {89}, locked for C4
6a. Cleanup: no 2,3 in R4C3

7. 13/2 at R6C6: no 1,2,3

8. 7/3 at R7C1 = {124}, locked for C1 and N7

9. 11/2 at R7C3 = {38|56} (no 7,9)

10. 7/2 at R7C7: no 7,8,9

11. Innie/outie difference N1: R4C1 = R2C3 + 8
11a. -> R4C1 = 9, R2C3 = 1
11b. Cleanup: no 6 in R3C23, no 8 in R3C7, no 2 in R4C4, no 3 in R3C6

12. Split 19/3 at R2C12+R3C1 = {(47|56)8} = ((4|5)..}
12a. 8 locked for N1
12b. 4 only in R2C2 -> no 7 in R2C2

13. 7/2 at R3C2 = {25|34} = ((4|5)..}
13a. -> Split 19/3 at R2C12+R3C1 and 7/2 at R3C2 form killer pair on {45} in N1
13b. -> no 4,5 elsewhere in N1

14. 9 in N1 locked in 18/3 at R1C23 = {(27|36)9}
14a. -> no 9 elsewhere in R1
14b. {29} only in R1C23 -> no 7 in R1C23

15. 7 in N1 locked in C1 -> not elsewhere in C1

16. Innie/outie difference N7: R7C2 = R9C4 + 7
16a. -> R7C2 = {89}, R9C4 = {12}

17. Naked Pair (NP) on {89} in R7 at R7C24 -> no 8,9 elsewhere in R7
17a. Cleanup: no 3 in R8C3

18. Outies N4: R4C4+R7C2 = 15/2
18a. -> R4C4 = {67}
18b. Cleanup: R4C3 = {45}

19. 1,2 in N5 locked in 23/5 at R4C5
19a. 23/5 at R4C5 cannot contain both of {89} due to R6C4
19b. -> 23/5 at R4C5 = {12(479|569|578)} (no 3)

20. Hidden Single (HS) in N5 at R4C6 = 3
20a. R3C6 = 9
20b. Cleanup: no 4 in R6C7

21. Innie N2: R1C6 = 1

22. Innie N3: R3C8 = 1
22a. Cleanup: no 8 in R2C7, no 6 in R7C7

23. 23/5 at R4C5 must contain exactly one of {45} (step 19b)
23a. Only other place for {45} in N5 is R6C6
23b. -> R6C6 = {45}
23c. cleanup: R6C7 = {89}

24. NP on {89} in R6 at R6C47 -> no 8,9 elsewhere in R6

25. Innies N6: R6C79 = 10/2
25a. -> R6C9 = {12}

26. Innie/outie difference N9: R8C7 = R6C9
26a. -> R8C7 = {12}

27. Split 20/3 at R1C78+R2C8 = (389|479|569|578) (no 2)

28. R8C7 and R6C9 are identical (step 26)
28a. -> cannot (both) be 2, as this would eliminate all candidate positions for 2 in N3
28b. -> R8C7 = 1, R6C9 = 1
28d. -> R6C7 = 9 (step 25)
28e. -> R6C6 = 4, R6C4 = 8
28f. -> R7C4 = 9
28g. -> R7C2 = 8
28h. -> R9C4 = 1 (step 16), R4C4 = 7 (step 18)
28i. -> R4C3 = 4
28j. Cleanup: no 3 in R3C2, no 6 in R7C8

Now for part 2:


29. HS in R7/C1/N7 at R7C1 = 1

30. HS in C3 at R5C3 = 8
30a. Cleanup: no 3 in R7C3

31. 11/2 at R7C3 = {56}, locked for C3 and N7
31a. Cleanup: no 2 in R3C2

32. HS in R5/C5/N5 at R5C5 = 9

33. HS in C5/N5 at R4C5 = 1

34. HS in R5/C2/N4 at R5C2 = 1

35. Split 14/3 at R5C1+R6C12 = {257|356) ({1489} unavailable)
35a. {257} blocked because {27} only in R6C2
35b. -> Split 14/3 at R5C1+R6C12 = {356), locked for N4

36. NS at R4C2 = 2
36a. -> R6C3 = 7

37. Split 14/3 at R4C89+R5C9 = {248|257|347|356}
37a. {248|257|347} blocked because {247} only in R5C9
37b. -> Split 14/3 at R4C89+R5C9 = {356}, locked for N6

38. NS at R4C7 = 8

39. NS at R6C8 = 2
39a. Cleanup: no 5 in R7C7

40. HS in N6 at R5C9 = 3

41. 15/4 at R7C5 = {2346} (1 unavailable), locked for N8

42. HS in R7 at R7C9 = 7
42a. Cleanup: Split 13/2 at R8C89 = {49|58} (no 2,3,6)

43. 6 in N9 locked in 17/3 at R9C7 = {(29|38)6} (no 4,5)

44. HS in R9 at R9C1 = 4
44a. -> R8C1 = 2

45. 2 in R9 locked in N9 -> not elsewhere in N9
45a. 7/2 at R7C7 = {34}, locked for R7 and N9
45b. Cleanup: Split 13/2 at R8C89 = {58}, locked for R8 and N9

46. NS at R8C3 = 6
46a. -> R7C3 = 5

47. NS at R8C6 = 7

48. HS in R8 at R8C2 = 9
48a. -> R9C23 = [73]
48b. -> R3C23 = [52]
48c. Cleanup: no 4,7 in R2C7

49. NS at R1C3 = 9
49a. Cleanup: Split 9/2 at R1C12 = {36}, locked for R1 and N1

50. NS at R2C2 = 4

51. 5 in C7 locked in N3 -> not elsewhere in N3

52. 4 in C9 locked in N3 -> not elsewhere in N3
52a. Cleanup: no 5 in R2C7

53. HS in C7/N3 at R1C7 = 5
53a. Cleanup: Split 15/2 at R12C8 = {78}, locked for C8 and N3
53b. Cleanup: 9/2 at R23C7 = {36}, locked for C7 and N3

54. NS at R3C9 = 4
54a. -> R12C9 = [29]

55. NS at R1C4 = 4
55a. Cleanup: Split 12/2 at R1C5+R2C4 = [75] (only possible permutation)

...and it's all naked singles from now on..

Hi all

This one went quickly for me. Not many difficult moves but maybe my definition of difficult is a bit off for other people.

Walk-through Assassin 54

1. 28(4) in R2C1 = {4789/5689}: no 1,2,3; 8 and 9 locked in 28(4) cage -->> R1C1: no 8,9

2. R23C7 = {18/27/36/45}: no 9

3. R3C23 and R7C78 = {16/25/34}: no 7,8,9

4. R34C6 = {39/48/57}: no 1,2,6

5. R4C34 and R78C3 = {29/38/47/56}: no 1

6. R6C767 = {49/58/67}: no 1,2,3

7. R67C4 = {89} -->> locked for C4
7a. Clean up: R4C3: no 2,3

8. R789C1 = {124} -->> locked for C1 and N7
8a. Clean up : R78C3: no 7,9; R2C2: no 7(only place for 4 in 28(4) cage)

9. 45 on R789: 2 outies and 1 innie: R6C49 = R7C2 + 1: Min R6C49 = 9 -->> Min R7C2 = 8; Max R7C2 = 9 -->> Max R6C49 = 10; Min R6C4 = 8 -->> Max R6C9 = 2: R7C2 = {89}; R6C9 = {12}
9a. R6C4 = R7C2 because of R7C4 -->> R6C9 = 1(step 9)

10. 45 on N9: 1 innie: R8C7 = 1
10a. Clean up: R7C78: no 6; R23C7: no 8

11. 45 on N1: 1 innie and 1 outie: R4C1 = R2C3 + 8 -->> R4C1 = 9 ; R2C3 = 1
11a. Clean up: R3C23: no 6; R3C6: no 3; R4C4: no 2

12. 45 on N6: 2 outies: R3C8 + R6C6 = 5 = [14] -->> R3C8 = 1; R6C6 = 4
12a. R1C6 = 1(hidden); R6C7 = 9; R67C4 = [89]; R7C2 = 8
12b. 8 in C1 locked for N1
12c. 9 in N1 locked for R1
12d. Clean up: R78C3 = {56} -->> locked for C3 and N7
12e. Clean up: R3C2: no 2; R3C6: no 8; R4C3: no 7; R4C4: no 5,6

13. 45 on N7: 1 outie: R9C4 = 1
13a. R7C1 = 1(hidden)

14. 45 on N5: 2 innies: R4C46 = 10 = {37} -->> locked for R4 and N5
14a. Clean up: R3C6: no 7

15. 45 on N4: 1 innie: R4C3 = 4
15a. R4C4 = 7; R34C6 = [93]; R5C5 = 9(hidden); R4C5 = 1(hidden); R5C2 = 1(hidden); R5C3 = 8(hidden)
15b. Clean up: R3C2: no 3
15c. 18(4) in R4C2 = 18 {27/36} -->> R4C2: no 5; R6C3: no 2

16. 21(4) in R8C6 = 1{578}(last possible combination) -->> R8C6 + R9C56 = {578} -->> locked for N8
16a. R7C9 = 7(hidden)
16b. 21(4) in R6C9 = 17{49/58} -->> R8C89 = {49/58}: no 2,3,6

17. Killer Pair {45} in R7C78 + R8C89 -->> locked for N9
17a. R9C1 = 4(hidden); R8C1 = 2
17b. 2 in N8 locked for R7
17c. Clean up: R7C78 = {34} -->> locked for R7 and N9
17d. R7C3 = 5(hidden); R8C3 = 6
17e. Clean up: R8C89 = {58} -->> locked for R8 and N9
17f. R8C6 = 7; R7C2 = 9(hidden); R1C3 = 9(hidden); R3C3 = 2(hidden); R3C2 = 5
17g. Clean up: R2C7: no 7

18. 18(3) in R1C1 = 9{36}(last possible combination) -->> R1C12 = {36} -->> locked for R1 and N1
18a. R2C2 = 4

19. 5 in R4 locked for N6

20. 15(3) in R1C9 = {249/348/456}: no {258} clashes with R8C9 -->> 4 locked in 15(3) cage for C9 and N3

21. 15(4) in R3C8 = 1{356} (last possible combination) -->> R5C9 = 3; R4C89 = {56} -->> locked for R4 and N6
21a. R4C2 = 2; R4C7 = 8; R6C3 = 7; R9C23 = [73]; R6C8 = 2

22. 15(3) in R1C9 = {249}: no {456} clashes with R4C9 -->> R123C9 = [294]
22a. R9C789 = [296]; R4C89 = [65]; R8C89 = [58]
22b. Clean up: R23C7 = {36} -->> locked for C7 and N3
22c. R7C78 = [43]; R5C78 = [74]; R1C7 = 5; R1C4 = 4; R8C45 = [34]

23. 17(4) in R1C4 = 14{57}(last possible combination): no 2,6,8

And the rest is all singles.

greetings

Para

As promised:

1. 7(3) r789c1 = {124} not elsewhere in c1/N7
-> 11(2) r78c3 = {38/56}

2. 17(2) r67c4 = {89} not elsewhere in c4.

3. Outies – Innies N1: r4c1 – r2c3 = 8 -> r4c1 = 9, r2c3 = 1

4. Innies N2: r13c6 = 10 = [19/28/37/64] -> 12(2) r34c6 = [93/84/75/48]

5. O-I N3: r1c6 – r3c8 = 0 -> r1c6 = r3c8: (1236)

6. O-I N7: r7c2 – r9c4 = 7 -> r7c2 = (89), r9c4 = (12)
-> NP {89} in r7c24, not elsewhere in r7 -> r8c3 <> 3
-> 20(4) r8c2+r9c234 must have 7 = {1379/2378/2567}

7. O-I N9: r6c9 – r8c7 = 0 -> r6c9 = r8c7

8. 7(2) r3c23 = {25/34}

9. 28(4) r2c12+r34c1 = {4789/5689} -> must have 8 within r2c12, r3c1

10. 18(3) r1c123 must have 9 -> {279/369} (Can’t be {459} else no options for 7(2))

11. Outies N47: r49c4 = 8 = [71/62] -> r4c3 = (45)

12. Innies N5: r4c46+r6c46 = 22 without either 1 or 2
-> 23(5) in N5 must have 1 and 2 = {12569/12578} -> 23(5) must also have 5.
-> r46c6 <> 5 -> r3c6 <> 7 -> r1c6 <> 3 -> r3c8 <> 3
-> r6c7 <> 8

13. Combination options for 22(4) r4c46+r6c46 = {3469/3478} -> must have 3 and 4
-> r4c6 = 3, r6c6 = 4
-> r3c6 = 9 -> r1c6 = 1, r3c8 = 1
-> r6c7 = 9 -> r6c4 = 8, r4c4 = 7 -> r4c3 = 4 (-> r3c3 <> 4 -> r3c2 <> 3), r7c4 = 9
-> r7c2 = 8 -> r9c4 = 1
-> 11(2) in N7 = {56} not elsewhere in c3
-> remaining 19(3) in N7 = {379}
-> only place left for 8 in N4 is r5c3
-> only place left for 9 in N5 is r5c5

14. Outies N8: r8c7 = 1 -> r7c1 = 1, r6c9 = 1, r4c5 = 1, r5c2 = 1
-> r4c2+ r6c3 of 18(4) in N4 = [27/63]
-> remaining 14(3) in N4 = {257/356}

15. Remaining 20(3) in N8 must be {578} -> 15(4) in N8 = {2346}
-> HS r7c9 = 7 -> r8c89 = {49/58}
-> 17(3) r9c789 must have 6: {269/368}
-> HS r9c1 = 4 -> r8c1 = 2

16. Row 9: 5 locked to r9c56 -> r8c6 <> 5

17. N8: 2 locked to r7c56 -> 7(2) r7c78 = {34} -> r8c89 = {58} -> 17(3) r9c789 = {269}
-> r7c56 = {26} -> r7c3 = 5, r8c3 = 6
-> r8c6 = 7, r8c2 = 9 -> r1c3 = 9
-> NP r69c3 {37} -> r3c3 = 2 -> r3c2 = 5
-> HS r2c2 = 4

18. UR: r4c89 + r8c89 -> r4c89 cannot both be {58}
Combination options for r4c89+r5c9 = {28}+4 / {56}+3
-> r4c7 = (58), r5c9 = (34)
-> 21(4) in N6 must have 7 = {3567/2478}.

19. 15(3) in N3 = {249} (Can’t be 348 else no options for r5c9, can’t be 258 else no options for r8c9, can’t be 456 else no options for r4c9)
-> r2c9 = 9 -> r9c8 = 9, r9c9 = 6, r9c7 = 2
-> 9(2) in N3 = {36} -> r7c7 = 4, r7c8 = 3, r5c8 = 4
-> remaining 20(3) in N3 = {578}

Straightforward from here with cage combos and singles.

Not too many typos considering the interruptions I had. Thanks to Andrew for checking and advising minor amendments.

Cathy

PS Just had a quick scan of Mike's and Para's walkthroughs. Interesting to see how different approaches / different order of steps leads to the same result

Good to see that Ed has posted a walkthrough for Mike's V2.

I've only done Ruud's V1 in the last couple of days and then worked through the 3 other posted walkthroughs which all followed somewhat different paths and had some nice moves.

The moves that I particularly liked were Mike's step 28a (my solution path was different so I never reached that position), Para's step 9a, Cathy's step 18 and the elimination of 456 in her step 19.

An alternative way of looking at the beginning of Cathy's step 18 is that r4c89 cannot be {58} because that would put 1 in the two other cells of the cage. Whichever way {58} is eliminated, Cathy's combination work in the rest of step 18 was neat.


I don't think any of the posted walkthroughs needed many advanced moves. After my key move (step 17), which wasn't used in any of the other walkthroughs, it flowed fairly easily.

Here is my walkthrough

Thanks Ed for the alternative way to do step 17 and for the typo correction to step 28b. I've also moved step 24 to become step 18b.

1. R23C7 = {18/27/36/45}, no 9

2. R3C23 = {16/25/34}, no 7,8,9

3. R34C6 = {39/48/57}, no 1,2,6

4. R4C34 = {29/38/47/56}, no 1

5. R67C4 = {89}, locked for C4, clean-up: no 2,3 in R4C3

6. R6C67 = {49/58/67}, no 1,2,3

7. R78C3 = {29/38/47/56}, no 1

8. R7C78 = {16/25/34}, no 7,8,9

9. R789C1 = {124}, locked for C1 and N7, clean-up: no 7,9 in R78C3

10. 28(4) cage at R2C1 = {4789/5689} = 89{47/56}, no 1,2,3 -> no 8,9 in R1C1
10a. 4 only in R2C2 -> no 7 in R2C2

11. 45 rule on N1 1 outie R4C1 – 8 = 1 innie R2C3 -> R2C3 = 1, R4C1 = 9, clean-up: no 6 in R3C23, no 3 in R3C6, no 8 in R3C7, no 2 in R4C4
11a. 28(4) cage at R2C1 (step 10) = 89{47/56} -> 8 locked for N1

12. Killer pair 4/5 in 28(4) cage at R2C1 and R3C23, locked for N1

13. 9 in N1 locked in R1C123 for R1
13a. R1C123 = 9{27/36}

14. 45 rule on N4 1 outie R7C2 – 4 = 1 remaining innie R4C3 -> R4C3 = {45}, R7C2 = {89}, clean-up: R4C4 = {67}

15. Naked pair {89} in R7C24, locked for R7, clean-up: no 3 in R8C3

16. 45 rule on N7 1 innie R7C2 – 7 = 1 outie R9C4 -> R9C4 = {12}

17. R79C4 = 10 as follows - if R7C2 = 8, R7C4 = 9 and R9C4 = 1 (step 16); if R7C2 = 9, R7C4 = 8 and R9C4 = 2 (step 16)
17a. 45 rule on N8 2 innies R79C4 – 9 = 1 outie R8C7 -> R8C7 = 1, clean-up: no 8 in R2C7, no 6 in R7C78
[Ed pointed out 45 rule on N78 2 innies R7C24 – 16 = R8C7 -> R8C7 = 1 which is possibly more obvious although it wasn’t what I spotted.]
17b. 21(4) cage at R8C6 = 1{389/479/569/578}, no 2

18. 45 rule on N9 1 innie R8C7 = 1 outie R6C9 -> R6C9 = 1
18a. 21(4) cage at R6C9 = 1{389/479/569/578}, no 2
18b. 8,9 only in R8C89 -> no 3 in R8C89

19. 45 rule on N6 1 remaining innie R6C7 – 8 = 1 outie R3C8 -> R3C8 = 1, R6C7 = 9, R6C6 = 4, R67C4 = [89], R7C2 = 8, clean-up: no 8 in R3C6, no 3 in R7C3, R9C4 = 1 (step 16)

20. Naked pair {56} in R78C3, locked for C3 and N7 -> R4C3 = 4, R4C4 = 7, clean-up: no 2,3 in R3C2, no 5 in R3C6

21. R7C1 = 1 (hidden single in C1)

22. R1C6 = 1 (hidden single in R1)
22a. 21(4) cage at R1C6 = 1{389/479/569/578}, no 2
22b. 9 only in R2C8 -> no 3,4,6 in R2C8

23. 45 rule on N2 1 remaining innie R3C6 = 9, R4C6 = 3

24. Moved to become step 18b

25. 22(4) cage at R5C1 = 8{257/356} = 58{27/36}, 5 locked for N4

26. R5C3 = 8 (hidden single in C3)
26a. 18(4) cage in N4 = 18{27/36}

27. 21(4) cage at R8C6 (step 17b) = {1578} (only remaining combination), locked for N8

28. R7C9 = 7 (hidden single in R7)
28a. 21(4) cage at R6C9 = 17{49/58}, no 6
28b. Killer pair 4/5 in R7C78 and R8C89, locked for N9
28c. R9C789 = 6{29/38}

29. R9C1 = 4 (hidden single in R9), R8C1, = 2

30. 2 in R9 locked in R9C789 (step 28c) = {269}, locked for R9 and N9, clean-up: no 5 in R7C78, no 4 in R8C89 (step 28a)
30a. Naked pair {34} in R7C78, locked for R7
30b. Naked pair {26} in R7C56, locked for R7 and N8 -> R78C3 = [56]
30c. Naked pair {58} in R8C89, locked for R8 -> R8C6 = 7

31. R8C2 = 9 (hidden single in R8)

32. R1C3 = 9 (hidden single in C3) -> R1C12 = {36}/[72], no 7 in R1C2

33. R5C5 = 9 (hidden single in R5) -> R4C5 = 1 (hidden single in C5), R5C2 = 1 (hidden single in C2)
[So many hidden singles in this puzzle! I think I saw most of them as soon as I could but R5C5 has been there since step 23 which would also have given R4C5 and R5C2 then. Probably woudn’t have made much difference to the solution path if I’d seen them earlier.]

34. 18(4) cage in N4 (step 26a) = 18{27/36}
34a. 7 only in R6C3 -> no 2 in R6C3

35. Naked pair {37} in R69C3, locked for C3 -> R3C3 = 2, R3C2 = 5, clean-up: no 7 in R1C1 (step 32), no 4,7 in R2C7
35a. Naked pair {36} in R1C12, locked for R1 and N1 -> R2C2 = 4
35b. Naked pair {78} in R23C1, locked for C1

36. 7 in N4 locked in R6C23, locked for R6
36a. 5 in R4 locked in R4C789, locked for N6

37. 22(4) cage at R5C1 (step 26) = {3568} (cannot be {2578} because 2,7 only in R6C2), locked for N4 -> R4C2 = 2, R6C3 = 7, R9C23 = [73]

38. R123C9 = {249/348/456} (cannot be {258} which clashes with R8C9) = 4{29/38/56}, 4 locked for C9 and N3, clean-up: no 5 in R2C7
38a. 9 only in R2C9 -> no 2 in R2C9

39. 21(4) cage at R1C6 (step 23a) = {1578} (only remaining combination), 5,7,8 locked for N3, clean-up: no 2 in R2C7
39a. Naked pair {36} in R23C7, locked for C7 and N3 -> R123C9 = [294], R7C78 = [43], R9C789 = [296], R5C7 = 7, R5C9 = 3, R5C8 = 4 (hidden single in R5)
39b. Naked pair {36} in R3C47, locked for R3

40. R4C8 = 6 (hidden single in R4), R4C9 = 5 (cage sum)

and the rest is naked singles, naked pairs and cage sums in N2

3x3::k:4864:4864:4864:4867:4867:4869:4869:4869:1800:4105:4105:4867:4867:5901:5901:3087:4869:1800:4105:2067:2067:5901:5901:2839:3087:6169:1800:4105:6172:2845:2845:6943:2839:4129:6169:6169:4644:6172:6172:6943:6943:6943:4129:4129:6169:4644:4644:6172:2096:6943:2866:2866:4129:5941:4662:4644:2360:2096:3898:3898:2364:2364:5941:4662:6464:2360:3898:3898:5444:5444:5941:5941:4662:6464:6464:6464:5444:5444:3150:3150:3150:

Congratulations Mike. Great to have another creator of V2's. It is a real good one too - lots of combining "45" moves to see contradictions. Forces you to have a whole-of-puzzle approach.

From Ruud's lead-in to Assassin 55, looks like he has developed a rating system. I wonder what this V2 would get.

Assassin 54 Version 2

1. 7(2)n3 = {124}: all locked for n3 & c9

2. "45" n3: r1c6 + 7 = r3c8
2a. r1c6 = {12}, r3c8 = {89}

3. 19(4)n2: {1279} blocked since 1 & 2 only in r1c6
3a. {18} & {29} combo's blocked from 19(4) by r3c8 (step 2)
3b. 19(4) = {1369/1567/2368}

4. 12(2)n3 = {39/57}(no 8)

5. "45" n6: r3c8 + 5 = r6c79
5a. r6c79 = 14/15
5b. no {123} in r6c79

6. 8(2)n5: no 4,8,9

7. "45" n8: r8c7 + 9 = r79c4
7a. max. r79c4 = [79] = 16
7b. -> max. r8c7 = 7

8. "45" n9: r6c9 + 1 = r8c7
8a. = [56/67]
8b. r6c9 = {56}, r8c7 = {67}

9. r79c4 = 15/16 (step 7)
9a. = [69/78]/[79]
9b. r7c4 = {67}, r9c4 = {89}
9c. r6c4 = {12}

10. "45" n7: r9c4 - 7 = r7c2
10a. r7c2 = {12}

11. "45" n36: r16c6 + 1 = r6c9
11a. ->r16c6 = 4/5 = [13/14/23]
11b. r6c6 = {34}, r6c7 = {78}

12. "45" n6: r3c8 + 5 = r6c79
12a. -> r6c79 = 13/14
12b. = [85/86] only ([76] blocked by 7 required in r8c7 when r6c9 = 6 step 8a.)
12c. r6c7 = 8, r6c6 = 3

13. "45" n69: r3c8 - 2 = r8c7 = [86] only ([97] clashes with 12(2)n3)
13a. r3c8 = 8, r8c7 = 6

14. "45" n9 -> r6c9 = 5

15. "45" n3 -> r1c6 = 1

16. 6 in c9 only in n6 in split-cage 16(3)
16a. 6 locked for n6 (including no 6 in r4c8)
16a. split-cage 16(3) = 6{19/37}(no 2,4)

17. "45" n8: r79c4 = 15 = [69/78]
17a. r679c4 = [269/178]

18. "45" n5: 3 innies = 15
18a. r6c4 = {12} -> r4c46 = 14/13
18b. r4c46: no 1, 2

19. "45" n12: r4c16 = 7 = h7(2)r4
19a. = [16/25/34]
19b. r4c1 = {123}, r4c6 = {456}

20. from step 18a.
i. r6c4 = 1 -> r4c46 = 14 = [95] ([86] blocked by 8 required in r9c4 when r6c4 = 1 (step 17a))
ii. r6c4 = 2 -> r4c46 = 13 = [85/76] ([94] blocked by 9 required in r9c4 when r6c4 = 2 (step 17a)
20a. r4c46 = [95/85/76]
20b. r4c4 = {789}, r4c6 = {56}-> r4c1 = {12}(step 19), r4c3 = {234}

21. [95] blocked from r4c46 since it forces 2 into both r4c13 (11(2) cage & h7(2)r4)

22. from step 20.ii: r4c46 = [85/76] = 13
22a. -> r6c4 = 2, r7c4 = 6
22b. r4c4 = {78},
22c. no 2 r4c3

23. "45" n8: r9c4 = 9

24. "45" n7: r7c2 = 2

25. 9(2)n9 = {45}: both locked for r7, n9

26. 12(3)n9 = {138/237} = 3{18/27}
26a. 3 locked for n9, r9

27. split-cage 18(3)r7c9 = 9{18/27}
27a. r8c8 = {12}

28. split-cage 16(3)r4c8 = {169/367}

29. {169} blocked from split-cage 16(3)r4c8. Here's how.
i. 1 in r4c8 -> r45c9 = {69}
11. 1 in r4c8 -> 2 in r8c8 -> r78c9 = {79} (step 27)
iii. but this means 2 9's c9
iv. {169} blocked from split-cage 16(3)r4c8

30. split-cage 16(3)r4c8 = {367}: all locked for n6

31. 11(2) n4 = [38/47] = [3/7..]

32. Killer pair 3/7 in 11(2)n4 and r4c8
32a. 3 & 7 locked for r4

33. r4c9 = 6

34. r4c6 = 5, r3c6 = 6

35. "45" n12: r4c1 = 2

36. "45" n1: r2c3 = 4

37. r4c348 = [387], r5c9 = 3

38. split-cage 15(3)r1c4 = {357} ({258} blocked by 2&8 only in r1c5)
38a. 3,5,7 locked for n2

39. r3c4 = 4
39a. rest of 23(3)n2 = {289}
39a. 8 only in r2: 8 locked for r2

40. split-cage: 15(3)r8c6 = {258}
40a. r9c5 = 5
40b. r89c6 {28}: both locked for n8 & c6

41. 9(2)n7 = {18}: both locked for n7 & c3

42. r27c6 = [97]

43. r23c5 = [82]

44. r123c9 = [421]

45. 8(2)n1 = [35](last valid combo)

The rest is hidden/naked singles and last valid combination

3x3::k:2560:2560:4866:4866:3332:3077:3077:4103:4103:2560:1802:4866:7180:3332:5902:3077:3344:4103:5394:1802:7180:7180:5910:5902:5902:3344:4378:5394:7180:7180:5910:5910:5910:5902:5902:4378:5394:3109:3109:3109:5910:2857:2857:2857:4378:5165:4910:4910:4912:4912:4912:6707:6707:4917:5165:5165:4910:4910:4912:6707:6707:4917:4917:3391:5165:3649:4910:2627:6707:5701:4917:4679:3391:3391:3649:3649:2627:5701:5701:4679:4679:

Hi all

What a work to fill 81 little cells. But in the end i liked the result. I keep missing obvious things but i get back to those some steps later. There's some nice steps in there. Especially one interesting 45 move.
I am fearing this monster Ruud has set up for us though.

Walkthrough Assassin 55

1. 10(3) in R1C1 = {127/136/145/235}: no 8, 9

2. 19(3) in R1C3 = {289/379/469/478/568}: no 1

3. R12C5 and R23C8 = {49/58/67}: no 1,2,3

4. R23C2 = {16/25/34}: no 7,8,9

5. 21(3) in R3C1 = {489/579/678}: no 1,2,3

6. 11(3) in R5C6 = {128/137/146/236/245}: no 9

7. R89C5 = {19/28/37/46}: no 5

8. 22(3) in R8C7 = {589/679}: no 1,2,3,4; 9 locked in 22(3) cage: R9C89: no 9

9. 45 on R5: 3 innie: R5C159 = 22 = {589/679}: no 1,2,3,4; 9 locked in R5C159 for R5

10. 45 on R89: 4 innies: R8C2468 = 13 = {1237/1246/1345}: no 8,9; 1 locked in R8C2468 for R8
10a. Clean up: R9C5: no 9

11. 45 on C89: 3 innies: R456C8 = 7 = {124} -->> locked for C8 and N6
11a. Clean up: R23C8: no 9

12. 45 on C12: 3 innies: R456C2 = 19 = {289/379/469/478/568}: no 1

13. 45 on N9: 2 outies and 1 innie: R6C9 + R9C6 = R7C7 + 14; Max R6C9 + R9C6 = 18 -->> Max R7C7 = 4; Min R7C7 = 1 -->> Min R6C9 + R9C6 = 15 (Min 6 and Max 9) Conclusion: R6C9 = {6789}, R7C7 = {1234}, R9C6 = {6789}
13a. R6C9 + R9C6 see all 9’s in N9: can’t both be 9 -->> Max R6C9 + R9C6 = 17 -->> R7C7: no 4
13b. 4 in N9 locked for C9

14. 45 on N1: 2 innies and 1 outie: R3C13 = R1C4 + 9; Max R3C13 = 17 -->> Max R1C4 = 8; Min R1C4 = 2 -->> Min R3C3 = 11; Conclusion: R3C13: no 1; R1C4: no 9
14a. 45 on N1: 4 innies: R123C3 + R3C1 = 28 = {4789/5689}: no 2,3

Hmm, missed these.
15. 45 on C1234: 2 innies: R46C4 = 17 = {89} -->> locked for C4 and N5
15a. Clean up: R5C19: no 5(no {89} in R5C5)(step 9)

16. 45 on C6789: 2 innies: R46C6 = 3 = {12} -->> locked for C6 and N5

17. 23(5) in R3C5 needs one of {89} in R4C4, one of {12} in R4C6 and one of {567} in R5C5 -->> 23(5) = {12479/12569/12578/13469/13478/13568/23459/23468}: can only have one of {89} and it goes in R4C4 -->> R3C5: no 8,9

18. 19(4) in R6C4 can only have one of {89} and it goes in R6C4 -->> R7C5: no 8,9

19. {89} in C5 locked in 13(2) in R1C5 and 10(2) in R8C5; they can’t have both so both need one of {89} -->> R12C5 = {49/58}: no 6,7; R89C5 = {28}/[91]: no 3,4,6,7

20. 23(5) in R3C5 only has {45} in R345C5. Can’t have both {45} because it clashes with R12C5 -->> 23(5): no {23459}; 23(5) = {12479/12569/1258/12469/12478/13568/23468}
20a. Killer Pair {45} in R12C5 + R345C5 -->> locked for C5

21. 45 on N9: 4 innies and 1 outie: R7C789 + R8C8 = R9C6 + 5: Max R9C6 = 9 -->> Max R7C789 + R8C8 = 14: Conclusion: R7C789 + R8C8: no 9;
21a. R1C8 = 9(hidden)
21b. Clean up: R2C5: no 4
21c. 16(3) in R1C8 = 9{16/25}: no 3,7,8

22. Killer Pair {56} in R12C9 + R23C8 -->> locked for N3

23. Follow up 45 on N9: R7C789 + R8C8= 11/12/13/14
23a. When 11: R78C8 = {35}, R7C79 = {12}
23b. When 12: R78C8 = {36}, R7C79 = {12}
23c. When 13: R78C8 = {35}, R7C79 = [14]; R7C78 = {37}, R7C79 = {12}
23d. When 14: R78C8 = {35}, R7C79 = [24]; R78C8 = {36}, R7C79 = [14]; R78C8 = [83], R7C79 = {12}(no {56} clashes with R23C8)
23e. Conclusion: R7C7: no 3; R7C9: no 3,5,6,7,8; R78C8 = {35/36/37}/[83]: 3 locked for N9

24. 3 in C9 locked in 17(3) cage in R3C9 -->> 17(3) = {359/368}: no 1,2,7
24a. Killer Pair {56} in R12C9 + 17(3) cage in R3C9 -->> locked for C9

25. 18(3) cage in R8C9 = [981/972/954]/{8[6]4}: R89C9: no 7
25a. R6C9 = 7(hidden)
25b. Clean up: R9C6: no 6,7(step 13(R6C9 + R9C6 is minimal 15)
25c. Killer Pair {89} in R89C5 + R9C6 -->> locked for N8

26. 26(5) in R6C7 needs 2 of {124} in R6C8 + R7C7 -->> 26(5) = {12689/14579/14678/23489/24569/24578}: Only 2 of {124} so no {124} anywhere else in 26(5) -->> R78C6: no 4; 26(5) needs one of {89} -->> only place for {89} is R6C7: R6C7 = {89}
26a. Naked Pair {89} in R6C47 -->> locked for R6

27. 19(4) in R6C4 needs one of {89} in R6C4, one of {36} in R6C5 and one of {12} in R6C6 -->> 19(4) = {1369/1378/2368}: Needs one of {12}, this goes in R6C6 so nowhere else in 19(4) cage: R7C5: no 1,2; 3 locked in R67C5 -->> locked for C5

28. Hidden Killer Pair {12} in C5 in R3C5 + R89C5: R3C5 = {12}
28a. 23(5) in R3C5 in needs both {12} in R3C5+ R4C6 -->> 23(5) = {12479/12569/12578}

29. 4 in N8 locked for C4
29a. 5 in R6 locked for N4
29b. 9 in R7 locked for N7

30. 11(3) in R5C6 = [731]/[461]/{36}[2]/[452]: R5C6: no 5; R5C7: no 8; R5C8: no 4

Missed something at step 26.
31. 26(5) in R6C7 can’t have both {89} -->> 26(5) = {14579/14678/24569/24578}: 4 locked in 26(5) -->> R6C8 = 4(only place in cage): R78C6 = {56/57/67}: no 3
31a. Naked Pair {12} in R4C68 -->> locked for R4

32. 45 on C9: 3 outies: R789C8 = 16; R78C8 = {35/36}/[83](step 23) -->> R9C8 = {578}
32a. 18(3) in R8C9 = [981/972/954] -->> R8C9 = 9; R9C9: no 8
32b. R9C6 = 9(hidden); R2C5 = 9; R1C5 = 4
32c. Clean up: R9C5: no 1
32d. R89C5 = {28} -->> locked for C5 and N8
32e. R3C5 = 1; R4C6 = 2; R45C8 = [12]; R6C6 = 1; R7C7 = 2(step 13)
32f. R5C3 = 1(hidden); R5C6 = 4(hidden); R5C7 = 5; R4C5 = 5(hidden)
32g. Clean up: R89C7 = {67} -->> locked for C7 and N9

33. 17(3) in R3C9 = {368} -->> locked for C9
33a. Clean up: R12C9 = {25} -->> locked for N3
33b. Clean up: R23C8 = {67}

34. 26(5) in R6C7 = 24{569/578} -->> 5 locked in R78C6 for C6 and N8

35. 12(3) in R5C2 = [813](last possible combination)
35a. R6C5 = 6; R5C5 = 7; R5C9 = 6; R5C1 = 9; R4C4 = 8; R6C4 = 9; R7C5 = 3
35b. R34C9 = [83]; R46C7 = [98]; R8C8 = 3(hidden); R3C3 = 9(hidden); R7C2 = 9(hidden)

36. 23(5) in R2C6 = {13469}(last possible combination) -->> R3C7 = 4(only place in cage); R23C6 = {36} -->> locked for C6 and N2
36a. R1C6 = 8(hidden); R2C3 = 8(hidden)

37. 19(3) in R1c3 = [658](last possible combination): R1C3= 6; R1C4 = 5
37a. R12C9 = [25]

38. 10(3) in R1C1 = {127}(last possible combination) -->> R2C1 = 2; R1C12 = {17} -->> locked for R1 and N1
38a. R12C7 = [31]; R23C4 = [72]; R23C8 = [67]; R23C6 = [36]; R23C2 = [43]; R3C1 = 5
38b. R4C1 = 7; R4C23 = [64]; R1C12 = [17]; R6C1 = 3
38c. R9C3 = 3(hidden); R9C7 = 7(hidden); R8C7 = 6;

39. R6C23 = {25} -->> locked for 19(5) cage in R6C2: 19(5) = {12457}: no 6
And now all singles.
39a. R7C3 = 7; R78C6 = [57]; R7C89 = [81]; R9C89 = [54]; R78C4 = [41]
39b. R7C1 = 6; R8C2 = 2; R6C23 = [52]; R9C12 = [81]; R8C1 = 4; R8C3 = 5
39c. R89C5 = [82]; R9C4 = 6

greetings

Para

Hi guys,


I hope not! How are you getting on?

In the meantime, thought I'd join Para in posting a V1 walkthrough, because I unfortunately won't be able to work on the V2 this week.

This V1 wasn't too bad. There was quite a narrow solving path at the beginning, which I suspect nearly everyone will take. There weren't any really difficult techniques required. Some of the earlier Assassins (e.g. 33,34) were worse IMO. I suspect Andrew's approach will be quite good on this one.

Having said all that, I thought the difficulty of this puzzle was about optimal for a V1. Very well chosen. Thanks, Ruud.

So, enough pre-amble, here's my A55V1 Walkthrough.


Assassin 55 Walkthrough

1. 10/3 at R1C1: no 8,9

2. 19/3 at R1C3: no 1

3. 13/2 at R1C5: no 1,2,3

4. 7/2 at R2C2: no 7,8,9

5. 13/2 at R2C8: no 1,2,3

6. 21/3 at R3C1: no 1,2,3

7. 11/3 at R5C6: no 9

8. 10/2 at R8C5: no 5

9. 22/3 at R8C7 = {(58|67)9}
9a. CPE: no 9 in R9C89

10. Innies C1234: R46C4 = 17/2 = {89}, locked for C4 and N5

11. Innies C6789: R46C6 = 3/2 = {12}, locked for C6 and N5
11a. {128} combo now blocked for 11/3 at R5C6, because none of these digits in R5C6
11b. -> no 8 in R5C67

12. Innies R5: R5C159 = 22/3 = {(58|67)9} = {(5|6)..}
12a. 9 locked for R5
12b. {89} only in R5C19 -> no 5 in R5C19

13. Innies C89: R456C8 = 7/3 = {124}, locked for C8 and N6
13a. Cleanup: no 9 in 13/2 at R2C8 = {58|67}
13b. no 7 in R3C9 (due to no {124} in R45C9)
13c. min. R45C9 = 9 -> max. R3C9 = 8
13d. -> no 9 in R3C9

14. 11/3 at R5C6 = {137|146|236|245) = {(3|4|5)..}
14a. {12} only in R5C8
14b. -> R5C8 = {12}
14c. 4 only in R5C6
14d. -> no 5 in R5C6

15. 12/3 at R5C2 = {138|147|237|156|246|345}
15a. {156} blocked by h22/3 R5 innies (step 12)
15b. {345} blocked by 11/3 (step 14)
15c. -> no 5 in 12/3 at R5C2

16. 5 in N5 locked in C5 -> not elsewhere in C5
16a. Cleanup: no 8 in 13/2 at R1C5 = {49|67} = {(4|6)..}

17. {46} for 10/2 at R8C5 blocked by 13/2 at R1C5 (step 16a)
17a. -> no 4,6 in R89C5

18. 19/4 at R6C4 can only contain 1 of {89} due to 19/4 cage sum, which must go in R6C4
18a. -> no {89} elsewhere in 19/4
18b. -> no 8,9 in R7C5

19. 23/5 at R3C5 cannot contain {12389}, because none of these digits are in R5C5
19a. -> 23/5 at R3C5 must contain exactly one of {89} (in R4C4)
19b. -> no {89} elsewhere in 23/5
19c. -> no 8,9 in R3C5

20. 8 in C5 now locked in 10/2 at R8C5
20a. -> 10/2 at R8C5 = {28}, locked for C5 and N8

21. 9 in C5 now locked in 13/2 at R1C5
21a. -> 13/2 at R1C5 = {49}, locked for C5 and N2

22. 4 in N5 locked in R5 -> not elsewhere in R5
22a. Cleanup: 4 in 12/3 at R5C2 only in R5C4 -> no 6 in R5C4 (see steps 15,15a)

23. 23/5 at R2C6: min. R23C6+R4C78 = 3+5+6+1 = 15
23a. -> no 9 in R3C7

24. 9 in R3 locked in N1 -> not elsewhere in N1
24a. -> 19/3 at R1C3 = {(47|56)8} (no 2,3)
24b. 8 locked in R12C3 for C3 and N1

25. 2 in C4 locked in 28/5 at R2C4 = {2..}
25a. -> cannot also contain a 1 (otherwise 28/5 cage sum unreachable)
25b. -> no 1 in 28/5 at R2C4 (otherwise 28/5 cage sum unreachable)
25c. 2 in 28/5 at R2C4 locked in R23C4
25d. -> no 2 in R3C3+R4C23

26. HS in N2 at R3C5 = 1
26a. -> R4C6 = 2
26b. -> R6C6 = 1
26c. Cleanup: no 6 in R2C2

27. HS in R4 at R4C8 = 1
27a. -> R56C8 = [24]

28. HS in C6 at R5C6 = 4
28a. -> R5C7 = 5 (last digit in cage)

29. 12/3 at R5C2 = {138} (only 12/3 combo that doesn't conflict w/ {245} at R5C678)
29a. -> R5C234 = [813] (only possible permutation)

30. HS in C5 at R7C5 = 3
30a. Split 15/2 at R6C45 = [87|96]
30b. -> no 5 in R6C5

31. HS in C5 at R4C5 = 5

32. 28/5 at R2C4 contains a 2 (step 25), but 8 now unavailable
32a. -> 28/5 at R2C4 = {24679}
32b. 9 locked in R3C3+R4C23
32c. -> no 9 in R6C3 (CPE)

33. Hidden triple (HT) on {235} in N4 at R6C123
33a. -> R6C123 = {235}, locked for R6

34. 21/3 at R3C1 = {579}
35a. R3C1 = 5 (HS in cage)
35b. -> {79} locked in R45C1 for C1 and N4
35c. Cleanup: no 2 in R23C2, no 8 in R2C8

36. HS in R3 at R3C3 = 9

37. Outie N1: R1C4 = 5
37a. -> Split 14/2 at R12C3 = {68}
37b. -> 6 locked in R12C3 for C3 and N1

38. NS at R4C3 = 4
38a. -> R4C2 = 6
38b. Cleanup: R23C4 = {27}, 7 locked for C4 and N2

39. HS in N1 at R1C2 = 7
39a. -> R12C1 = {12}, locked for C1 and N1
39b. -> 7/2 at R23C2 = {34}, locked for C2

40. NS at R6C1 = 3

41. NP on {25} in 19/5 at R6C23
41a. -> no 2,5 in R7C3

42. NS at R7C3 = 7

43. Outie N7: R9C4 = 6
43a. -> Split 8/2 at R89C3 = {35}, locked for C3 and N7

44. NS at R6C3 = 2
44a. -> R6C2 = 5

45. 13/3 at R8C1 = {148} (only remaining combo, due to {237} being unavailable)
45a. -> R9C2 = 1 (HS)
45b. -> R89C1 = {48}, locked for C1/N7

46. NS at R7C1 = 6

Now switch attention to right-hand side of grid...

47. deleted

48. I/O diff. N9: R6C9 + R9C6 = R7C7 + 14
48a. -> max. R6C9 + R9C6 = 18
48b. -> max. R7C7 = 4
48c. -> no 8,9 in R7C7
48d. min. R7C7 = 1
48e. -> min. R6C9 + R9C6 = 15
48f. -> no 5 in R9C6

49. 8 in R7 locked in R7C89 -> not elsewhere in N9
49a. no 8 in R6C9

50. 22/3 at R8C7 = {679}
50a. -> R8C7 = 6 (HS)
50b. -> R9C67 = {79}, locked for R9

51. 18/3 at R8C9 must have 1 of {35} (due to R9C8), w/ {68} unavailable
51a. -> 18/3 at R8C9 = {459}
51b. -> R9C8 = 5, R89C9 = [94]

Next 2 moves just to get down to all naked singles...

52. 13/2 at R2C8 = {67}, locked for C8 and N3

53. HS in C9 at R2C9 = 5

As promised, rest is naked singles now

At last! It fell fairly quickly after I'd eventually made the first placements in step 35 of 38. Step 28 was a real marathon of combination analysis but it proved important.

Now I'll go see how Para and Mike did it.

1. Only initial candidate entry!
22(3) r8c7+r9c67 must have 9 -> r9c89 can see all cells of this 22(3) therefore <> 9

2. Innies c12: r456c2 = 19 (no 1)

3. Innies c89: r456c8 = 7 = {124} -> 13(2) r23c8 = {58/67}

4. Innies r5: r5c159 = 22 = {589/679}, 9 not elsewhere in r5.

5. Innies c1234: r46c4 = 17 = {89}, not elsewhere in c4/N5.
-> split 22(3) in r5c159: r5c19 <> 5

6. Innies c6789: r46c6 = 3 = {12}, not elsewhere in c6/N5 -> since r1c6 <> 1 or 2, r12c7 of 12(3) <> 9.

7. Outies – Innies N7: r6c1 + r9c4 – r7c3 = 2.

8. O-I N9: r6c9 + r9c6 – r7c7 = 14 (15-1, 16-2, 17-3, 18-4)
-> r7c7 = (1234); r6c9 + r9c6 = 15 {69/78} or 16 {79/88} or 17 {89} or 18 {99}
-> r6c9, r9c6 = (6789)

9. 11(3) r5c678: r5c6 (34567), r5c7 (35678), can’t make 7 from these candidates without repetition -> r5c8 <> 4
-> combination options: {137/146/236/245} -> r5c6 <> 5, r5c7 <> 8.

10. Innies r12: r2c2468 = 20.

11. Innies r89: r8c2468 = 13 (must have 1 which is thus locked to r8c24) not elsewhere in r8 -> r9c5 <> 9. Options: {1237/1246/1345}
a) If {1237} r8c24 = {12}, r8c68 = {37}
b) If {1246} r8c24 = {12}, r8c68 = [46]
c) If {1345} r8c2 = (1345), r8c4 = (1345), r8c6 = (345), r8c8 = (35)
-> r8c24 = (12345), r8c6 = (3457), r8c8 = (3567)

12. Innies N1: r3c1 + r123c3 = 28 = {4789/5689} -> 10(3) in N1 cannot have 4 since if split 28(4) = {5689}, 7(2) would be {34} -> options for 10(3): {127/136/235}

13. Innies N3: r3c9 + r123c7 = 16.

14. O-I N5: r5c46 – r37c5 = 3 -> r5c46 is max 13, min 7; r37c5 is max 10, min 4.

15. 17(3) r345c9: min from r45c9 = 9 -> r3c9 <> 9.

16. 16(3) in N3 can’t be {178/268/457} else no options for 13(2). Remaining options: {169/259/349/358/367}

17. Outies c1: r1789c2 = 19.

18. Outies c9: r1789c8 = 25 = {3589/3679}

19. 12(3) r5c234: options {138/147/156/237/246/345}. Analysis -> r5c3 <> 8

20. 19(4) r67c9 + r78c8: minimum from r6c9 + r78c8 = 14 -> r7c9 = max 5.
Options: {1369/1378/1567/2359/2368/3457} Analysis -> r7c9 <> 3,5

21. Pointing pair: r8c24 must have 1 -> r7c3 <> 1.

22. 17(3) r345c9: can’t make 10 from candidates in r45c9 without repetition -> r3c9 <> 7.

23. 18(3) in N9: max from r9c89 = 15 -> r8c9 <> 2.

24. 19(4) r6c456 + r7c5 must have at least one of (12) and (89) -> r6c46 = 9, 10 or 11
a) if r6c46 = [81] -> r67c5 = 10 = {37/46}
b) if r6c46 = [82/91] -> r67c5 = 9 = {27/36/45}
c) if r6c46 = [92] -> r67c5 = 8 = {17/35}
-> r7c5 <> 8, 9.

25. 23(5) r345c5 + r4c46 must have at least one of (12) and (89) in r4c46 -> r345c5 = 12, 13 or 14
a) if r4c46 = [81] -> r345c5 = 14 = {257/347/356}
b) if r4c46 = [82/91] -> r345c5 = 13 = {157/247/346}
c) if r4c46 = [92] -> r345c5 = 12 = {147/156/345}
-> r3c5 <> 7, 8, 9

26. Innies c5: r34567c5 = 22 = {13567/23467}-> 3, 6, 7 not elsewhere in c5 -> r89c5 <> 4
-> 7 locked to r456c5 -> r5c46 <> 7

27. 26(5) r6c78 + r7c67 + r8c6:
Can’t have both {12} as can’t make 23 {689} from remaining candidates
Max from r6c8 + r7c7 + r8c6 = 14 [437] leaving r6c7 + r7c6 = 12 -> Conflict as no options remaining
-> Max from r6c8 + r7c7 + r8c6 = 12 {435} leaving r6c7 + r7c6 = min 14 {59/68}
-> r6c7 <> 3, r7c6 <> 3,4

28. Analysis of options for 23(5) r345c5 + r4c46:
a) if r4c46 = [81], r345c5 = {257} -> r12c5 = {49}, r89c5 = {28}, Conflict – no options for r67c5 of 19(4)
b) if r4c46 = [81], r345c5 = {347} -> r12c5 = {58}, r89c5 = [91], r67c5 = [62] Conflict in 19(4)
c) if r4c46 = [81], r345c5 = {356} -> r12c5 = {49}, r89c5 = {28}, r67c5 = [71] OK
d) if r4c46 = [82], r345c5 = {157}-> r12c5 = {49}, r89c5 = {28}, r67c5 = {36} OK
e) if r4c46 = [82], r345c5 = {346} -> r12c5 = {58}, r89c5 = [91], r67c5 = [72] OK
f) if r4c46 = [91], r345c5 = {247} -> r12c5 = {58}, r89c5 = [91], r67c5 = {36} OK
g) if r4c46 = [91], r345c5 = {346} -> r12c5 = {58}, r89c5 = [91], r67c5 = [72] Conflict in 19(4)
h) if r4c46 = [92], r345c5 = {147} -> r12c5 = {58}, r89c5 = [91] -> Conflict as no options left for remaining 10(2) in r67c5.
i) if r4c46 = [92], r345c5 = {156} -> r12c5 = {49}, r89c5 = {28}-> r67c5 = [73] OK
j) if r4c46 = [92], r345c5 = 345 -> Conflict as no options left for 13(2) in r12c5.

Summary:
a) if r4c46 = [81], r345c5 = {356}, r67c5 = [71]
b) if r4c46 = [82], r345c5 = {157/346}, r67c5 = {36}/[72]
c) if r4c46 = [91], r345c5 = {247}, r67c5 = {36}
d) if r4c46 = [92], r345c5 = {156}, r67c5 = [73]

Conclusion: 13(2) in c5 must have at least one of (45), r345c5 of 23(5) must have at least one of (45) (but can’t have both) -> 45 not elsewhere in c5: r6c5 = (367), r7c5 = (1236)

29. 12(3) r5c234 = {138|147|237|246|345} – can’t be {156} else no options for split 22(3) in r5.

30. 11(3) r5c678 = {137|146|236|245}. Must have 3 or 4 -> 12(3) r5c234 can’t be {345}
-> eliminate 5 from r5c234
-> 5 locked to r45c5 in N5 -> eliminate 5 from r123c5 -> 13(2) = {49} not elsewhere in N2/c5
-> 10(2) in c5 = {28} not elsewhere in N8/c5
-> 4 locked to r5 in N5 -> r5c23 <> 4

31. 12(3) r5c234 = {138/147/237/246} Analysis -> r5c23 <> 3, r5c4 <> 6

32. Multi-colouring 1s: r3c5 <=> r7c5, r8c2 <=> r8c4
r7c5 and r8c4 both in N8 -> buddy of r3c5 and r8c2 <> 1
-> r3c2 <> 1 -> r2c2 <> 6
(I hope this is correct for nice loop notation:
[r3c5]=1=[r7c5]-1-[r8c4]=1=[r8c2]-1-[r3c2])

33. Multi-colouring 1s: r3c5 <=> r4c6 <=> r6c6 <=> r7c5; r8c2 <=> r8c4
r7c5 and r8c4 both in N8 -> r6c1 (buddy of r6c6 and r8c2) <> 1
-> 1 locked to r456c3 -> r9c3 <> 1
[r6c6]=1=[r7c5]-1-[r8c4]=1=[r8c2]-1-[r6c1]

34. Cage 23/5 r23c6 + r34c7 + r4c8 = {12389/12569/12578/13469/13478/13568/14567/23459/23468/23567}
For any option with 9, must be in r4c7 -> r3c7 <> 9 -> 16(3) in N3 must have 9
-> 16(3) <> 7,8
-> 9 locked to r3c13 -> r12c3 <> 9

35. 19(3) r1c34 + r2c3 = {478/568} (Must have 8)
-> 8 not elsewhere in c3
-> r1c4 <> 2, 3
-> 2 locked to r23c4 in N2/cage 28(5) -> r4c23 <> 2
-> Cage 28(5) must have 2 -> can’t also have 1 thus r23c4 <> 1, r4c3 <> 1
-> First placement!! HS r3c5 = 1
-> r4c6 = 2, r6c6 = 1, r4c8 = 1, r5c8 = 2, r6c8 = 4
-> r5c3 = 1

36. HS r5c6 = 4 -> r5c7 = 5, r5c4 = 3, r5c2 = 8, r4c5 = 5, r7c5 = 3

37. 21(3) r345c1 = {579} not elsewhere in c1.
-> r3c1 = 5 -> {79} not elsewhere in N4; r23c2 <> 2

38. O-I N1: r3c3 – r1c4 = 4 -> r3c3 = 9, r1c4 = 5
-> r12c3 = {68} -> eliminate 6 from elsewhere in N1/c3
-> 7(2) in N1 = {34} not elsewhere in c2, 10(3) in N1 = {127}
-> r4c2 = 6 …

Fairly straightforward from here.

Edit: I've added what I hope is correct nice loop notation for steps 32 and 33 where I used multi-colouring. Please let me know if it's not right.

Just managed to finish V1 before we move from Calgary to Lethbridge, about 2 hours drive south of Calgary. Will be packing up my computer as soon as I've posted this message and won't be on-line again for about a week so I don't know when I'll get to try Assassin 56.

When I first saw the comments in this thread, I was expecting this puzzle to be harder than it turned out to be. It was still a very long solution so, while it was a V1, it was definitely one of the harder ones. I hope that Ruud keeps up his excellent mixture of hard puzzles like this one and slightly easier ones.

I've worked through Para's and Mike's walkthroughs this evening but haven't had time for more than a glance at Cathy's walkthrough. Sorry Cathy! I'll look through it when my computer is in the new home.

[Edit. As promised I've now worked through Cathy's walkthrough, after our move, apart from the multi-colouring and nice loops which are techniques that I haven't yet learned.

Ed commented that Cathy and Mike were the only ones that found the narrow solution path. Not sure if I did.]

Here is my walkthrough. There is one heavy combination analysis step with a summary at the end for those who don't want to work through the details. Initially this step was even heavier but then I spotted the locked 5 in C5 which simplified the step and provided further eliminations.

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

2. R23C2 = {16/25/34}, no 7,8,9

3. R23C8 = {49/58/67}, no 1,2,3

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

5. 10(3) cage in N1 = {127/136/235} (cannot be {145} which clashes with R23C2), no 4,8,9

6. 19(3) cage at R1C3 = {289/379/469/478/568}, no 1

7. R345C1 = {489/579/678}, no 1,2,3

8. R5C678 = {128/137/146/236/245}, no 9

9. 22(3) cage at R8C7 = 9{58/67} -> no 9 in R9C89

10. 18(3) cage in N9 = {189/279/369/378/459/468/567}
10a. 9 only in R8C9 -> no 1,2 in R8C9

11. 19(5) cage at R6C2 = 1{2349/2358/2367/2457/3456}

12. 45 rule on R5 3 innies R5C159 = 22 = 9{58/67}, 9 locked for R5

13. 45 rule on N9 2 outies R6C9 + R9C6 – 14 = 1 innie R7C7
13a. Min R6C9 + R9C6 = 15 -> no 1,2,3,4,5 in R6C9 and R9C6
13b. Max R6C9 + R9C6 = 18 -> max R7C7 = 4

14. 45 rule on R89 4 innies R8C2468 = 13 = 1{237/246/345}, no 8,9, 1 locked for R8, clean-up: no 9 in R9C5

15. 45 rule on C12 3 innies R456C2 = 19 = {289/379/469/478/568}, no 1

16. 45 rule on C89 3 innies R456C8 = 7 = {124}, locked for C8 and N6, clean-up: no 9 in R23C8

17. 45 rule on C1234 2 innies R46C4 = 17 = {89}, locked for C4 and N5
17a. 19(3) cage at R1C3 (step 6) = {289/379/469/478/568}, 8,9 only in R12C3 -> no 2 in R12C3

18. 45 rule on C6789 2 innies R46C6 = 3 = {12}, locked for C6 and N5

19. 19(4) cage at R6C4 has R6C4 = {89}, R6C6 = {12}, valid combinations {1279/1369/1378/1459/1468/2359/2368/2458} -> no 8,9 in R7C5

20. R5C159 (step 12) = 9{58/67}
20a. 8,9 only in R5C19 -> no 5 in R5C19

21. R5C678 (step 8) = {137/146/236/245} (cannot be {128} because 1,2 only in R5C8), no 8
21a. 1,2 only in R5C8 -> no 4 in R5C8
21b. 4 only in R5C6 -> no 5 in R5C6

22. R5C234 = {138/147/237/246} (cannot be {156/345} which clash with all combinations in R5C678), no 5
22a. 1 only in R5C3 -> no 8 in R5C3

23. 5 in N5 locked in R456C5, locked for C5, clean-up: no 8 in R12C5

24. 45 rule on N1 2 innies R3C13 – 9 = 1 outie R1C4
24a. Min R1C4 = 2 -> min R3C13 = 11, no 1 in R3C3

25. 45 rule on C9 2 outies R19C8 – 6 = 2 innies R67C9
25a. Min R67C9 = 7 -> min R19C8 = 13 -> no 3 in R19C8

26. 3 in C8 locked in R78C8, locked for N9

27. 19(4) cage at R6C9 = 3{169/178/259/268/457}
27a. 1,2,4 only in R7C9 and each combination requires 1/2/4 -> R7C9 = {124}

28. 18(3) cage in N9 = {189/279/459/468/567}
28a. If 18(3) cage = {468/567} there must be a 9 in R89C7 (if R9C6 = 9 R89C7 = {58/67} which would clash with 18(3) cage) -> no 9 in R7C8

29. If R8C9 = 9, R6C9 <> 9
If R8C9 <>9, 9 in R89C7 (step 27a) -> R9C6 <>9
-> R6C9 + R9C6 <> 18 -> no 4 in R7C7 (step 13)

30. 4 in N9 locked in R789C9, locked for C9

31. 23(5) cage at R3C5 has R4C4 = {89}, R4C6 = {12}, consider the options
31a. R4C46 = [81] -> R345C5 = 14 = {257/347/356} (cannot be {149/158/167/248} which clash with R4C46, cannot be {239} because 2, 9 only in R3C5), also R6C46 = [92] -> R67C5 = 8 = [53/62/71]
31aa. R345C5 = {257} clashes with R67C5
31ab. R345C5 = {347} clashes with R12C5
31ac. R345C5 = {356} -> R67C5 = [71] -> R12C5 = {49} -> R89C5 = {28}
31b. R4C46 = [82] -> R345C5 = 13 = {157/346} (cannot be {148/238/247/256} which clash with R6C46, cannot be {139} because 1,9 only in R3C5), also R6C46 = [91] -> R67C5 = 9 = {36}/[54/72]
31ba. R345C5 = {157} -> R67C5 = {36} -> R12C5 = {49} -> R89C5 = {28}
31bb. R345C5 = {346} clashes with R12C5
31c. R4C46 = [91] -> R345C5 = 13 = {247/256/346} (cannot be {139/148/157} which clash with R6C46, cannot be {238} because 2,8 only in R3C5), also R6C46 = [82] -> R67C5 = 9 = {36}/[54]
31ca. R345C5 = {247} clashes with R12C5
31cb. R345C5 = {256} clashes with R67C5
31cc. R345C5 = {346} clashes with R67C5
31d. R4C46 = [92] -> R345C5 = 12 = {147/156/345}(cannot be {129/237/246} which clash with R6C46, cannot be {138} because 1,8 only in R3C5), also R6C46 = [81] -> R67C5 = 10 = {37/46}
31da. R345C5 = {147} clashes with R67C5
31db. R345C5 = {156} -> R67C5 = {37} -> R12C5 = {49} -> R89C5 = {28}
31dc. R345C5 = {345} clashes with R67C5

Summary R4C46 = [81/82/92] (no valid combinations for R345C5 with [91]) -> R6C46 = [81/91/92]
R345C5 = {156/157/356} -> no 2,4,8,9 in R3C5, no 4 in R4C5
For {157}, 1 only in R3C5 -> no 7 in R3C5
R67C5 = {36/37}/[71], no 2,4,5
R12C5 = {49}, no 6,7, naked pair {49} locked for C5 and N2
R89C5 = {28}, no 1,3,4,6,7,9, naked pair {28} locked for N8

32. 4 in N5 locked in R5C46, locked for R5

33. R5C234 (step 22) = {138/147/237/246}
33a. 4 only in R5C4 -> no 6 in R5C4

34. 12(3) cage at R1C6, min R1C6 = 3 -> max R12C7 = 9, no 9

35. 26(5) cage at R6C7, max R6C8 + R7C7 = 6 -> min R6C7 + R78C6 = 20 = {389/…}
35a. 8 only in R6C7 and only other 9 in R7C6 -> no 3 in R6C7 and R7C6
35b. Valid combinations for 26(5) cage with R6C8 = {124} and R7C7 = {12} {12689/14579/14678/23489/24569/24578}
35c. All combinations with 4 must have 4 in R6C8 -> no 4 in R78C6

36. R5C6 = 4 (hidden single in C6) -> R5C78 = 7 = [52/61] (step 21), no 3,7 in R5C7

37. Killer pair 5/6 in R5C159 and R5C7, locked for R5

38. R5C234 (step 22) = {138/237} = 3{18/27}

39. 3 in N6 locked in R4C79, locked for R4

40. R345C5 (step 31 summary) = {156/157/356}
40a. 1,3 only in R3C5 -> no 6 in R3C5

41. 16(3) cage in N3 = {169/259/358/367} (cannot be {178/268} which clash with R23C8)

42. R345C9 = {179/269/278/359/368}
42a. 1,2 only in R3C9 -> no 7 in R3C9
42b. 1,2 only in R3C9 and {359} requires 9 in R5C9 -> no 9 in R3C9

43. 23(5) at R2C6 has {124} = R4C8, valid combinations {12389/12569/12578/13469/13478/13568/14567/23459/23468/23567} (cannot be {12479} because 1,2,4 only in R3C7 and R4C8)
43a. All combinations with 9 require {12/14/24} which must be in R3C7 and R4C8 -> no 9 in R3C7

44. 9 in N3 locked in 16(3) cage = 9{16/25}, no 3,7,8

45. Killer pair 5/6 in 16(3) cage and R23C8, locked for N3

46. 3 in C9 locked in R34C9 -> R345C9 (step 42) = 3{59/68}, no 1,2 7
46a. 5 only in R4C9 -> no 9 in R4C9

47. R1C8 = 9 (hidden single in C8, not sure how long that has been there) -> R12C5 = [49], R12C9 = {16/25}

48. Killer pair 5/6 in R12C9 and R345C9, locked for C9

49. R7C7 = {12}, R7C9 = {124} -> 1/2/4 in 18(3) cage (step 28) = {189/279/459/468} (cannot be {567})
49a. 2 only in R9C9 and 9 only in R8C9 -> no 7 in R89C9

50. R6C9 = 7 (hidden single in C9)

51. R67C5 (step 31 summary) = {36/37} (cannot now be [71]) -> no 1 in R7C5

52. R3C5 = 1 (hidden single in C5) -> R46C6 = [21], clean-up: no 6 in R2C2

53. 19(4) cage at R6C9 (step 27) = 37{18/45}, no 2,6

54. 22(3) cage at R8C7 = {679} (cannot be {589} which must have 5,8 in R78C7 and would then clash with the part of the 19(4)cage that is in N9), no 5,8 -> no 6,7 in R9C8
54a. 6,7 in N9 locked in R89C7 = {67}, locked for C7 -> R9C6 = 9 -> R8C9 = 9 (hidden single in N9), R5C7 = 5, R5C8 = 2, R6C8 = 4, R4C8 = 1

55. Naked triple {358} in R789C8, locked for C8 and N9 -> R23C8 = {67}, locked for N3, clean-up: no 1 in R12C9 (step 47) = {25}, locked for C9 and N3

56. R7C7 = 2 (hidden single in C7)

57. 1 in C7 locked in R12C7 -> 12(3) cage at R1C6 = 1{38/47} (cannot be {156} because 5,6 only in R1C6), no 5,6

58. 26(5) cage at R7C7 = {24569/24578} (cannot be {23489} because 8,9 only in R7C7) = 245{69/78}, no 3
58a. 5 locked in R78C6, locked for C6 and N8

59. 3 in C6 locked in R123C6, locked for N2

60. R5C3 = 1 (hidden single in R5, saw that a long time ago but forgot about it) -> R5C2 = 8, R5C4 = 3, R5C159 = [976], R67C5 = [63], R4C5 = 5, R46C4 = [89] (cage sums), R34C9 = [83], R4C7 = 9, R6C7 = 8, R78C6 = {57} (step 58), locked for C6 and N8

61. R8C8 = 3 (hidden single in C8)

62. 23(5) cage at R2C6 = {13469} (only remaining combination), no 8 -> R3C7 = 4, R23C6 = {36}, locked for C6 and N2 -> R1C6 = 8, clean-up: no 3 in R2C2

63. 6 in R1 locked in R1C123, locked for N1, clean-up: no 1 in R2C2
[Alternatively X-wing in 6 on R23C68]

64. 1 in N1 locked in 10(3) cage = 1{27/36}, no 5

65. R5C1 = 9 -> R34C1 = [57/84]

66. 6 in R4 locked in R4C23 in 28(5) cage

67. R3C13 – 9 = R1C4 (step 24), R1C4 = {257} -> R3C13 = 11, 14 or 16
67a. If R3C13 = 11 -> [83]
67b. If R3C13 = 14 -> [59]
67c. R3C13 cannot total 16
-> R3C3 = {39}, R1C4 = {25}

68. 7 in C4 locked in R23C4 -> no 7 in R4C23 -> R4C1 = 7 (hidden single in R4), R3C1 = 5, R3C3 = 9 (hidden single in R3), R1C4 = 5 (step 67b), R12C9 = [25], clean-up: no 2 in R23C2 = [43], R4C23 = [64]

69. 10(3) cage in N1 = {127} (only remaining combination) -> R1C2 = 7, R12C1 = [12], R12C3 = [68], R12C7 = [31], R23C4 = [72], R23C8 = [67], R23C6 = [36], R6C1 = 3

70. R7C2 = 9 (hidden single in C2)

71. R6C1 + R7C2 = [39] -> R7C1 + R8C2 = 8 = [62], R6C23 = [52], R9C2 = 1, R9C9 = 4, R9C8 = 5 (cage sum), R7C89 = [81], R7C4 = 4, R89C5 = [82], R89C1 = [48], R89C4 = [16], R89C7 = [67], R9C3 = 3, R8C3 = 5 (cage sum), R7C3 = 7, R78C6 = [57]

and the rest is naked singles, naked pairs and cage sums in N2

I haven't had enough time to check it properly. If you find any typos or other errors, please tell me by PM and I'll make the necessary corrections. I'm sure there will be a number of things that I ought to have seen earlier so no need to point those out to me.
[Edit. A few minor corrections have been made. I didn't think they were significant enough to colour code them.]

3x3::k:5120:5120:2818:2818:1540:3333:3333:5895:5895:5120:2058:2818:6156:1540:6926:3333:1040:5895:3858:2058:6156:6156:8982:6926:6926:1040:4378:3858:6156:6156:8982:8982:8982:6926:6926:4378:3858:2341:2341:2341:8982:3369:3369:3369:4378:4397:7982:7982:4400:4400:4400:6963:6963:5429:4397:4397:7982:7982:4400:6963:6963:5429:5429:4159:4397:4161:7982:2883:6963:3141:5429:3143:4159:4159:4161:4161:2883:3141:3141:3143:3143:

This ran as a “tag” solution, by Ed, Cathy and Mike, on the forum.

I never tried it at the time, we were getting ready to move as I commented when I posted by walkthrough for the V1, so I’ve now decided to have a go at it years later. I’m pleased that I did; I enjoyed this puzzle, even though it was a very tough one.

Since it was originally solved as a “tag” I didn’t hold back from using forcing chains but I tried to avoid using contradiction moves for as long as possible; originally I used four contradiction moves but I've since managed to re-write three of those steps using forcing chains. Edit. I've added an alternative step 35 which replaces the final contradiction move with a longer forcing chain.

Here is my walkthrough for A55 V2.

Prelims

a) R12C5 = {15/24}
b) R23C2 = {17/26/35}, no 4,8,9
c) R23C8 = {13}
d) R89C5 = {29/38/47/56}, no 1
e) 20(3) cage in N1 = {389/479/569/578}, no 1,2
f) 11(3) cage at R1C3 = {128/137/146/236/245}, no 9
g) 23(3) cage in N3 = {689}
h) 9(3) cage at R5C2 = {126/135/234}, no 7,8,9
i) 35(5) cage at R3C5 = {56789}

Steps resulting from Prelims
1a. Naked pair {13} in R23C8, locked for C8 and N3
1b. Naked triple {689} in 23(3) cage, locked for N3
1c. Naked quint {56789} in 35(5) cage at R3C5, CPE no 5,6,7,8,9 in R6C5
[The 35(5) cage looks potentially useful for ALS blocks.]

2. 45 rule on R5 3 innies R5C159 = 23 = {689}, locked for R5
2a. Naked triple {689} in R125C9, locked for C9
2b. Min R5C1 = 6 -> max R34C1 = 9, no 9 in R34C1
2c. 23(3) cage in N3 = {689}, R125C9 = {689} -> R1C8 = R5C9
[Added because it may be useful for analysis or clean-up.]

3. 9(3) cage at R5C2 = {135/234}, 3 locked for R5
3a. 13(3) cage at R5C6 = {157/247}

4. 45 rule on C9 4 outies R1789C8 = 28 = {4789/5689}, no 2, 8,9 locked for C8

5. 2 in C8 only in R456C8, locked for N6
5a. 45 rule on C89 3 innies R456C8 = 13 = {247/256}
5b. 4 of {247} must be in R46C8 (R5C8 cannot be 4 because R456C8 = {247} clashes with 13(3) cage at R5C6 = [274], CCC) -> no 4 in R5C8
5c. 13(3) cage at R5C6 (step 3a) = {157/247}
5d. 2 of {247} must be in R5C8 (cannot be [247] which clashes with R456C8 = {247}, CCC) -> no 2 in R5C6

6. 13(3) cage at R1C6 = {157/247/256} (cannot be {139/148/238/346} because 1,3,6,8,9 only in R1C6), no 3,8,9
6a. 1,6 of {157/256} must be in R1C6 -> no 5 in R1C6

7. 45 rule on C1234 2 innies R46C4 = 13 = {58/67}/[94], no 1,2,3,9 in R6C4

8. 45 rule on C6789 2 innies R46C6 = 11 = {56}/[74/83/92], no 1,7,8,9 in R6C6
8a. 9 in N5 only in R4C456 + R5C5, locked for 35(5) cage at R3C5 -> no 9 in R3C5

9. 45 rule on N5 2 outies R37C5 = 2 innies R5C46 + 7
9a. Min R5C46 = 3 -> min R37C5 = 10, no 1 in R7C5

10. 20(3) cage in N1 = {389/479/569/578}
10a. 3 of {389} must be in R1C12 (R12 cannot be {89} which clashes with 23(3) cage in N3, ALS block) -> no 3 in R2C1

11. 20(3) cage in N1 = {389/479/569/578}
11aa. 20(3) cage = {389/479/569} => caged X-wing for 9 in 20(3) cage in N1 and 23(3) cage in N3 for R12, no other 9 in R12
11ab. 20(3) cage = {578} => R3C3 = 9 (hidden single in N1) => no 9 in R2C4
11b. -> no 9 in R2C4
[I missed the simpler 9 in R3 only in R3C346, CPE no 9 in R2C4.]

12. R46C4 (step 7) = {58/67}/[94], R46C6 (step 8) = {56}/[74/83/92]
12a. Consider placements for R5C5
12aa. R5C5 = 6 => R46C6 cannot be {56}
12ab. R5C8 = 8 => no 3 in R89C5, R46C4 = {67}/[94]
12abi. R46C4 = {67} => R46C6 cannot be {56} which clashes with R46C4
12abii. R46C4 = [94], 3 in C5 only in R67C5 => 17(4) cage at R6C4 = {1349/2348} => no 5,6 in R6C6
12ac. R5C5 = 9 => R46C4 = {58/67} => R46C6 cannot be {56} which clashes with R46C4
12b. -> no 5,6 in R6C6, clean-up: no 5,6 in R4C6

13. R89C5 = {29/38/47} (cannot be {56} which clashes with 35(5) cage at R3C5, ALS block), no 5,6

14. 12(3) cage at R8C9 = {129/138/237/246/345} (cannot be {147/156} which clash with R1789C8, CCC because no 1 in R9C8)
14a. 7 of {237} must be in R9C8 -> no 7 in R89C9
[At one stage I eliminated {345} here, mistakenly thinking that it clashed with R1789C8, so I’ve had to do a lot of re-work from here.]

15. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 1 innie R7C7
15a. Min R6C9 + R9C6 = 2 -> min R7C7 = 2
15b. Max R7C7 = 9 -> max R6C9 + R9C6 = 9, no 9 in R9C6

16. 17(3) cage at R3C9 = {179/278/359/458/467} (cannot be {269/368} because 6,8,9 only in R5C9)
[If I’d spotted step 25 here, some of the following steps might have been simplified.]

17. 21(4) cage at R6C9 = {1389/1479/1569/1578/2469/2478/2568/3468/3567} (cannot be {2379/3459} which clash with 12(3) cage at R8C9)
17a. Hidden killer pair 5,7 in 17(3) cage at R3C9, R67C9 and R89C9 for C9, 17(3) cage at R3C9 contains one of 5,7 -> R67C9 must contain one of 5,7 or R89C9 must contain 5
17b. From the previous sub-step, if R67C9 doesn’t contain one of 5,7 then 5 must be in R89C9 -> 12(3) cage at R8C9 (step 14) = [345/543] which can block combinations for 21(4) cage at R6C9 not containing 5 or 7 in R67C9
17c. 21(4) cage at R6C9 = {1479/1569/1578/2469/2478/2568/3567} (cannot be {1389/3468} which clash with 12(3) cage at R8C9 = [345/543])
17d. 3,7 of {3567} must be in R67C9 (R67C9 cannot be {35} because R78C8 = {67} clashes with combinations for R1789C8) => R78C8 = {56}, locked for C8 => R456C8 (step 5a) = {247}, locked for N6 => 3 of {3567} must be in R6C9 -> no 3 in R7C9

18. R1789C8 (step 4) = {4789/5689}
18a. Consider the combinations for R1789C8
18aa. R1789C8 = {4789} => 21(4) cage at R6C9 cannot be {2469}
18ab. R1789C8 = {5689} => 5 in R78C8 or in R9C8
18abi. 5 in R78C8 => 21(4) cage at R6C9 cannot be {2469}
18abii. 5 in R9C8 => 12(3) cage at R8C9 = [354/453], 4 locked for C9 => 21(4) cage at R6C9 cannot be {2469}
18b. -> 21(4) cage at R6C9 cannot be {2469}
-> 21(4) cage at R6C9 (step 17c) = {1479/1569/1578/2478/2568/3567}

19. R1789C8 (step 4) = {4789/5689}
19a. Consider the combinations for R1789C8
19aa. R1789C8 = {4789}, locked for N9
19ab. R1789C8 = {5689} => R78C8 must contain at least one of 5,6 (cannot be {89} because no remaining combination for 21(4) cage at R6C9 contains both of 8,9) => no 4 in R67C9 (because no remaining combination for 21(4) cage at R6C9 contains 4 and one of 5,6)
19b. -> no 4 in R7C9

20. R1789C8 (step 4) = {4789/5689}
20a. 4 or 7 of {4789} must be in R9C8 (R78C8 cannot be {47} because 21(4) cage at R6C9 doesn’t contain 8 or 9 in C9) => 12(3) cage at R8C9 = {237/345}, 3 and either 2 or 5 locked for C9
[This might be useful later.]

[I’d been concentrating so much on C89, particularly before the error and re-work, that I’ve only just thought of looking at C5 again.]

21. R12C5 = {15/24}
21a. Consider the combinations for R12C5
21aa. R12C5 = {15} => hidden killer triple 2,3,4 in R6789C5, R6C5 = {234}, R89C5 must contain one of 2,3,4 => R7C5 = {234}, R6C56 + R7C5 = {234} = 9 => R6C4 = 8
21ab. R12C4 = {24}, locked for C5 => R89C5 = {38}, locked for C5, R7C5 = 1 (hidden single in C5), 8 in 35(5) cage at R3C5 only in R4C46, locked for N5, no 8 in R6C4 => no 5 in R4C4 => 5 in 35(5) cage at R3C5 only in R34C5
21b. From steps 21aa and 21ab
21ba. 8 must be in R6C4 or R89C5, CPE no 8 in R457C5
21bb. 5 must be in R12C5 or R34C5, locked for C5
21c. 17(4) cage at R6C4 must be 8{234} or contain 1 at R7C5 -> 17(4) cage = {1259/1349/1367/1457/2348} (cannot be {1358} because 5,8 only in R6C4, cannot be {1268} because 8 must be in R4C46 and also in R89C5 when R7C5 = 1)

[I hope the next step is acceptable as a hidden killer. I first saw it as a contradiction move and then tried to find a way to write it in a more acceptable way.]
22. Hidden killer pair 2,4 in R5C46 and R6C456 for N5, R5C46 cannot contain both of 2,4 (because 2,4 in R5 must both be in 9(3) cage at R5C2 or both in 13(3) cage at R5C6) -> R6C456 must contain at least one of 2,4
22a. 17(4) cage at R6C4 (step 21c) = {1259/1349/1457/2348} (cannot be {1367} which doesn’t contain either of 2,4), no 6, clean-up: no 7 in R4C4 (step 7)
22b. 6 in C5 only in R345C5, locked for 35(5) cage at R3C5, no 6 in R4C4, clean-up: no 7 in R6C4 (step 7)
22c. 6 in N5 only in R45C5, locked for C5

23. Hidden killer pair 2,3 in R5C4 and R6C456, R5C4 cannot contain more than one of 2,3 -> R6C456 must contain at least one of 2,3
23a. 17(4) cage at R6C4 (step 22a) = {1259/1349/2348} (cannot be {1457} which doesn’t contain either of 2,3), no 7
23b. Consider the combinations for the 17(4) cage
23ba. 17(4) cage = {1259} => R6C4 = 5
23bb. 17(4) cage = {1349} => R5C4 = 2 (hidden single in N5)
23bc. 17(4) cage = {2348} = 8{234} => R4C4 = 5 (step 7)
23c. -> no 5 in R5C4

24. 13(3) cage at R5C6 (step 3a) = {157/247}, R456C8 (step 5a) = {247/256}
24a. Consider the combinations for the 13(3) cage
24aa. 13(3) cage = {157} => R5C8 = {57}
24aai. R5C8 = 5 => R456C8 = {256} => no 7 in R46C8
24aaii. R5C8 = 7 => no 7 in R46C8
24ab. 13(3) cage = {247} => R456C8 = {256} (cannot be {247} which clashes with 13(3) cage, CCC) => no 7 in R46C8
24b. -> no 7 in R46C8

25. R456C8 (step 5a) = {247/256}
25a. Consider the combinations for R456C8
25aa. R456C8 = {247}, locked for N6 => no 4,7 in R4C9
25ab. R456C8 = {256}, locked for N6 => no 6 in R5C9
25b. -> 17(3) cage at R3C9 (step 16) = {179/278/359/458} (cannot be {467}), no 6, clean-up: no 6 in R1C8 (step 2c)
[I saw this as a short forcing chain but it can probably be considered to be a combo blocker.]

26. R1789C8 (step 4) = {4789/5689}
26a. Consider the combinations for R1789C8
26aa. R1789C8 = {4789} => R9C8 = {47} (step 20a)
26ab. R1789C8 = {5689} => R9C8 = {56} or R78C8 = {56} = 11 => R67C9 = 10 = [37] => 12(3) cage at R8C9 cannot be [183/381]
26b. -> no 8 in R9C8

27. R1789C8 (step 4) = {4789/5689}
27a. Consider the combinations for R1789C8
27aa. R1789C8 = {4789} => R9C8 = {47} (step 20a)
27aai. R78C8 = {48/49} => R67C9 = {17/27} (from combinations for 21(4) cage at R6C9, step 18b), no 5 in R67C9
27aaii. R9C8 = 4 => R89C9 = {35} (step 14), 5 locked for C9
27ab. R1789C8 = {5689}, 5 locked for N9
27b. -> no 5 in R7C9

28. 13(3) cage at R5C6 (step 3a) = {157/247}
28a. Consider combinations for the 13(3) cage
28aa. 13(3) cage = {157} => R5C8 = {57}, R456C8 (step 5a) = {247/256}
28aai. R456C8 = {247} => R46C7 + R5C9 = {689} (hidden triple in N5)
28aaii. R456C8 = {256}, locked for C8 => R1789C8 = {4789} => R9C8 = {47} and 3 locked for C9 (step 20a) => R46C7 + R5C9 = {389} (hidden triple in N5)
28ab. 13(3) cage = {247} => R456C8 (step 5a) = {256} (cannot be {247} which clashes with 13(3) cage, CCC), locked for C8 => R1789C8 = {4789} => R9C8 = {47} and 3 locked for C9 (step 20a) => R46C7 + R5C9 = {389} (hidden triple in N5)
28b. -> R46C7 + R5C9 = {389/689}, no 1,4,5,7 in R46C7

[I originally did the next three steps as contradiction moves, two of which ended with no possible combination for the 17(3) cage at R3C9 and the other one with an ALS block involving the 17(4) cage at R6C4. I’ve now re-written all three steps as forcing chains based on these 17(3) and 17(4) cages.]

29. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
29a. Consider combinations for the 17(3) cage
29aa. 17(3) cage = {179} = [719], naked triple {245} in R123C7, locked for C7 => R5C7 = 7, R456C8 (step 5a) = {256} => R6C9 = 4 (hidden single in N6)
29ab. 17(3) cage = {278/359/458}, 2 or 5 locked for C9 => R67C9 cannot be [52]
29b. -> R67C9 cannot be [52] -> 21(4) cage at R6C9 (step 18b) = {1479/1569/1578/2478/3567} (cannot be {2568} = [52]{68})

30. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
30a. Consider combinations for the 17(3) cage
30aa. 17(3) cage = {179/359/458}, 1 or 5 locked for C9 => R67C9 cannot be [51]
30ab. 17(3) cage = {278} = [278] => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
30b. -> R67C9 cannot be [51] or no 6 in R78C8 -> 21(4) cage at R6C9 (step 29b) = {1479/1578/2478/3567} (cannot be {1569} = [51]{69})

31. 17(4) cage at R6C4 (step 23a) = {1259/1349/2348}
31a. Consider combinations for the 17(4) cage
31aa. 17(4) cage = {1259} = [5129] => R5C4 = 3 (hidden single in N4) => R5C6 = 4 (hidden single in N4) => 13(3) cage at R5C4 (step 3a) = {247} = [472] => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
31ab. 17(4) cage = {1349} => R7C5 = 9, 3 locked for R6 => no 3 in R6C9
31ac. 17(4) cage = {2348} can have 3 in R6C56 or R7C5
31aci. 3 in R6C56, locked for R6 => no 3 in R6C9
31acii. 3 in R7C5 => R6C56 = {24}, locked for R6 => R6C8 = {56} => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
31b. -> no 3 in R6C9 or no 6 in R78C8 -> 21(4) cage at R6C9 (step 30b) = {1479/1578/2478} (cannot be {3567} = [37]{56}), no 3,6

32. R1789C8 (step 4) = {4789/5689}
32a. 4 or 7 of {4789} must be in R9C8 (step 20a), 6 of {5689} only in R9C8 -> R9C8 = {467}, no 5,9
32b. 12(3) cage at R8C9 (step 14) = {237/246/345}, no 1

33. R1789C8 (step 4) = {4789/5689}
33a. Consider the combinations for R1789C8
33aa. R1789C8 = {4789}, locked for C8 => R456C8 (step 5a) = {256}, locked for N6, no 5 in R6C9
33ab. R1789C8 = {5689} => 5 must be in R78C8 => 5 in 21(4) cage at R6C9 = {1578} (step 31b) must be in R78C8 => no 5 in R6C9
33b. -> no 5 in R6C9
[Alternatively there’s a short contradiction move starting with R6C9 = 5 ...]

34. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
34a. Consider combinations of 17(3) cage which contain 5
34aa. 17(3) cage = {359} = [539]
34ab. 17(3) cage = {458}, locked for C9 => R89C9 = {23}, R9C8 = 7 (step 32b) => R456C8 (step 5a) = {256}, locked for N6
34b. -> no 5 in R4C9
34c. 4 of {458} must be in R4C9 -> no 4 in R3C9
34d. 4 in N3 only in R123C3, locked for C3

[I’ve managed to replace all the contradiction moves except for this one, which can probably be considered to be “brute force” although the clash at the end of the chain is an interesting one. I tried looking at forcing chains starting from 17(3) cage at R3C9, 17(4) cage at R6C4 and from R46C7 + R5C9 = {389/689}, step 28b, but couldn’t make any of them work except possibly as contradictions. After further thought I managed to come up with alternative step 35, which is quite a lot longer but is a forcing chain rather than a contradiction move.]

35. 13(3) cage at R5C6 (step 3a) = {157/247} cannot be {247}, here’s how
35a. 13(3) cage = {247} => R5C8 = 2, R5C6 = 4, R5C7 = 7, R3C9 = 7 (hidden single in N3) => 17(3) cage at R3C9 (step 25b) = {179} => R4C9 = 1, R5C9 = 9, R46C7 = {38} (hidden pair in N6), 17(4) cage at R6C4 (step 23a) = {1259} (cannot be {2348} = 8{23}4 clashes with R6C7) => R6C456 = [512], R4C4 = 8 (step 7), R5C4 = 3 (hidden single in N5), R23C6 = {38} (hidden pair in N2) clashes with R4C7
35b. -> 13(3) cage at R5C6 = {157}, locked for R5
35c. 9(3) cage at R5C2 = {234}

[Alternative step 35.
35. 13(3) cage at R5C6 (step 3a) = {157/247}, R456C8 (step 5a) = {247/256}
35a. Consider the combinations for 17(4) cage at R6C4 (step 23a) = {1259/1349/2348}
35aa. 17(4) cage = {1259} = [5129], R4C4 = 8 (step 7), R5C4 = 3 (hidden single in N5) => R23C6 = {38} (hidden pair in N2), locked for 27(5) cage at R2C6 => R4C7 = {69}
35aai. R4C7 = 6 => R456C8 = {247} => 13(3) cage at R5C6 = {157} (cannot be {247}, CCC)
35aaii. R4C7 = 9 => R5C9 = 8 => 17(3) cage at R3C9 (step 25b) = {278/458} => R3C9 = {25} => 7 in N3 only in R123C7, locked for C7 => R5C7 = {15} => 13(3) cage at R5C6 = {157} (only remaining combination)
35ab. 17(4) cage = {1349}, 4 locked for N5 => R5C6 = {157} => 13(3) cage at R5C6 = {157}
35ac. 17(4) cage = {2348} = 8{23}4/8{24}3/8{34}2
35aca. 17(4) cage = 8{23}4, 3,8 locked for R6 => R6C7 = {69}
35acai. R6C7 = 6 => R456C8 = {247} => 13(3) cage at R5C6 = {157} (cannot be {247}, CCC)
35acaii. R6C7 = 9 => R5C9 = 8 => 17(3) cage at R3C9 (step 25b) = {278/458} => R3C9 = {25} => 7 in N3 only in R123C7, locked for C7 => R5C7 = {15} => 13(3) cage at R5C6 = {157} (only remaining combination)
35acb. 17(4) cage = 8{24}3/8{34}2, 4 locked for N5 => R5C6 = {157} => 13(3) cage at R5C6 = {157}
35b. -> 13(3) cage at R5C6 = {157}, locked for R5
35c. 9(3) cage at R5C2 = {234}]

36. R456C8 (step 5a) = {247/256}
36a. R5C8 = {57} -> no 5 in R46C8

[Now for a “two-pronged attack” which I found interesting.]
37. 31(5) cage at R6C2 must contain 9
37a. First consider if 12(3) cage at R8C7 = {129}, 9 locked for C7 and N9 => R5C9 = 9 (hidden single in N6) => 17(3) cage at R3C9 (step 25b) = {179/359} => R3C9 = {57}, 2 in N3 only in R123C7, locked for C7 => R89C7 = {19}, locked for C7 => R5C6 = 1 (hidden single in R5) => 17(4) cage at R6C4 (step 23a) = {2348} => no 9 in R7C5
37b. Now consider placements for 9 in 31(5) cage
37ba. 9 in 31(5) cage in R6C23 => 9 in R5 only in R5C59
37bai. R5C5 = 9 => no 9 in R7C5
37baii. R5C9 = 9 => 9 in C7 only in R789C7 => either R7C7 = 9 => no 9 in R7C5 or 9 in 12(3) cage at R8C7 => no 9 in R7C5 (step 37a)
37bb. 9 in 31(5) cage in R7C3 + R78C4 => no 9 in R7C5
37c. -> no 9 in R7C5
[Looks like the puzzle may now be cracked.]

38. 17(4) cage at R6C4 (step 23a) = {2348} (only remaining combination), no 1 -> R6C4 = 8, R4C4 = 5 (step 7), R3C5 = 8 (hidden single in 35(5) cage at R3C5), clean-up: no 3 in R6C6 (step 8), no 3 in R89C5

39. R12C5 = {15} (hidden pair in C5), locked for N2

40. R5C6 = 1 (hidden single in N5)
40a. Naked pair {57} in R5C78, locked for N6
40b. 1 in C7 only in R89C7, locked for N9

41. 17(3) cage at R3C9 (step 25b) = {179/359/458} (cannot be {278} because 2,7 only in R3C9), no 2
41a. 2 in C9 only in R789C9, locked for N9

42. 21(4) cage at R6C9 (step 31b) = {1479/1578/2478}, 7 locked for N9
42a. 12(3) cage at R8C9 (step 32b) = {246/345}, 4 locked for N9
42b. 17(3) cage at R3C9 (step 41) = {179/359} (cannot be {458} which clashes with 12(3) cage at R8C9, ALS block because 12(3) cage must contain 4 or 5 in R89C9) => R5C9 = 9, R5C5 = 6, R5C1 = 8, R1C8 = 9 (step 2c), R4C9 = {13}
42c. Naked pair {79} in R4C56, locked for R4
42d. R4C7 = 8 (hidden single in R4)

43. Naked quad {2346} in R6C5678, locked for R6 -> R6C9 = 1, R4C9 = 3, R3C9 = 5 (step 42b), R6C7 = 6, clean-up: no 3 in R2C2

44. Naked pair {24} in R89C9, locked for N9 => R7C9 = 7, R9C8 = 6

45. Naked pair {58} in R78C8, locked for C8 and N9 -> R5C8 = 7, R5C7 = 5

46. 1 in N9 only in 12(3) cage at R8C7 = {129/138} (cannot be {147} because 4,7 only in R9C6), no 4,5,7
46a. 2,8 only in R9C6 -> R9C6 = {28}

47. R6C5 = 3 (hidden single in R6)
47a. Killer pair 2,4 in R7C5 and R89C5, locked for N8 -> R9C6 = 8 -> 12(3) cage at R8C7 (step 46) = {138}, no 9

49. R7C7 = 9 (hidden single in C7)
49a. 5 in N8 only in R78C6 -> 27(5) cage at R6C7 = 69{345} (only remaining combination) -> R6C8 = 4, R78C6 = {35}, locked for C6 and N8, R4C8 = 2, R6C6 = 2, R7C5 = 4, clean-up: no 7 in R89C5

50. Naked pair {29} in R89C5, locked for C5 and N8 -> R4C56 = [79]
50a. Naked triple {467} in R123C6, locked for N2

51. Naked pair {23} in R12C4, locked for C4, CPE no 2,3 in R2C3
51a. R3C4 = 9, R5C4 = 4

52. R2C1 = 9 (hidden single in N1)
52a. R1C12 = 11 = [38]/{47/56}, no 3 in R1C2

53. R5C1 = 8 -> R34C1 = 7 = [16/34/61], no 2,4,7 in R3C1

54. 6 in N8 only in R78C4 -> 31(5) cage at R6C2 = {16789} (only remaining combination, cannot be {35689} because 3,8 only in R7C3) -> R7C3 = 8, R6C23 = {79}, R78C4 = {16}, R78C8 = [58], R78C6 = [35]
54a. R6C1 = 5 (hidden single in R6), clean-up: no 6 in R1C2 (step 53)

55. R9C4 = 7 (hidden single in C4) -> R89C3 = 9 = [45/63], no 1,2,9, no 3 in R8C3, no 4 in R9C3

56. R6C3 = 9 (hidden single in C3), R6C2 = 7, clean-up: no 4 in R1C1 (step 52a), no 1 in R23C2
56a. Killer pair 2,3 in R23C2 and R5C2, locked for C2

57. R8C1 = 7 (hidden single in R8) -> R9C12 = 9 = [45], clean-up: no 6 in R1C1, no 4 in R1C2 (both step 52a), no 3 in R3C1 (step 53)

58. R1C12 = [38], R1C4 = 2, R12C3 = 9 = {45}, locked for C3

and the rest is naked singles.


Rating Comment. I won't try to decide whether my solving path was harder or easier than that used in the "tag" solution (steps 1 to 38 and 55 onward); I'll leave that to anyone who may decide to work through both of them. However I will say that they should probably both be in the same rating range.

3x3::k:2816:5889:5889:1539:1539:1539:4870:4870:2824:2816:3082:5889:3084:4621:4366:4870:3600:2824:2816:3082:3084:3084:4621:4366:4366:3600:2824:5659:3082:3869:3869:4621:4640:4640:3600:3619:5659:5659:3869:6183:3880:3369:4640:3619:3619:3117:3117:3117:6183:3880:3369:4915:4915:4915:3126:3126:6183:6183:3880:3369:3369:2877:2877:3126:3648:3393:6722:6722:6722:2629:6214:2877:3648:3648:3393:3393:6722:2629:2629:6214:6214:

Hi all

I think this walk-through typically shows how i tackle solving killers. Going through the basics of initial cage-combos and 45-test till i hit some conflicting combinations and run with it (and clean up properly), skipping the remaining 45-tests still in the grid. That is mostly the reason i sometimes don't use 45-tests to solve killers, just because there's something more interesting to work with (i like working cage combo's over 45-tests).

[edit]
Ok counted wrong on a 45-test. See that is why i rather not do 45 test. Feeling like a rookie again. I have to recheck my steps. Thanks Glyn.
It would be soooooo much handier when you make a mistake the puzzle actually turns up wrong instead off ending up with the proper solution.

[edit]
Here is the redraft with a little trick to replace the busted 45-test. This trick's pretty easy to see but even easier to overlook.

Walkthrough assassin 56

1. 23(3) in R1C2 = {689} -->> locked for N1

2. R1C456 = {123} -->> locked for R1 and N2

3. 19(3) in R1C7 and R6C7 = {289/379/469/478/568}: no 1

4. 22(3) in R4C1 = {589/679}: no 1,2,3,4; 9 locked for N4

5. 13(4) in R5C6 = {1237/1246/1345}: no 8,9

6. 26(4) in R8C4 = {2789/3689/4589/4679/5678}: no 1

7. 24(3) in R8C8 = {789} -->> locked for N9

8. 45 on R89: 2 innies: R8C19 = 3 = {12} -->> locked for R8

9. 45 on R123: 3 outies: R4C258 = 8 = {125/134}: no 6,7,8,9; 1 locked for R4

10. 45 on C12: 3 outies: R126C3 = 16 = {169/268}(only possible combinations) -->> R6C3 = {12}; R12C3 = {68/69} -->> 6 locked in R12C3 for C3

11. 45 on C1234: 2 innies: R18C4 = 10 = [19/28/37]: R8C4 = {789}

12. 45 on C89: 3 outies: R126C7 = 22 = {589/679}: no 2,3,4; 9 locked for C7
12a. 45 on C89: 1 innie and 1 outie: R6C7 = R1C8 + 3 -->> R6C7 = {789}; R1C8 = {456}
12b. R12C7 = {58/59/67/69} -->> 19(3) in R1C7 = {469/568}: no {478} clashes wih R12C7 -->> R12C7 = {58/69}: no 7; R1C8 = {46} -->> R6C7: no 8(step 12a); 6 locked in 19(3) for N3

13. 45 on C6789: 2 innies: R18C6 = 10 = [19/28/37] -->> R8C6 = {789}
13a. Naked Triple {789} in R8C468 -->> locked for R8

14. 45 on N1: 1 innie and 1 outie: R4C2 = R3C3 + 1 -->> R4C2 = {2345}; R3C3 = {1234}

15. 45 on N3: 1 innie and 1 outie; R3C7 = R4C8 + 1: R3C7: no 1,7,8; R4C8: no 5

16. 12(3) in R2C4 = {147/156/246/345}: no {129/138/237} needs 2 of {456789} in R23C4 -->> R3C3 = {123}(only place for {123}); R23C4 = {45/46/47/56}: no 8,9
16a. Clean up: R4C2: no 5(step 14); R4C5 and R4C8: no 2(step 9(When {125} R4C2 = 2(no {15}); R3C7: no 3(step 15)

17. 11(3) in R1C9 = {137}: {128} clashes with R8C9; {245} clashes with R3C7 -->> R1C9 = 7; R23C9 = {13} -->> locked for C9 and N3
17a. R8C9 = 2; R8C1 = 1

18. 11(3) in R1C1 = {245} -->> locked for C1 and N1
18a. Clean up: R4C2: no 3(step 14); R4C58: no 4(step 9(When {134} R4C2 = 4: no {13})); R3C7: no 5(step 15)

Little Trick on step 14, put it to replace the broken 45-test:
19. 45 on N1: “R4C2 = 2 -->> R3C3 = 1” blocked by R6C3 (R6C3 = {12}sees both cells, so they can’t contain both {12}) -->> R4C2: no 2; R3C3: no 1
19a. R4C2 = 4; R3C3 = 3; R23C9 = [31]; R23C2 = [17]
19b. Clean up: R23C4 = {45}(step 16) -->> locked for C4 and N2

Alternative step 19:
19. 12(3) in R6C1 = {38}[1]/[741]/[651]/{37}[2]: [642] blocked by R4C2 -->> R6C2: no 1,2,6
19a. 1 in N4 locked for C3
19b. R3C3 = 3; R4C2 = 4(step 14); R23C9 = [31]; R23C2 = [17]
19c. Clean up: R23C4 = {45}(step 16) -->> locked for C4 and N2

20. 45 on N4: 1 outie: R4C4 = 8
20a. 15(3) in R4C3 = 8{25}(last possible combination) -->> R45C3 = {25} -->> locked for C3 and N4
20b. R6C3 = 1; R8C3 = 4
20c. R12C3 = {69} (step 10) -->> locked for C3 and N1
20d. R1C2 = 8
20e. Clean up: R1C4: no 2(step 11); R2C7: no 5(step 12b); R4C5: no 5(step 9)

21. 12(3) in R6C1 = [831](last possible combination)

22. Naked Pair {78} in R79C3 -->> locked for N7

23. 11(3) in R7C8 = {45}2/[63]2 -->>R7C89 = {45}/[63]: R7C8: no 1,6
23a. 12(3) in R7C1 = [92]1: [65]1 clashes because R7C12 = [65] with R7C89 -->> R7C12 = [92]
23b. R5C2 = 9(hidden); R9C1 = 3(hidden)

Didn't see this whole cascade coming.
24. 19(3) in R6C7 = {469} (last possible combination) -->> R6C7 = 9; R6C89 = {46} -->> locked for R6 and N6
24a. R4C9 = 5; R5C9 = 8; R5C8 = 1; R4C8 = 3; R4C5 = 1; R45C3 = [25]; R45C7 = [72]
24b. R45C1 = [67]; R4C6 = 9; R3C7 = 4; R1C78 = [56]; R2C7 = 8; R12C3 = [96]
24c. R1C1 = 4; R23C4 = [45]; R23C1 = [52]; R23C8 = [29]; R2C6 = 7
24d. R3C6 = 6; R23C5 = [98]; R6C89 = [46]; R7C89 = [54]; R9C9 = 9; R8C6 = 8
24e. R89C8 = [78]; R8C4 = 9; R79C3 = [87]; R9C4 = 2; R6C4 = 7; R7C5 = 7(hidden)

And the last bit: let’s finish with a 45-test.
25. 45 on N9: 1 innie and 1 outie: R7C7 = R9C6 = 1(only common value)
25a. R89C7 = [36]; R89C2 = [65]; R89C5 = [54]; R7C46 = [63]; R1C456 = [132]
25b. R5C4 = 3; R56C5 = [62]; R56C6 = [45]


greetings

Para

2nd time OK

In contrast to Para, I do use innies and outies as much as possible! Steps 20 and 21 should have been spotted earlier but this is in the order I did it.

1. 23(3) N1 = {689}, not elsewhere in N1

2. 6(3) r1c456 = {123}, not elsewhere in r1/N2

3. 22(3) N4 must have 9, not elsewhere in N4

4. 24(3) N9 = {789}, not elsewhere in N9

5. Innies r89: r8c19 = 3 = {12}, not elsewhere in r8

6. Outies r123: r4c258 = 8 = {125/134}, 1 not elsewhere in r4

7. Innies r6: r6c456 = 14

8. Outies r6789: r5c456 = 13 -> r4c456 = 18

9. Outies – Innies N1: r4c2 – r3c3 = 1 -> r4c2 <> 1, r3c3 = (1234)

10. O-I N3: r3c7 – r4c8 = 1 -> r3c7 = (23456)

11. O-I N4: r4c4 – r4c2 = 4 -> r4c4 = (6789)

12. O-I N6: r4c6 – r4c8 = 6 -> r4c8 = (123), r4c6 = (789) -> r3c7 = (234)

13. O-I N7: r7c3 – r9c4 = 6 -> r7c3 = (789), r9c4 = (123)

14. O-I N9: r9c6 – r7c7 = 0 -> r9c6 = r7c7 = (123456)

15. O-I c12: r1c2 – r6c3 = 7 -> r1c2 = (89), r6c3 = (12)
6 locked to r12c3, not elsewhere in c3
-> r2c7 <> 2 else conflict with r1c2

16. O-I c89: r6c7 – r1c8 = 3 -> r6c7 = (789), r1c8 = (456)

17. 19(3) in N3 can’t have 1. 14(3) r234c8 has max 3 in r4c8 -> no 1 in r23c8.
-> 11(3) r123c9 must have 1 in r23c9 (1 + {37/46}, (11(3) = {128} is blocked by r8c9) 1 not elsewhere in c9
-> r8c9 = 2, r8c1 = 1 -> r9c6 <> 2
-> 11(3) r123c1 = {245}, not elsewhere in N1/c1
-> r3c3 = (13), r23c2 = (137), 7 not elsewhere in c2 -> r4c2 = (24) -> r4c4 = (68)
-> r5c2 <> 8

18. Split 8(3) r4c258: If {125}, r4c5 <> 2; if {134}, r4c5 <> 4

19. Outies N2: r3c37 + r4c5 = 8 = [143/341/125]-> r3c7 <> 3 -> r4c8 <> 2 -> r4c6 <> 8
-> split 18(3) r4c456 = [657/639/819]

20. Innies c1234: r18c4 = 10 -> r8c4 = (789)

21. Innies c6789: r18c6 = 10 -> r8c6= {789}
-> NT {789} r8c468, not elsewhere in r8

22. 17(3) r23c6 + r3c7 = {269/278/458/467} -> r23c6 = (56789)

23. 12(3) r3c3 + r23c4 = {147/156/345} -> r23c4 = (4567)

24. 18(3) r234c5 = {189/369/378/549/567}

25. 14(3) r234c8 = {149/158/329/356} ({167/347} blocked by options for 11(3)) -> r23c8 = {29/49/56/58}, no 3 or 7
no 3 in r12c7 since max 6 in r1c8 of 19(3) (step 16)
-> 11(3) in N3 = {137}, not elsewhere in c9 -> r1c9 = 7

26. 14(3) in N6 = {149/158/248/356} -> r5c8 = (123)

27. Innies c5: r189c5 = 12 = {138/147/156/237/246/345} -> r9c5 = (45678)

28. 13(3) r89c3 + r9c4 = [391/382/481/472/571] -> r9c3 = (789)
-> NQ (6789) in r1279c3 -> other cells in c3 max 5.
-> NT (789) in r9c389 -> other cells in r9 max 6 -> 14(3) in N7 = {356}
-> 5 locked to r89c2, not elsewhere in c2 -> 22(3) in N4 = {679}
-> r8c3 = 4 -> r9c3 <> 9, r9c4 <> 3 -> r7c3 <> 9 -> NP 7/8 in r79c3 not elsewhere in c3 -> r1c2 = 8
-> r7c2 = 2, r7c1 = 9 -> r5c2 = 9
-> r9c1 = 3, r6c1 = 8, r6c2 = 3, r6c3 = 1, r4c2 = 4, r3c3 = 3 … several more singles.

Fairly straightforward from here.

Edit: Finally had time to clarify step 25 and make other minor edits. Thanks to Para and Glyn.

Guess I was luckier. My mistake did take me to an impossible position.

Second time was OK. The earlier parts of my walkthrough are more like Cathy's solution path than Para's one. There were several naked singles that I haven't included so I just added a general comment at the end of step 45.

1. R1C456 = {123}, locked for R1 and N2

2. 23(3) cage in N1 = {689}, locked for N1

3. R123C9 = {128/137/146/236/245}, no 9

4. 19(3) cage in N3 = {289/379/469/478/569}, no 1

5. 22(3) cage in N4 = 9{58/67}, 9 locked for N4

6. R6C789 = {289/379/469/478/569}, no 1

7. 10(3) cage at R8C7 = {127/136/145/235}, no 8,9

8. 24(3) cage in N9 = {789}, locked for N9
[Didn’t include the 11(3) cage as a step because the 24(3) cage does more eliminations.]

9. 13(4) cage at R5C6 = 1{237/245/345}, no 8,9

10. 26(4) cage in N8 = {2789/3689/4589/4679/5678}, no 1

11. 45 rule on N1 1 outie R4C2 – 1 = 1 innie R3C3 -> R4C2 = {234568}

12. 45 rule on N3 1 innie R3C7 – 1 = 1 outie R4C8 -> no 1 in R3C7, no 9 in R4C8

13. 45 rule on N7 1 innie R7C3 – 6 = 1 outie R9C4 -> R7C3 = {789}, R9C4 = {123}

14. 45 rule on N7 3 innies R789C3 = 19 = {289/379/469/478/569}, no 1

15. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 -> no 7 in R9C6

16. 45 rule on N4 1 outie R4C4 – 4 = 1 innie R4C2 -> R4C2 = {2345}, R4C4 = {6789}, clean-up: R3C3 = {1234} (step 11)

17. 45 rule on N6 1 outie R4C6 – 6 = 1 innie R4C8 -> R4C6 = {789}, R4C8 = {123}, clean-up: R3C7 = {234} (step 12)

18. 45 rule on R89 2 innies R8C19 = 3 = {12}, locked for R8

19. 11(3) cage in N9 = {146/236/245}, R8C9 = {12} -> no 1,2 in R7C89

20. 45 rule on C12 1 innie R1C2 – 7 = 1 outie R6C3 -> R1C2 = {89}, R6C3= {12}
20a. 6 in N1 locked in R12C3, locked for C3

21. R6C123 = {138/147/237/246} (cannot be {156} which clashes with 22(3) cage), no 5
21a. R6C3 = {12} -> no 1,2 in R6C12

22. 45 rule on C89 1 outie R6C7 – 3 = 1 innie R1C8 -> R1C8 = {456}, R6C7 = {789}

23. 19(3) cage in N3 max R1C8 = 6 -> min R12C7 = 13, no 2,3 in R2C7

24. 45 rule on N2 3 outies R3C37 + R4C5 = 8, min R3C37 = 3 -> max R4C5 = 5
24a. 45 rule on R123 3 outies R4C258 = 8 = {125/134}, 1 locked for R4

25. R234C8 = {149/158/167/239/248/257/347/356}
25a. Max R4C8 = 3 -> min R23C8 = 11, no 1

26. 1 in N3 locked in R23C9, locked for C9 -> R8C9 = 2, R8C1 = 1, clean-up: no 2 in R9C6 (step 15)
26a. R123C9 (step 3) = 1{37/46} (only remaining combinations), no 5,8
26b. 1,3 only in R23C9 -> no 7 in R23C9
26c. 1 in N9 locked in R79C7, locked for C7

27. 11(3) cage in N9 = 2{36/45} -> R789C7 = 1{36/45} -> 10(3) cage at R8C7 = 1{36/45} from R7C7 = R9C6 (step 15)
27a. 1 in 10(3) cage at R8C7 locked in R9C67, locked for R9, clean-up: no 7 in R7C3 (step 13)

28. Killer pair 8/9 in R12C3 and R7C3, locked for C3

29. 13(3) cage at R8C3 = {247} (only remaining combination) -> R9C4 = 2, R89C3 = {47}, locked for C3 and N7, clean-up: R7C3 = 8 (step 13), no 5 in R4C2 (step 11), no 9 in R4C4 (step 16)

30. R12C3 = {69} -> R1C2 = 8, clean-up: R6C3 = 1 (step 20), no 2 in R4C2 (step 11), no 6 in R4C4 (step 16)

31. 2 in N7 locked in R7C12 -> R7C12 = {29} (only remaining combination for 12(3) cage in N7), locked for R7 and N7

32. 5 in C3 locked in R45C3, locked for N4, clean-up: no 8 in 22(3) cage in N4 (step 5)
32a. 22(3) cage in N4 = {679}, locked for N4

33. R6C123 = [831] (only remaining permutation) -> R4C2 = 4, R3C3 = 3 (step 11), R4C4 = 8 (step 16), clean-up: no 5 in R1C8 (step 21), no 2 in R4C8 (step 17)

34. R23C2 = {17} (only remaining combination for R234C2), locked for C2 and N1

35. R123C1 = {245}, locked for C1 -> R7C12 = [92]

36. R45C1 = {67}, locked for C1 -> R5C2 = 9, R9C1 = 3, clean-up: no 3 in R7C7 (step 15)

37. 10(3) cage at R8C7 (step 27) = 1{36/45}
37a. 3 only in R8C7 -> no 6 in R8C7

38. R4C2 = 4 -> R4C58 = {13} (step 23a), locked for R4

39. R123C9 = [731] (only remaining permutation, {146} clashes with R1C8) -> R23C2 = [17]

40. R3C3 = 3 -> R23C4 = 9 = {45} (only remaining combination), locked for C4 and N2

41. Naked triple {245} in R3C147, locked for R3

42. 24(4) cage at R5C4 = {1689/3678} = 68{19/37}
42a. 6 locked in R567C4, locked for C4
42b. Killer pair 1/3 in R1C4 and R57C4, locked for C4

43. 7 in R7 locked in R7C456, locked for N8 -> R8C4 = 9

44. R567C4 (step 42) = {367}, locked for C4 -> R1C4 = 1

45. R6C789 = {469} (only remaining combination) -> R6C7 = 9, R6C89 = {46}, locked for R6 and N6 -> R6C4 = 7, R4C6 = 9, R6C56 = {25}, locked for N5, R45C9 = [58], R45C3 = [25], R4C7 = 7, R45C1 = [67], clean-up: R4C8 = 3 (step 17), R4C5 = 1, R3C7 = 4 (step 12), R23C4 = [45], R123C1 = [452], R1C8 = 6, R12C3 = [96], R6C89 = [46], R7C8 = 5, R7C9 = 4, R8C7 = 3, clean-up: no 4,5 in R9C6 (step 15) and a few more naked singles

46. R2C8 = 2, R5C8 = 1 (hidden singles in C8) -> R3C8 = 9 (cage sum), R9C9 = 9

47. R4C5 = 1 -> R23C5 = 17 = [98]

and the rest is naked singles

3x3::k:3584:3841:3841:2307:2307:2307:3590:3590:5384:3584:3850:3841:5132:3341:3598:3590:3088:5384:3584:3850:5132:5132:3341:3598:3598:3088:5384:3355:3850:4381:4381:3341:4640:4640:3088:3875:3355:3355:4381:4903:3624:5161:4640:3875:3875:3629:3629:3629:4903:3624:5161:4147:4147:4147:5942:5942:4903:4903:3624:5161:5161:2621:2621:5942:2624:3393:5698:5698:5698:4165:4678:2621:2624:2624:3393:3393:5698:4165:4165:4678:4678:

Here is the full Walk-through for Assassin 56V2. A very, very difficult puzzle with lots of contradiction moves and extensive combination conflicts. Had to restart many times to get a valid solution. Must have spent 15 hours on this one . Perfect for a V2. Thanks again Para.

Please let me know of any corrections or clarifications needed. The original steps 1-16 have been revised for clarity.

Cheers
Ed

Assassin 56 V2
1. 23(3)n7 = {689}: all locked for n7

2. 10(3)n7 = {127/145/235}

3. "45" n7: r789c3 = 12 = h12(3)n7
3a. = {147/237/345}

4. "45" n36: r4c6 - 6 = r3c7
4a. r4c6 = {789}, r3c7 = {123}

5. "45" n3: r4c8 - 2 = r3c7
5a. r4c8 = {345}

6. 20(3)n2: no 1,2

7. "45" n1: r4c2 + 1 = r3c3
7a. r4c2 = 2..8

8. "45" r123: r4c258 = 12 = h12(3)r4
8a. min. r4c8 = 3 -> max. r4c25 = 9
8b. min. r4c2 = 2 -> max. r4c5 = 7

9. "45" n4: r4c4 + 1 = r4c2
9a. r4c4 = 1..7

10. "45" n6: r4c6 - 4 = r4c8
10a. r4c68 = [73/84/95]

11. h12(3)r4 = {147/246/345}. Others blocked. Here's how.
i. r4c258 = {138}: blocked since can only be = [813], but including "45" moves for r4c4 (1 less than r4c2 step 9) & r4c6 (4 more than r4c8 step 10) = [87173]: but this means 2 7's r4
ii.{147} = [714] only
iii. {156}: blocked since can only be [615] but including "45" moves for r4c4 & r4c6 = [65195]: but this means 2 5's r4
iv. {237}: blocked since both [273/723] require 7 in r4c6 but this means 2 7's r4
v. {246} = [264/624]
vi. {345} = [345/354]. Others are blocked. [435/453] by 3 required in r4c4; [534/543] by 4 required in r4c4

12. In summary: h12(3) r4 = {147/246/345} = 4{..}
12a. 4 locked for r4
12b. = [714/264/624/345/354]
12c. r4c2 = {2367} -> r3c3 = {3478} (step 7) and r4c4 = {1256}(step 9)
12d. r4c5 = {12456}(no 3,7)
12e. r4c8 = {45} -> r4c6 = {89}(step 10) & r3c7 = {23}(step 5)

13. "45" on n14: 3 innies r345c3 = 19 & remembering that r789c3 = h12(3)n7 = {147/237/345}
13a. r45c3 "sees" r3c3, r4c2 and r4c4: these 3 are all linked through "45" moves and = [321/432/765/876]
13b. r3c3 + r4c2 + r4c4 = [321]
i. -> r45c3 = 16 = {79} blocked: r345c3 = [3]{79}: clashes with h12(3)n7
13c. r3c3 + r4c2 + r4c4 = [432]
i. -> r45c3 = 15 = {69} -> r345c3 = [4]{69}
ii............... = 15 = {78} blocked: r345c3 = [4]{78} but clashes with h12(3)n7
13d. r3c3 + r4c2 + r4c4 = [765]
i.-> r45c3 = 12 = {39} blocked: r345c3 = [7]{39}: clashes with h12(3)n7
ii............. = 12 = {48} blocked: r345c3 = [7]{48}: clashes with h12(3)n7
iii............ = 12 = {57} blocked by 5 in r4c4
13e. r3c3 + r4c2 + r4c4 = [876]
i. -> r45c3 = 11 = {29} -> r345c3 = [8]{29}
ii................= 11 = {38}: blocked by 8 in r3c3
iii...............= 11 = {47}: blocked by 7 in r4c2
iv................= 11 = {65}: blocked by 6 in r4c4

14. In summary: r345c3 = [4]{69}/[8]{29} = 9{..}
14a. 9 must be in r45c3: 9 locked for c3 and n4
14b. r45c3 = 9{26}
14c. r4c4 = {26}
14d. r4c2 = {37}

15. 20(3)n2 = r3c3 + r23c4 = [4]{79}/[8]{39/57} (no 4,6,8 r23c4)
15. = {389/479/578}

16. from step 12b. h12(3)r4 = [714/345/354] = {147/345}
16a. r4c5 = {145}

Continuing on.
17. from step 16: remembering r4c4 is 1 less than r4c2 & r4c6 is 4 more than r4c8
17a. -> r4c456 = [618/249/258]

18. "45" n4: 3 innies = 18 = h18(3)n4 = 9{36/27}
18a. 14(3) n4 = {158/248/257/356} ({167/347} blocked by h18(3)n4)

19. 17(3)n4 = {269}
19a. -> no 2,6 in r4c1

20. "45" r6: 3 innies r6c456 = 15 = h15(3)r6
20a. = {168/249/357} ({159/258/267/348/456} all blocked by r4c456! step 17a)

Now: moving across to n36 for the same tricks.
21. r3c7 + 6 = r4c6: r4c6 - 4 = r4c8 -> {r45c7} = 10/9
21a. [r3c7 = 2][r4c6 = 8][r4c8 = 4] = [284]
i. -> r45c7 = 10 = {19/37}
21b. [r3c7 = 3][r4c6 = 9][r4c8 = 5] = [395]
i. -> r45c7 = 9 = {18/27}
21c. In summary: r45c7 = {18/19/27/37}(no 456)
21d. 18(3)n5 = {189/279/378}

22. "45" n6: 3 innies = 14 = h14(3)n6 = {149/347/158/257}

Now for lots of combo. crunching.
23. 16(3)n6 = {169/349/367}(no 2,5,8). Others blocked. Here's how.
23a. {178/457} blocked by h14(3)n6 step 22
23b. {268/259/358} blocked by 14(3)n4 (step 18a.)

24. 14(3)n4 = {158/248/257}(no 3,6) ({356} blocked by 16(3)n6 step 23.)

25. deleted

26. Generalized X-wing on 2 required in 17(3)n4 (only in r45) & 2's in n6 (only in r45)
26a. -> no 2 elsewhere in r45

27. 13(3)n4 = {148/157/346}

28. "45" r12345: r5c456 = 15 = h15(3)r5 (note: h15(3)r6 shows all the blocked combinations for 15(3) by r4c456. See step 20a)
28a. r5c456 = {168/357}(no 4,9)
28b. = [1/5,1/7..]
28c. -> no 5 & 7 in r4c1 as 13(3) must be {157}
28d. but r5c12 = {17/15} will clash with h15(3)r5 (step 28b)
28e. no 5,7 r4c1

29. h15(3)r5 = [5/8..](step 28a)
29a. -> from step 17 r4c456 = [618/249](no 5) ([258] blocked by h15(3)r5)

30. 5 in r4 only in n6
30a. 5 locked for n6

31. 13(3)n4:{346} combo is the only combo with 3
31a. -> no 3 r5c12

32. 3 in n4 only in r4
32a. -> 3 locked for r4

Now to n1.
33. 15(3)r2c2 must have 3/7 in r4c2
33a. = {267/357}(no 1,4,8,9) ({348} blocked by 4 required in r3c3 when r4c2 = 3 step 7)
33b. = 7{26/35}
33c. 7 locked for c2

34. "45" n1: 3 innies = 16 = h16(3)
34a. r3c3 + r23c2 = [4]{57}/[8]{26/35}
34b. = {268/358/457}

35. 15(3)r1c2 = {159/168/249/267/357} ({258/348/456} blocked by h16(3)n1 step 34b.)

36. "45" c12: r126c3 = 14 = h14(3)c3 & remembering h12(3)n7 = {147/237/345}(step 3a)
36a. h14(3)c3 = {158/167/356} (no 2,4) ({248} blocked by r3c3; {257/347} blocked by h12(3)n7)
[Alternate: Andrew used killer pair 2,4 in h19(3)r3c3 and h12(3)r7c3. Thanks Andrew!]

37. "45" c12: r6c3 + 1 = r1c2
37a. r1c2 = {2,6,8,9}

38. 15(3)r1c2 = {159/168/267}(no 3) ({357} blocked by no candidates in r1c2)

39. 3 in c7 only in n7
39a. 3 locked for n7
39b. h12(3)n7 must have 3 = 3{27/45}(no 1)

40. from step 34b. h16(3)n1 = {268/358/457}
40a. ->14(3)n1 = {149/239/347}(no 5,6,8) ({158/248/257/356} blocked by h16(3)n1; {167} blocked by 15(3)r1c2)

41. {346} combo. blocked from 13(3)n4. Here's how.
41a. {346} combo. -> r4c1 = 3 -> 14(3)n1 = {149} -> r5c1 = 6: but {69} r1235c1 clashes with r78c1.
41b. 13(3)n4 = {148/157} (no 3,6) = 1{..}
41c. 1 locked for n4
41d. -> no 2 r1c2 ("45" c12)

42. r4c2 = 3 (hsingle n4)
42a. r4c4 = 2 ("45" n4)
42b. r3c3 = 4 ("45" n1)

43. r23c2 = {57}: both locked for n1 & c2

44. r45c3 = {69}: both locked for c3

45. r12c3 = {18}: both locked for n1 & c3
45a. r1c2 = 6
45b. r6c3 = 5 ("45" c12)

46. 14(3)n1 = {239}: all locked for c1

47. r78c1 = {68}: both locked for c1 & n7
47a. r7c2 = 9

48. r6c2 = 2, r9c1 = 5 (hsingles)

49. 13(3)n4 = [148](only valid combo)

50. r6c1 = 7, r4c58 = [45]

51. r4c6 = 9, r3c7 = 3 ("45" n3,n6)

52. r45c3 = [69]

53. r23c4 = {79}: both locked for n2, c4

54. r23c6 = {56}: both locked for n2,c6

55. 9(3)n2 = {234}: all locked for n2 & r1

56. r23c8 = {16}: both locked for c8 & n3

57. "45" c89: r6c7 - 2 = r1c8
57a. r6c7 = 9, r1c8 = 7
57b. r12c7 = [52]

The rest is naked or last valid combo

Another variant which I've just tried for the first time; in spring and summer 2007 I didn't try variants because we were moving from Calgary to Lethbridge.

Prelims

a) 9(3) cage at R1C4 = {126/135/234}, no 7,8,9
b) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
c) 20(3) cage at R2C4 = {389/479/569/578}, no 1,2
d) 23(3) cage at R7C1 = {689}
e) 10(3) cage at R7C8 = {127/136/145/235}, no 8,9
f) 10(3) cage at R8C2 = {127/136/145/235}, no 8,9

1. Naked triple {689} in 23(3) cage at R7C1, locked for N7

2. 45 rule on R89 2 innies R8C19 = 11 = [65/83/92]

3. 45 rule on N4 1 innie R4C2 = 1 outie R4C4 + 1, no 1 in R4C2, no 9 in R4C4

4. 45 rule on N6 1 outie R4C6 = 1 innie R4C8 + 4, no 1,2,3,4 in R4C6, no 6,7,8,9 in R4C8

5. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 1, no 1,7,9 in R9C4

6. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 1, no 1 in R7C7, no 9 in R9C6

7. 45 rule on N7 3 innies R789C3 = 12 = {147/237/345}
7a. 2 of {237} must be in R89C3 (cannot be 2{37} because 13(3) cage at R8C3 cannot be {37}3), no 2 in R7C3, clean-up: no 3 in R9C4 (step 5)

8. 45 rule on N1 1 innie R3C3 = 1 outie R4C2 + 1, no 9 in R4C2, clean-up: no 8 in R4C4 (step 3)

9. 45 rule on N3 1 outie R4C8 = 1 innie R3C7 + 2, R3C7 = {123}, R4C8 = {345}, clean-up: no 5,6 in R4C6 (step 4)

10. 45 rule on R123 3 outies R4C258 = 12 = {138/147/156/237/246/345} (cannot be {129} because R4C8 only contains 3,4,5), no 9
10a. R4C258 = {147/156/237/246/345} (cannot be {138} = [813] because R4C24 = [87], step 3, clashes with R4C68 = [73], step 4), no 8, clean-up: no 9 in R3C3 (step 8), no 7 in R4C4 (step 3)
10b. R4C258 = {147/237/246/345} (cannot be {156} = [615] because R4C24 = [65], step 3, clashes with R4C8)
10c. R4C258 = {147/246/345} (cannot be {237} = {27}3 which clashes with R4C68 = [73]), 4 locked for R4, clean-up: no 5 in R4C2 (step 3), no 6 in R3C3 (step 8)
10d. 1 of {147} must be in R4C5 -> no 7 in R4C5
10e. 3 of {345} must be in R4C2 (cannot be 4{35} which clashes with R4C24 = [43], step 3), 7 of {147} must be in R4C2, 4 of {246} must be in R4C8 -> no 4 in R4C2, no 3 in R4C58, clean-up: no 5 in R3C3 (step 8), no 1 in R3C7 (step 9), no 3 in R4C4 (step 3), no 7 in R4C6 (step 4)

11. 45 rule on N3 3 innies R2C8 + R3C78 = 10 = {127/136/235} (cannot be {145} because R3C7 only contains 2,3), no 4,8,9
11a. 2 of {235} must be in R3C7 (cannot be [235/532] which clash with 12(3) cage at R2C8 = [255/525]), no 2 in R23C8

12. 45 rule on C1234 2 innies R18C4 = 7 = {16/25/34}, no 7,8,9

13. 45 rule on C6789 2 innies R18C6 = 6 = {15/24}

14. 45 rule on C5 3 innies R189C5 = 18 = {189/279/369/378/459/468/567}
14a. 1,2 of {189/279} must be in R1C5 -> no 1,2 in R89C5

15. 45 rule on C12 1 innie R1C2 = 1 outie R6C3 + 1, no 1 in R1C2, no 9 in R6C3

16. 45 rule on C89 1 outie R6C7 = 1 innie R1C8 + 2, no 1,2 in R6C7, no 8,9 in R1C8

17. 45 rule on N36 3 outies R234C6 = 20 = {389/479/569/578}, no 1,2
17a. 8 of {389/578} must be in R4C6 (cannot be {38}9 because 14(3) cage at R2C6 cannot be {38}3), no 8 in R23C6

18. 45 rule on N14 3 outies R234C4 = 18 = {189/279/369/459/567} (cannot be {378} because no 3,7,8 in R4C4, cannot be {468} because 20(3) cage at R2C4 cannot be {48}8)
18a. 6 of {369/567} must be in R4C4 (cannot be {67}5 because 20(3) cage at R2C4 cannot be {67}7), no 6 in R23C4

19. 45 rule on N14 3 innies R345C3 = 19 = {289/469} (cannot be {379/478} which clash with R789C3, cannot be {568} = 8{56} because 17(3) cage at R4C3 cannot be {56}6), no 1,3,5,7, 9 locked for C3 and N4, clean-up: no 2,6 in R4C2 (step 8), no 1,5 in R4C4 (step 3)
19a. R3C3 = {48} -> no 4,8 in R45C3
19b. Killer pair 2,4 in R345C3 and R789C3, locked for C3, clean-up: no 3,5 in R1C2 (step 15)
19c. 17(3) cage at R4C3 = {269} (only remaining combination), CPE no 2,6 in R4C1

20. R4C258 (step 10c) = {147/345} (cannot be {246} because 2,6 only in R4C5), no 2,6

21. 20(3) cage at R2C4 = {389/479/578}
21a. R3C3 = {48} -> no 4,8 in R23C4

22. 8 in N2 only in R23C5 -> 13(3) cage at R2C5 = {148} (only remaining combination, cannot be {238} because no 2,3,8 in R4C5), locked for C5
22a. Killer pair 1,4 in 9(3) cage at R1C4 and 13(3) cage, locked for N2

23. 2 in N2 only in 9(3) cage at R1C4, locked for R1, clean-up: no 1 in R6C3 (step 15), no 4 in R6C7 (step 16)
23a. 9(3) cage = {126/234}, no 5, clean-up: no 2 in R8C4 (step 12), no 1 in R8C6 (step 13)

24. 45 rule on N6 3 innies R4C78 + R5C7 = 14
24a. 45 rule on N36 3 innies R345C7 = 12 = {129/138/237} (cannot be {147/156} because R3C7 only contains 2,3, cannot be {246} = 2{46} because R4C78 + R5C7 cannot be {46}4, cannot be {345} = 3{45} because R4C78 + R5C7 cannot be {45}5), no 4,5,6

25. 45 rule on N5789 3 innies R4C456 = 15 = {168/249}
25a. 45 rule on R6789 3 outies R5C456 = 15 = {168/249/357} (cannot be {159/258/267/348/456} which clash with R4C456)
25b. 45 rule on R6 3 innies R6C456 = 15 = {168/249/357} (cannot be {159/258/267/348/456} which clash with R4C456)
[Step 25 reminded me of the top row of A74 Brick Wall, which also had three 15(3) cages. This step was actually more powerful until I realised that I’d overlooked a killer pair which reduced the number of combinations in step 25. In its original form, this step was longer but reduced the three 15(3) hidden cages to the same combinations. That’s why I’ve kept it in even though, in the simplified form, there aren’t any immediate candidate eliminations.]

26. 45 rule on N1 3 innies R2C2 + R3C23 = 16
26a. 15(3) cage at R2C2 = {267/357} (cannot be {159/168/249/258/456} because R4C2 only contains 3,7, cannot be {348} = {48}3 because R2C2 + R3C23 cannot be {48}4), no 1,4,8,9, 7 locked for C2, clean-up: no 6 in R6C3 (step 15)

27. 45 rule on C12 3 outies R126C3 = 14 = {158/167/356}
27a. 3 of {356} must be in R6C3 (cannot be {36}5 because 15(3) cage at R1C2 cannot be 6{36}), no 3 in R12C3
27b. R126C3 = {158/167} (cannot be {356} = {56}3) which clashes with 15(3) cage at R2C2), no 3, 1 locked for C3 and N1, clean-up: no 4 in R1C2 (step 15), no 2 in R9C4 (step 5)

28. R789C3 (step 7) = {237/345}, 3 locked for N7
28a. 4 of {345} must be in R89C3 (cannot be 4{35} because 13(3) cage at R8C3 cannot be {35}5), no 4 in R7C3, clean-up: no 5 in R9C4 (step 5)

29. 10(3) cage at R8C2 = {127/145}
29a. 7 of {127} must be in R9C1 -> no 2 in R9C1

30. 15(3) cage at R1C2 = {159/168}, no 7, 1 locked for N1
30a. Killer pair 5,6 in 15(3) cage at R1C2 and 15(3) cage at R2C2, locked for N1
30b. 14(3) cage at R1C1 = {239/347} (cannot be {248} which clashes with R3C3), no 8, 3 locked for C1 and N1

31. 13(3) cage at R4C1 = {148/157/238/247} (cannot be {256} which clashes with R45C3, ALS block, cannot be {346} because no 3,4,6 in R4C1), no 6
31a. 14(3) cage at R6C1 = {158/167/248/257/356} (cannot be {347} which clashes with R4C2)
31b. Variable hidden killer quad 1,4,5,8 in 13(3) cage and 14(3) cage for N4 -> each cage must contain two of 1,4,5,8 or one cage must contain one and the other three
31c. 14(3) cage = {158/248/257/356} (cannot be {167} because 13(3) cage cannot contain all of 4,5,8)
31d. 3 of {356} must be in R6C2 -> no 6 in R6C2
31e. 13(3) cage = {148/157/247} (cannot be {238} which clashes with 14(3) cage), no 3

[At this stage I originally used a short forcing chain
14(3) cage at R6C1 (step 31c) = {158/248/257/356}
Consider placement for 6 in 23(3) cage at R7C1
6 in R78C1 -> 14(3) cage = {158/248/257}
or 6 in R7C2 -> no 6 in R1C2, no 5 in R6C3 (step 15) -> 14(3) cage = {158/248/257}
-> 14(3) cage = {158/248/257}, no 3,6
Then, when I went through Ed’s walkthrough, I realised that he’d done more work in N6 than I had; I hadn’t looked at the 15(3) and 16(3) cages, so …]

32. 14(3) cage at R6C1 (step 31c) = {158/248/257/356}
32a. 18(3) cage at R4C6 = 8{19/37}/9{18/27}
32b. 16(3) cage at R6C7 = {169/349/367} (cannot be {178} which clashes with 18(3) cage at R4C6, cannot be {259/268/358/457} which clash with 14(3) cage at R6C1), no 2,5,8, clean-up: no 3,6 in R1C8 (step 16)
32c. 14(3) cage at R6C1 = {158/248/257} (cannot be {356} which clashes with 16(3) cage at R6C7), no 3,6

33. R4C2 = 3 (hidden single in N4), R3C3 = 4 (step 8), R4C4 = 2 (step 3)
33a. Naked pair {69} in R45C3, locked for C3
[Cracked. The rest is straightforward, routine clean-ups have been omitted.]

34. R3C3 = 4 -> R23C4 = 16 = {79}, locked for C4 and N2

35. R4C258 (step 20) = {345} (only remaining combination) -> R4C5 = 4, R4C8 = 5, R4C6 = 9 (step 4), R45C3 = [69], R3C6 = 3 (step 9)
35a. R4C8 = 5 -> R23C8 = 7 = {16}, locked for C8 and N3, clean-up: no 7,8 in R6C7 (step 16)

36. Naked pair {56} in R23C6, locked for C6 and N2, clean-up: no 1 in R1C6 (step 13)
36a. Naked pair {24} in R18C6, locked for C6

37. Naked pair {18} in R23C5, locked for N2
37a. Naked triple {234} in 9(3) cage at R1C4, locked for R1 -> R1C8 = 7, R6C7 = 9 (step 16), R1C1 = 9, clean-up: no 2 in R8C9 (step 2)
37b. Naked pair {68} in R78C1, locked for C1 and N7 -> R7C2 = 9

38. R1C23 = [61] (hidden pair in R1), R2C3 = 8 (cage sum)

39. R2C1 = 3 (hidden single in N1), R3C1 = 2 (cage sum)

40. Naked pair {58} in R1C79, locked for N3 -> R3C9 = 9, R2C9 = 4, R1C9 = 8 (cage sum), R12C7 = [52]

41. R4C7 = 8 (hidden single in R4), R5C7 = 1 (cage sum), R4C19 = [17]
41a. R4C9 = 7 -> R5C89 = 8 = [26], R6C89 = [43], R7C8 = 3, R8C9 = 5, R7C9 = 2 (cage sum), R8C1 = 6 (step 2), R9C9 = 1

[And now to use those hidden 15(3) cages before it’s too late ]
42. R6C456 (step 25b) = {168} (only remaining combination) -> R6C5 = 6, R6C46 = {18}, locked for R6 and N5

43. R6C5 = 6 -> R57C5 = 8 = [35]

and the rest is naked singles.

I'll rate my walkthrough for A56 V2 at 1.5. There was a lot of analysis of interactions between hidden cages.

3x3::k:3584:3584:1794:1794:2308:3333:3333:3335:3335:3584:5130:3339:3339:2308:1806:1806:4880:3335:5130:5130:1556:1556:4374:2839:2839:4880:4880:5130:4380:2589:2589:4374:3872:3872:4642:4880:4380:4380:5926:5926:5926:5926:5926:4642:4642:4380:5422:4399:4399:2353:1586:1586:6708:4642:5422:5422:2616:2616:2353:2107:2107:6708:6708:5422:4672:1601:1601:2371:3140:3140:3654:6708:4672:4672:2634:2634:2371:1869:1869:3654:3654:

I have to agree with JC. I thought more combination analysis would be required from Ruud's comment on the puzzle page. Still fun to solve!

1. Innies c5: r5c5 = 1

2. 17(2) r34c5 -> {89} not elsewhere in c5.

3. 17(2) r6c34 -> {89} not elsewhere in r6.

4. NP {89} in r4c5/r6c4, not elsewhere in N5
-> 15(2) r4c67 = [69/78]
-> NP {89} r4c57, not elsewhere in r4
-> 10(2) r4c34 = {37/46}
-> Killer pair with r4c6 -> 6,7 not elsewhere in r4 -> r4c1289 = (12345)

5. 6(2) r6c67 = {24}/[51]

6. Innies r12: r2c28 = 14 = {59/68}

7. Innies r89: r8c19 = 14 = {59/68}

8. Innies r1234: r4c28 = 6 = {15/24} (no 3)
-> r4c159 = 14 = {158/248/149/239}

9. Innies r6789: r6c19 = 7 = {16/34} ({25} blocked by 6(2))
-> r6c258 = 15

10. Innies c34: r5c34 = 11 -> r5c67 = 11

11. If split 14(2) r8c19 = {59} -> 12(2) r8c67 = {48} -> CONFLICT as no options for 6(2) r8c34
-> r8c19 = {68}, not elsewhere in r8 -> r8c67 <> 4, r9c5 <> 3

12. 13(2) r2c34 = {49/67} ({58} blocked by split 14(2) r2c28)
-> 13(2) r2c34 and split 14(2) r2c28 form killer pair on 6 and 9, not elsewhere in r2
-> r1c5 <> 3, r2c67 <> 1

13. r2c159 = 11 = {128/137}
-> r12c5 <> 4,5
-> Analysis of 11(3) options -> r2c19 <> 2.

14. Split 7(2) r6c19 and 6(2) r6c67 form killer pair on 1 and 4, not elsewhere in r6
-> r7c5 <> 5
-> 4 locked to r789c5, not elsewhere in N8
-> r79c3 <> 6, r8c3 <> 2, r9c7 <> 3

15. Split 15(3) r6c258 = {267/357}

16. 1 locked to r13c4, not elsewhere in c4 -> r79c3 <> 9, r8c3 <> 5

17. 1 locked to r2c19 and 1 locked to r4c12/r6c1 in N4
If r2c1 = 1 -> r4c2 = 1 -> r4c9 <> 1
If r2c9 = 1 -> r4c9 <> 1
Either case r4c9 <> 1
-> Analysis of split 14(3) r4c159 (see step 8) -> r4c1 <> 5

18. Outies N9: r789c6 + r6c8 = 22
Must have 1 in r79c6
Options: {1579/1669} – only place for 9 is r8c6 -> r8c7 = 3
-> r79c6, r6c8 <> 2,3
Clean up
-> r7c7 <> 6, r7c67 <> 5, r79c3 <> 1, r9c7 <> 4,5, r1c7 <> 4, r2c6 <> 4, r3c6 <> 8, r3c7 <> 2


19. 8 locked to r79c4, not elsewhere in c4
-> r6c4 = 9, r6c3 = 8, r4c5 = 8, r3c5 = 9, r4c7 = 9, r4c6 = 6
Clean up -> 8(2) r7c67 = {17} not elsewhere in r7, r9c4 <> 2, r9c7 <> 1 and 10(2) r4c34 = {37} not elsewhere in r4; r7c3 <> 3, r7c4 <> 2,3, r7c5 <> 3

20. HS r1c6 = 8 -> r1c7 = 5
-> r2c6 <> 2, r1c34 <> 2, r2c2 <> 9

21. HS r9c4 = 3 -> r4c4 = 7, r4c3 = 3
-> r2c3 <> 6, r5c3 <> 4, r9c3 <> 3, r5c7 <> 4, r7c5 <> 2

Straightforward from here

I agree with the comments from J-C and Cathy. A bit easier than some recent ones and definitely less combination work than suggested by Ruud's introduction to this puzzle. However there were far less opportunities than usual to apply the 45 rule. I only used it on rows and columns; I didn't find any useful application for it on the nonets although Cathy did have a useful application for it on N9.

Here is my walkthrough, before people start working on Ed's V2X.

First the preliminary steps

1. R1C34 = {16/25/34}, no 7,8,9

2. R12C5 = {18/27/36/45}, no 9

3. R1C67 = {49/58/67}, no 1,2,3

4. R2C34 = {49/58/67}, no 1,2,3

5. R2C67 = {16/25/34}, no 7,8,9

6. R3C34 = {15/24}

7. R34C5 = {89}, locked for C5, clean-up: no 1 in R12C5

8. R3C67 = {29/38/47/56}, no 1

9. R4C34 = {19/28/37/46}, no 5

10. R4C67 = {69/78}

11. R6C34 = {89}, locked for R6

12. R67C5 = {27/36/45}, no 1

13. R6C67 = {15/24}

14. R7C34 = {19/28/37/46}, no 5

15. R7C67 = {17/26/35}, no 4,8,9

16. R8C34 = {15/24}

17. R89C5 = {27/36/45}, no 1

18. R8C67 = {39/48/57}, no 1,2,6

19. R9C34 = {19/28/37/46}, no 5

20. R9C67 = {16/25/34}, no 7,8,9

21. 26(4) cage at R6C8 = {2789/3689/4589/4679/5678}, no 1

And now for the early present

22. 45 rule on C5 1 innie R5C5 = 1 [Alternatively hidden single in C5, after doing the 2-cell cages], clean-up: no 9 in R4C3, no 5 in R6C7

23. Naked pair {89} in R4C5 and R6C4, locked for N5, clean-up: no 1,2 in R4C3, no 6,7 in R4C7

24. Naked pair {89} in R4C57, locked for R4, clean-up: no 2 in R4C4

25. 45 rule on R1 3 outies R2C159 = 11 = {128/137/146/236/245}, no 9

26. 45 rule on R12 2 innies R2C28 = 14 = {59/68}
[I should then have seen that this eliminates {58} from R2C34]

27. 45 rule on R1234 2 innies R4C28 = 6 = {15/24}

28. 45 rule on R123 3 outies R4C159 = 14, min R4C5 = 8 -> max R4C19 = 6, no 6,7
[Alternatively, after step 24, could have used killer pair 6/7 in R4C34 and R4C6, locked for R4]
28a. Valid combinations 8{15}/8{24}/9{23} (cannot be 9{14} which clashes with R4C28)
[The order of steps 27 and 28 has been exchanged for clarity]

29. 45 rule on R6789 2 innies R6C19 = 7 = {16/34} (cannot be {25} which clashes with R6C67), no 2,5,7

30. 45 rule on R89 2 innies R8C19 = 14 = {59/68}
30a. If R8C19 = {59} => R8C34 = {24} clash with all combinations for R8C67 so R8C19 cannot be {59}
30b. R8C19 = {68}, locked for R8, clean-up: no 4 in R8C67, no 3 in R9C5

31. 8 in N8 locked in R79C4, locked for C4 -> R6C34 = [89], R34C5 = [98], R4C67 = [69], clean-up: no 4 in R1C6, no 4,7 in R1C7, no 4,5 in R2C3, no 5 in R2C4, no 1 in R2C7, no 2 in R3C6, no 2,5 in R3C7, no 4 in R4C34, no 1 in R7C3, no 2 in R7C4, no 3 in R7C5, no 2 in R7C7, no 3 in R8C6, no 1 in R9C3, no 2 in R9C4, no 1 in R9C7
[I missed the fact that R8C6 = 9 (hidden single in C6) after R34C5 were fixed.]

32. Naked pair {37} in R4C34, locked for R4
[Alternatively 3 could be eliminated from R4C19 using step 28a with R4C5 = 8]

33. 7 in R6 locked in R6C258
33a. 45 rule on R789 3 outies R6C258 = 15 = 7{26/35}, no 1,4, clean-up: no 5 in R7C5
[Note that if 7 wasn’t already locked in this split 15(3) cage, then it would have to be 7{26/35} because {456} clashes with R6C67.]

34. 45 rule on C34 2 innies R5C34 = 11 = {47}/[65/92], no 2,3,5 in R5C3, no 3 in R5C4

35. 45 rule on C67 2 innies R5C67 = 11 = {47}/[56/83], no 2, in R5C6, no 2,3,5 in R5C7

36. 45 rule on R9 3 outies R8C258 = 13 = {139/157/247}
36a. If R8C258 = {139}, 3 must be in R8C5 -> no 3 in R8C28

37. R2C28 (step 26) = {59/68}
37a. If R2C28 = {59} => R2C34 = {67} => R2C67 = {34} => R2C159 = {128} => R2C5 = 2, R2C19 = {18}
37b. If R2C28 = {68} => R2C34 = [94] => R2C67 = {25} => R2C159 = {137} => R2C5 = {37}, R2C19 = {137}
Summary
R2C28 and R2C34 unchanged
R2C67 = {25/34}, no 1,6
R2C19 = {1378}, no 2,4,5,6
R2C5 = {237}, clean-up: R1C5 = {267}

38. 1 in C6 locked in R79C6, locked for N8, clean-up: no 9 in R7C3, no 5 in R8C3, no 9 in R9C3
38a. If R9C6 = 1 => R9C7 = 6, if R7C6 = 1 => R7C7 = 7 -> no 6 in R7C7, clean-up: no 2 in R7C6

39. 14(3) cage in N9 = {149/158/239/248/257/347/356} (cannot be {167} which clashes with R79C7)

40. 4 in C5 locked in R789C5, locked for N8, clean-up: no 6 in R7C3, no 2 in R8C3, no 6 in R9C3, no 3 in R9C7

41. 8 locked in R79C4 (step 31) -> 2 locked in R79C3, locked for C3 and N7, clean-up: no 5 in R1C4, no 4 in R3C4

42. 1 in C4 locked R13C4
42a. If R3C4 = 1 => R3C3 = 5, if R1C4 = 1 => R1C3 = 6 -> no 5 in R1C3, clean-up: no 2 in R1C4

43. R3C3 = 5 (hidden single in C3), R3C4 = 1, clean-up: no 6 in R1C3, no 9 in R2C8 (step 26), no 6 in R3C7

44. 5 in N2 locked in R12C6, locked for C6, clean-up: no 6 in R5C7 (step 35), no 1 in R6C7, no 3 in R7C7, no 7 in R8C7, no 2 in R9C7

45. R7C7 = 1 (hidden single in C7), R7C6 = 7, R8C67 = [93], R9C6 = 1 (hidden single in C6), R9C7 = 6 [I should then have put R8C19 = [68] here but maybe I forgot to eliminate the 6 from R8C9 at this stage; I do my eliminations manually. It gets done in step 48.], clean-up: no 4 in R2C6, no 8 in R3C6, no 4 in R3C7, no 4 in R5C7 (step 35), no 2 in R6C5, no 3 in R7C34, no 2 in R89C5, no 3,4 in R9C3
[Having missed that R8C6 was a hidden single in step 31, it seems a bit ironic that I have now fixed that cell by using the hidden single R7C7!]

Now for several naked pairs

46. Naked pair {24} in R6C67, locked for R6, clean-up: no 3 in R6C19 (step 29)

47. Naked pair {16} in R6C19, locked for R6
[Just noticed that I could have reduced to this pair after the clean-up in step 44.]

48. Naked pair {34} in R35C6, locked for C6 -> R6C67 = [24], R2C67 = [52], R1C67 = [85], R8C19 = [68], R6C19 = [16], clean-up: no 7 in R1C5, no 2 in R4C2, no 5 in R4C8 (both step 27)
[R1C6 had been a hidden single for a while. Must get better at spotting them!]

49. Naked pair {45} in R89C5, locked for C5 and N8 -> R8C34 = [42] , R89C5 = [54], R7C34 = [28], R67C5 = [36], R12C5 = [27], R8C8 = 7, R8C2 = 1, R6C8 = 5, R6C2 = 7, R9C34 = [73], R4C34 = [37], R1C34 = [16], R2C34 = [94], R5C34 = [65], R5C6 = 4, R5C7 = 7 (step 35), R3C67 = [38]

50. Naked pair {68} in R2C28, locked for R2 -> R2C1 = 3, R2C9 = 1, R1C12 = [74]

51. Naked pair {49} in R7C89, locked for R7 and N9 -> R7C12 = [53], R9C89 = [25]

52. Naked pair {29} in R5C12, locked for R5 and N4

and the rest is naked singles

Any corrections will be welcome by PM. I know from working through J-C's and Cathy's walkthroughs that I missed some things or might have seen them earlier.

3x3::k:4864:4864:4610:4610:1540:5381:5381:4615:4615:4864:5642:4610:4610:1540:5381:5381:5392:4615:5642:5642:3860:3860:1558:1815:1815:5392:5392:5642:5148:1309:1309:1558:3104:3104:5154:5392:5148:5148:6950:6950:6950:6950:6950:5154:5154:5148:4910:2351:2351:3633:818:818:4660:5154:4910:4910:2616:2616:3633:4155:4155:4660:4660:4910:2624:5441:5441:3139:5956:5956:3398:4660:2624:2624:5441:5441:3139:5956:5956:3398:3398:

Thanks Ruud for a nice variant, after J-C had suggested that A57 wasn't as hard as usual.

There was only one walkthrough posted at the time, started by Cathy and completed by Para, so I think it deserves another one.

I spotted Cathy's UR but deliberately avoided using it, as I've commented after step 19. My breakthrough (step 22) was from a position fairly similar to the one from which Para made his breakthrough.

Here is my walkthrough for A57 V1.5.

Prelims

a) R12C5 = {15/24}
b) R3C34 = {69/78}
c) R34C5 = {15/24}
d) R3C67 = {16/25/34}, no 7,8,9
e) R4C34 = {14/23}
f) R4C67 = {39/48/57}, no 1,2,6
g) R6C34 = {18/27/36/45}, no 9
h) R67C5 = {59/68}
i) R6C67 = {12}
j) R7C34 = {19/28/37/46}, no 5
k) R7C67 = {79}
l) R89C5 = {39/48/57}, no 1,2,6
m) 19(3) cage in N1 = {289/379/469/478/568}, no 1
n) 10(3) cage in N7 = {127/136/145/235}, no 8,9

Steps resulting from Prelims
1a. Naked pair {12} in R6C67, locked for R6, clean-up: no 7,8 in R6C34
1b. Naked pair {79} in R7C67, locked for R7, clean-up: no 5 in R6C5, no 1,3 in R7C34

Now for the starting “present”
2. 45 rule on C5 1 innie R5C5 = 7, clean-up: no 5 in R4C7, no 5 in R89C5

3. Naked quad {1245} in R1234C5, locked for C5, clean-up: no 9 in R6C5, no 8 in R89C5

4. Naked pair {39} in R89C5, locked for N8 -> R7C67 = [79], clean-up: no 3 in R4C6

5. Killer pair 6,8 in R7C34 and R7C5, locked for R7

6. 45 rule on R12 2 innies R2C28 = 8 = {17/26/35}, no 4,8,9

7. 45 rule on R1234 2 innies R4C28 = 10 = {19/28/37/46}, no 5

8. 45 rule on R6789 2 innies R6C19 = 12 = {39/48/57}, no 6

9. 45 rule on R789 3 outies R6C258 = 21 = {489/678} (cannot be {579} because no 5,7,9 in R6C4), no 3,5, 8 locked for R6, clean-up: no 4 in R6C19 (step 8)

10. 45 rule on R89 2 innies R8C19 = 11 = {38/47/56}/[92], no 1, no 2 in R8C1

11. 45 rule on C34 2 innies R5C34 = 12 = {39/48}, no 1,2,5,6

12. 45 rule on C67 2 innies R5C67 = 8 = {26/35}, no 1,4,8,9

13. 45 rule on R123 3 outies R4C159 = 18
13a. Max R4C5 = 5 -> min R4C19 = 13, no 1,2,3 in R4C19

14. 45 rule on N7 4(1+3) outies R6C2 + R789C4 = 15, min R789C4 = {124} = 7 -> max R6C2 = 8

15. R6C258 (step 9) = {489/678}
15a. 9 of {489} must be in R6C8 -> no 4 in R6C8

16. 45 rule on N7 3 innies R789C3 = 1 outie R6C2 + 16
16a. Min R6C2 = 4 -> min R789C3 = 20, no 1,2, clean-up: no 8 in R7C4

17. 45 rule on N9 2 remaining innies R89C7 = 1 outie R6C8 + 5
17a. Min R6C8 = 6 -> min R89C7 = 11, no 1,2 in R89C7

18. 23(4) cage at R8C6 = {2678/3578/4568}, no 1

19. 1 in N8 only in R89C4, locked for C4, clean-up: no 4 in R4C3
19a. 21(4) cage at R8C3 must contain 1 = {1389/1479/1569/1578}, no 2
19b. 1,4 of {1479} must be in R89C4 -> no 4 in R89C3
[Note. There’s a UR move 21(4) cage cannot be {1389} because wouldn’t be able to uniquely place 3,9 for R89C3 and R89C5. However I don’t use UR moves because I prefer to solve puzzles completely rather than rely on the solution being unique to make eliminations. In my opinion using UR isn’t solving a whole puzzle.]

20. 45 rule on N1 3 innies R123C3 = 1 outie R4C1 + 4
20a. Min R123C3 = {126} = 9 -> min R4C1 = 5
20b. Max R123C3 = 13, min R3C3 = 6 -> no 6,7,8,9 in R12C3 (R123C3 cannot be {16}6)

21. 45 rule on N1 4(3+1) outies R123C4 + R4C1 = 29 -> min R123C4 = 20, no 2 in R12C4

22. 9 in N5 only in R4C6 + R5C4 -> R4C67 = [93] or R5C34 = [39] (locking cages) -> no 3 in R4C23+ R5C789, clean-up: no 2 in R4C4, no 5 in R5C6 (step 12)
[I originally saw this step as a very short forcing chain but, while checking my walkthrough, I realised that it’s actually locking cages so I’ve re-written it that way.
Thanks Ed for pointing out some more eliminations from this step.]

23. R7C4 = 2 (hidden single in C4), R7C3 = 8, R67C5 = [86], clean-up: no 7 in R3C4, no 4 in R4C7, no 4 in R5C34 (step 11), no 3 in R8C9 (step 10)

24. Naked pair {39} in R5C34, locked for R5, clean-up: no 5 in R5C7 (step 12)
24a. Naked pair {26} in R5C67, locked for R5
24b. 3 in N5 only in R456C4, locked for C4

25. X-Wing for 2 in R5C67 and R6C67, no other 2 in C67, clean-up: no 5 in R3C67
25a. 2 in N2 only in R123C5, locked for C5, clean-up: no 4 in R3C5

26. 7 in N2 only in 18(4) cage at R1C3 = {1278/1467/2367/2457}, no 9

27. R4C159 (step 13) = {189/468/567} (cannot be {459} which clashes with R4C6)
27a. 4,5 of {468/567} must be in R4C5 -> no 4,5 in R4C19
27b. 5 in R4 only in R4C56, locked for N5, clean-up: no 4 in R6C3

28. R124C3 = {124} (hidden triple in C3), 4 locked for N1 and 18(4) cage at R1C3, no 4 in R12C4
28a. 18(4) cage at R1C3 (step 26) = {1467/2457}, no 8
28b. Min R123C3 = {14}6 (cannot be {12}6 which clashes with R4C3) = 11 -> min R4C1 = 7 (step 20)

29. 45 rule on N7 3(1+2) remaining outies R6C2 + R89C4 = 13 and must contain 1 in R89C4 -> R6C2 + R89C4 = 4{18}/7{15}, no 6 in R6C2, no 4 in R89C4
29a. 4 in N8 only in R89C6, locked for C4 and 23(4) cage at R8C6, no 4 in R89C7, clean-up: no 3 in R3C7, no 8 in R4C7
29b. 23(4) cage at R8C6 (step 18) must contain 4 = {4568} (only remaining combination), no 3,7, 6 locked for C7 and N9 -> R5C67 = [62], R6C67 = [21], R3C7 = 4, R3C6 = 3, clean-up: no 3 in R6C3, no 5 in R8C1 (step 10)

30. Naked pair {34} in R46C4, locked for N5 -> R5C4 = 9, R5C3 = 3, R4C6 = 5, R4C7 = 7, R4C5 = 1, R3C5 = 5, R4C3 = 2, R4C4 = 3, R6C4 = 4, R6C3 = 5, R6C2 = 7, R6C1 = 9, R6C89 = [63], R4C1 = 8, R4C9 = 9, R4C8 = 4, R4C2 = 6, clean-up: no 2 in R2C2, no 1,2 in R2C8 (both step 6), no 6 in R3C3, no 2,4 in R8C9 (step 10)

31. Naked pair {48} in R89C6, locked for C6, N8 and 23(4) cage at R8C6, no 8 in R89C7

32. R3C4 = 8 (hidden single in N2), R3C3 = 7
32a. Naked pair {69} in R89C3, locked for N7, clean-up: no 5 in R8C9 (step 10)

33. R9C1 = 7 (hidden single in N7) -> R89C2 = 3 = {12}, locked for C2 and N7 -> R3C2 = 9, R5C12 = [14]

34. 19(3) cage in N1 = {568} (only remaining combination) -> R1C2 = 8, R12C1 = {56}, locked for C1 and N1 -> R2C2 = 3, R2C8 = 5 (step 6)

and the rest is naked singles.


Rating Comment. I'll rate my walkthrough for A57 V1.5 at Easy 1.5. My hardest step was locking cages (step 22), which I originally saw as a very short forcing chain.

3x3:d:k:3328:3328:1794:1794:2564:3077:3077:4359:4359:3328:5642:2819:2819:2564:3598:3598:3856:4359:5642:5642:3348:3348:4374:4102:4102:3856:3856:5642:4124:3348:3348:4374:1824:1824:4898:3856:4124:4124:3366:3366:4374:2601:2601:4898:4898:4124:4910:3631:3631:2353:1842:1842:5428:4898:4910:4910:5688:5688:2353:2611:2611:5428:5428:4910:5184:5688:5688:2371:2884:2884:4678:5428:5184:5184:2634:2634:2371:769:769:4678:4678:

Well done team. Am very relieved that soft hints were not needed. Some really interesting moves you found..and 1 you didn't..

Here's a simplified walk-through for Assassin 57V2X. It includes a couple of short-cuts and a really nice (fish?) move. However, the basic solution is very similiar. An alternative path would have been much longer.- though there are some good leads that could show another way. Incidentally, this puzzle has a unique solution without the diagonals (X) if you want some torture.

BTW - I found the way Andrew did the innies hypothetical in n7 (original WT step 28) very easy to follow. I like easy - so have used his format for the 1 innies move I use in n3 rather than Mikes AIC's. Not that I have anything against AIC's! But this way seemed a fraction simpler.

Please let me know of any corrections or simplifications. Thanks. Ed


Simplified walk-through for Assassin 57V2X (1-9 cannot repeat on the diagonals)

0. Cage 7(2) n12 - no 789
0a. Cage 10(2) n2 - no 5
0b. Cage 12(2) n23 -no 126
0c. Cage 11(2) n12 - no 1
0d. Cage 14(2) n23 = {59}/{68}
0e. Cage 13(4) n1245 - no 89
0f. Cage 16(2) n23 = {79}
0g. Cage 7(2) n56 @r4c6 - no 789
0h. Cage 13(2) n45 - no 123
0i. Cage 10(2) n56 - no 5
0j. Cage 14(2) n45 - = {59}/{68}
0k. Cage 9(2) n58 - no 9
0l. Cage 7(2) n56 @r6c6 - no 789
0m. Cage 10(2) n89 - no 5
0n. Cage 20(3) n7 - no 12
0o. Cage 9(2) n8 - no 9
0p. Cage 11(2) n89 - no 1
0q. Cage 10(2) n78 - no 5
0r. Cage 3(2) n89 ={12}

1. 3(2)n8 = {12}: both locked for r9
1a. -> no 7 or 8 in r8c5
1b. and 10(2)n7 = {37/46}(no 8,9)

2. 16(2)n2 = {79}: both locked for r3

3. {179} combo. in 17(3)n3 blocked by {79} at r3c7
3a. no 1 in 17(3)n3

4. 1 in n3 now only in 15(4)n3
4a. no 1 in r4c9
4b. 15(4) can't be {2346}

5. 14(2)n2 = [5/6..]
5a. -> 11(2)n1 = {29/38/47}(no 5,6) ({56}combo blocked by 14(2)n2)

6. "45" n3: outies r123c6 + r4c9 = 29
6a. max. r123c6 = {789} = 24 -> min. r4c9 = 5

7. "45" n7: outies r789c4 + r6c2 = 26
7a. max r789c4 = {789} = -> min r6c2 = 2

8. "45" r6789: innies r6c19 = 7 = h7(2)r6
8a. = {16/25/34}(no 789)

9. "45" r789: outies r6c258 = 17 = h17(3)r6
9a. must have 7 for r6
9b. = 7{19/28}(no 3456) ({467} blocked: forces both 7(2)n5 and h7(2)r6 to {25})
9c. -> no 3,4,5,6 r7c5

10. 9(2)n5 = {18/27} = [1/7,2/8..]
10a. -> 10(2) n2 = {19/37/46/}(no 2,8) ({28} blocked by 9(2)n5)

11. 17(3)n2 = {269/359/368/458/467}(no 1) ({179}/{278} blocked by 9(2)n5 step 10.)

12. {467} combo in 17(3)n2 blocked: forces the 2 9(2) cages in c5 to {18}
12a. 17(3) n2 = {269/359/368/458}(no 7)

13. "45" r12: innies r2c28 = 6 = h6(2)r2
13a. = {15/24}

14. "45" r1: outies r2c159 = 14 = h14(3)r2
14a. = {167/239/347}(no 5,8). Other combo's blocked. Here's how.
i.{149} -> 11(2)n1 = {38}: but no [8/9] left for 14(2)n2
ii. {158} clashes with [5/8] needed by 14(2)n2
iii.{248} blocked by [2/4/8] needed for 11(2)n1
iv. {257} -> by 11(2)n1 = {38}:but no [5/8] left for 14(2)n2
v. {356} blocked by [5/6] needed for 14(2)n2

A really cool move.
15. no 1 in r4c6 because of 1 required in 13(4)n1. Here's how. Maybe this is a nice fish move. Don't know which one [edit: Turbot fish! Thanks Mike].
15a. 13(4)n1 must have 1.
i.1 in r3c34 -> 1 in n3 in r2c8 -> no 1 r4c6
ii. 1 in r4c34 -> no 1 in r4c6

15b. -> no 1 r4c6, no 6 r4c7

16. 1 in D/ only in r7c3 or r2c8:
16a. cross-over -> no 1 r7c8

17. no 1 in r2c2 or r8c8 because of 1's in D/. Here's how.
17a. 1 in D/ only in r7c3 or r2c8
i. 1 in r7c3 -> 1 in n3 in r3 -> 1 required in 13(3)n1 only in r4c4 -> no 1 in r2c2 or r8c8 (on same D\ as r4c4)
ii. 1 in D/ in r2c8 -> no 1 in r2c2 or r8c8
17b. no 5 r2c8 (h6(2)r2)
17c. {189} combo blocked from 18(3)n9 (no 1 available)

[edit out redundant step]

A nice shortcut compared to the big innies move in the original WT.
Alternate step 18: better way as suggested by Mike from following post. Original step in TT below.
18. 13(3)n1 can only contain at most one of {79} due to cage sum.
18a. only other place for {79} in n1 is r2c3
18b. -> r2c3 = {79} and...
18c. ...13(3)n1 = {(7/9)..} (only other place in n1)
18d. 13(3) = {157/247/139} (no 6,8)
18e. 8 in n1 only in r3c12.
18f. 8 locked for r3 and no 8 r4c1

18. 8 locked in r3c12 because of innies n1. Here's how.
18a. "45" n1:6 innies {r2c2, r3c12} + {r123c3} = 32(6) and can only contain at most 2 of {789} from r3c12=[8] or r2c3={789}. No other place available for {789} in innies n1.
18a. -> 32(6) n1 = {135689/145679/234689/235678/245678}
18b. However, the {145679} is blocked since 7 & 9 only in r2c3.
18c. = {135689/234689/235678/245678}
18d. Each of these combinations has exactly 2 of {789} -> the 8 in r3c12 must be one of them. The [7/9] must come from r2c3.
18e. since 8 must be in r3c12: 8 locked for n1, r3 and no 8 in r4c1
18f. -> r2c3 = {79}, r2c4 = {24}

19. h6(2)r2c28 = [51] ({24} blocked by r2c4)

20. 14(2)n2 = {68}:both locked for r2
20a. no 4 or 9 in r1c5

21. split-cage 17(3)r3c1 = 8{27/36}(no 1,4,9)
21a. 7 only in r4c1 -> no 2 r4c1

22. 13(3)n1 = {139/247} = [2/3..] [edit out invalid combination]

23. deleted: redundant from new step 18

24. 8 in n2 only in c6: 8 locked for c6
24a. -> no 2 in r57c7
24b. and no 3 r8c7

25. 1 in r3 only in r3c34 in 13(4)
25a. -> no 1 r4c34
note: there are lots of neat eliminations now that there are only 2 1s in r4: but can't use it to solve the puzzle.
25c. 1 in n4 only in 16(4)
25d. {2347/2356} combo's blocked [edit step numbers]

26. {368} combo blocked from 17(3)n3 by r2c7
26a. {467} combo blocked since r12c7 = [4/6] (8 in n2 is only in r12c6)
26b. = {269/278/359/458} = [2/5..]
[edit:26a added]

Now an innies move for n3 rather than using AIC's - simpler?[edit: step 27a. because of added step 26a]
27. "45" n3: 5 innies = 27 = 27(5)
27a. = {23679/24678/34569/34578} ({23589/24579} blocked by 17(3) step 26b)
27b. r3c89 = 5..9 -> r3c89 = 9..5 [edit: Not sure what this sub-step is trying to say!]

i. r4c9 = 5 -> r3c89 = 9 = {36} only: blocked. Here's how.
r3c89 = {36} -> r123c7 = blocked: from combinations in 27(5) step 27a. can't make {279} from candidates in r2c7
-> no 5 r4c9

ii. r4c9 = 6 -> r3c89 = 8 = {35} -> from combinations in 27(5) step 27a.
-> r123c7 = {469}: blocked: can only be [469] which forces r123c6 = [887]:but 2 8's c6
................= {478} = [487]
-> innies n3 = {34578}

iii. r4c9 = 7 -> r3c89 = 7 = {25}: blocked. NO combinations with {25} in 27a.

iv. r4c9 = 7 -> r3c89 = 7 = {34}:Blocked. Here's how.
r3c89 = {34} -> from combinations in 27(5) step 27a-> r123c7 = {569} = [569]: but this forces 2 7's into c6;
...............................................................................................= {578} = [587]-> r123c6 = [769]:but this means no 8 for r12c6
-> no 7 r4c9

v. r4c9 = 8 -> r3c89 = 6 = {24}
r3c89 = {24}-> from combinations in 27(5) step 27a.-> r123c7 = {678} = [867]
-> innies n3 = {24678}

vi. r4c9 = 9 -> r3c89 = 5 = {23}
r3c89 = {23}-> from combinations in 27(5) step 27a.r123c7 = {679} = [769/967]
-> innies n3 = {23679}

28. In summary from steps 27ii,v,vi: innies n3 = {23679/24678/34578} = 7{..}
28a. 7 locked for n3 and c7
28b. no 3 r57c6, no 4 r8c6

29. in Summary from steps 27ii,v,vi:: r3c89 = {35/24/23}(no 6)
29a. split-cage 14(3)r2c8 = {239/248/356}
29b. r4c9 = {689}

30. In summary from steps 27ii,v,vi:: r123c7 = [487/867/769/967]
30a. r1c7 = {4789}(no 3,5)
30b. r1c6 = {3458}(no 7,9)

31. 17(3)n3 = {269/359/458} = [4/9..]
31a. {58} only in r1c89 -> no 4 r1c89

32. no 9 in r3c7. Here's how.
32a. 17(3)n3 = [4/9..](step 31)
i. if [4] -> r1c89 = {58} -> 12(2)n2 = [39]-> no 9 r3c7
ii. if [9] -> no 9 r3c7

33. r3c67 = [97]
33a. Cleanup: no 1 in r1c5 or r57c7, no 2 in r8c7, no 5 in r1c6,

34. 9 in c5 only in r45c5 in 17(3): 9 locked for n5
34a. no 4 r5c3, no 5 r6c3
34a. 17(3)n2 = 9{26/35} - no 4, 8

35. 9 in c4 only in r78c4 in 22(4)n78: 9 locked for 22(4)
35a. no 9 r67c3

36. 8 in D\ only in n9: 9 locked for n9
36a. cleanup - no 3 at r8c6

37. 9(2)n8 can't be {36} - blocked by [3/6..] needed for 10(2)n2
37a. no 3,6 r89c5

38. 3 in n8 now only in c4: 4 locked for c4

39. 7 in n2 only in 10(2) = {37}: both locked for n2 & c5
39a. no 9 r1c7

40. 9(2)n5 = {18} locked for c5

41. 9(2)n8 = {45}: both locked for c5, n8
41a. no 6 r9c3, no 6 r7c7 or r7c8

42. 3 in c6 only in r46c6

42a. -> either 7(2)r4c6 = [34] or 7(2)r6c6 = [34]
42b. -> 4 locked in r46c7 for c6 and n6
42c. Cleanup: no 6 in r57c6, no 8 in r1c6, no 7 in r8c6

43. r12c7 = [86], r12c6 = [48], r4c6 = 5 (HS c6), r4c7 = 2, r9c67 = [21], r8c67 = [65],

and on you go!

This was originally solved as a “tag” started by Richard, then I joined and Mike also did a bit later. I’m now trying it again by myself.

Prelims

a) R1C34 = {16/25/34}, no 7,8,9
b) R12C5 = {19/28/37/46}, no 5
c) R1C67 = {39/48/57}, no 1,2,6
d) R2C34 = {29/38/47/56}, no 1
e) R2C67 = {59/68}
f) R3C67 = {79}
g) R4C67 = {16/25/34}, no 7,8,9
h) R5C34 = {49/58/67}, no 1,2,3
i) R5C67 = {19/28/37/46}, no 5
j) R6C34 = {59/68}
k) R67C5 = {18/27/36/45}, no 9
l) R6C67 = {16/25/34}, no 7,8,9
m) R7C67 = {19/28/37/46}, no 5
n) R89C5 = {18/27/36/45}, no 9
o) R8C67 = {29/38/47/56}, no 1
p) R9C34 = {19/28/37/46}, no 5
q) R9C67 = {12}
r) 20(3) cage at R8C2 = {389/479/569/578}, no 1,2
s) 13(4) cage at R3C3 = {1237/1246/1345}, no 8,9

Steps resulting from Prelims
1a. Naked pair {79} in R3C67, locked for R3
1b. Naked pair {12} in R9C67, locked for R9, clean-up: no 7,8 in R8C5, no 8,9 in R9C34
1c. R2C34 = {29/38/47} (cannot be {56} which clashes with R2C67), no 5,6

2. 45 rule on R12 2 innies R2C28 = 6 = {15/24}

3. 45 rule on R6789 2 innies R6C19 = 7 = {16/25/34}, no 7,8,9
3a. Killer triple 4,5,6 in R6C19, R6C34 and R6C67, locked for R6, clean-up: no 3,4,5 in R7C5

4. 7 in R6 only in R6C258
4a. 45 rule on R789 3 outies R6C258 = 17 = {179/278}, no 3, clean-up: no 6 in R7C5
[Alternatively 4 in R6 only in R6C19 (step 3) = {34} or R6C67 = {34}, 3 locked for R6, locking cages]

5. 17(3) cage at R1C8 = {269/278/359/368/458/467} (cannot be {179} which clashes with R3C7), no 1

6. 45 rule on N3 4(3+1) outies R123C6 + R4C9 = 29
6a. Max R123C6 = 24 -> min R4C9 = 5
6b. Min R4C9 = 5 -> max R2C8 + R3C89 = 10, no 8 in R3C89

7. 45 rule on R1 3 outies R2C159 = 14 = {167/239/347} (cannot be {149/257} which clash with R2C28, cannot be {158/356} which clash with R2C67, cannot be {248} which clashes with R2C34), no 5,8, clean-up: no 2 in R1C5

8. 45 rule on N7 4(1+3) outies R6C2 + R789C4 = 26
8a. Max R789C4 = 24 -> min R6C2 = 2

9. R12C5 = {19/37/46} (cannot be [82] which clashes with R67C5), no 2,8

10. 17(3) cage at R3C5 = {269/359/368/458} (cannot be {179/278} which clash with R67C5, cannot be {467} because both 9(2) cages in C5 cannot be {18}), no 1,7

11. 20(3) cage at R8C2 = {389/479/569/578}
11a. 4 of {479} must be in R8C2 + R9C1 (R8C2 + R9C1 cannot be {79} which clashes with R3C7) -> no 4 in R9C2

[I really ought to have spotted this earlier.]
12. Hidden killer pair 7,9 in 13(3) cage at R1C1 and R2C3 for N1, 13(3) cannot contain both of 7,9 -> 13(3) cage must contain one of 7,9 and R2C3 = {79}, R2C4 = {24}
12a. 13(3) cage = {139/157/247} (other combinations don’t contain 7 or 9), no 6,8

13. R2C28 (step 2) = {15} (cannot be {24} which clashes with R2C4), locked for R2, clean-up: no 9 in R1C5, no 9 in R2C67
13a. R2C28 = {15}, CPE no 1,5 in R5C5 + R8C28 using the diagonals

14. Naked pair {68} in R2C67, locked for R2, clean-up: no 4 in R1C5

15. 13(3) cage at R1C1 (step 12a) = {139/247} (cannot be {157} which clashes with R2C2), no 5
15a. 2 of {247} must be in R1C12 (R2C13 cannot be [29] which clashes with R2C34 = [92]), no 2 in R2C1
[Alternatively looking at the effect of R1C12 = {47} would give R1C89 = {68} which clashes with R2C7.]

16. R2C159 (step 7) = {239/347}
16a. 2 of {239} must be in R2C9 -> no 9 in R2C9

17. 17(3) cage at R1C8 (step 5) = {269/278/359/458/467} (cannot be {368} which clashes with R2C7)
17a. 3 of {359} must be in R2C9 -> no 3 in R1C89

18. 1 in N3 only in 15(4) cage at R2C8 = {1239/1248/1257/1347/1356}
[Delete step 18a, which was my hardest step, and step 18b. I subsequently found that they aren’t needed; the eliminations are made later by simpler steps.
18a. 6 of {1356} must be in R4C9 (cannot be {136}5 which clashes 17(3) cage at R1C8 + R2C7 because 2 in N3 only in 17(3) cage or 15(4) cage), no 6 in R3C89
18b. 6,7 of {1257/1356} must be in R4C9 -> no 5 in R4C9]

19. 8 in N1 only in R3C12, locked for R3 and 22(4) cage at R2C2, no 8 in R4C1
19a. 22(4) cage = {1678/2578/3568} (cannot be {2389/3478} because R2C2 only contains 1,5, cannot be {1489} = {148}9 which clashes with 13(3) cage at R1C1), no 4,9
19b. R2C2 = {15} -> no 1,5 in R3C12 + R4C1
19c. 7 of {2578} must be in R4C1 -> no 2 in R4C1

20. 8 in N2 only in R12C6, locked for C6, clean-up: no 2 in R5C7, no 2 in R7C7, no 3 in R8C7
20a. 8 only in R1C67 = [84] or R2C67 = [86] -> 17(3) cage at R1C8 (step 17) = {269/278/359/458} (cannot be {467}, locking-out cages)
20b. 4 of {458} must be in R2C9 -> no 4 in R1C89

21. Deleted
[Thanks Ed for pointing out that this step was flawed. I’d been analysing from previous steps in my walkthrough file without looking at them properly on my worksheet diagram.]

22. 15(4) cage at R2C8 (step 18) = {1239/1248/1347/1356} (cannot be {1257} which clashes with 17(3) cage at R1C8)
22a. 17(3) cage at R1C8 (step 20a) = {269/278/359/458}, 15(4) cage = {1239/1248/1347/1356} -> combined cage 17(3) + 15(4) must contain 3, locked for N3, clean-up: no 9 in R1C6

23. 13(3) cage at R1C1 (step 15) = {139/247}
23a. R2C159 (step 16) = {239/347}
23b. 9 of {239} must be in R2C1 (cannot be [392] because 13(3) cage = {139} clashes with R12C5 = [19]), no 9 in R2C5, clean-up: no 1 in R1C5
23c. 1 in N2 only in R13C4, locked for C4

24. 9 in C5 only in 17(3) cage at R3C5, locked for N5, clean-up: no 4 in R5C3, no 1 in R5C7, no 5 in R6C3
24a. 17(3) cage (step 10) = {269/359}, no 4,8
24b. Killer pair 3,6 in R12C5 and 17(3) cage, locked for C5

25. 9 in R2 only in R2C13, locked for N1
25a. 13(3) cage at R1C1 (step 12a) = {139/247}
25b. 9 of {139} must be in R2C1 -> no 3 in R2C1

26. 9 in R1 only in R1C789, locked for N3 -> R3C7 = 7, placed for D/, R3C6 = 9, clean-up: no 5 in R1C6, no 3 in R5C6, no 3 in R7C6, no 1 in R7C7, no 4 in R8C6, no 2 in R8C7
26a. 9 in N8 only in 22(4) cage at R7C3, no 9 in R78C3

27. 17(3) cage at R1C8 (step 20a) = {269/359/458}
27a. 2 of {269} must be in R2C9 -> no 2 in R1C89

28. Hidden killer pair 8,9 in R1C67 and 17(3) cage at R1C8 for R1, 17(3) cage contains one of 8,9 -> R1C67 must contain one of 8,9 = [39/48/84], no 5,7

29. 7 in N2 only in R12C5 = {37}, locked for C5 and N2, clean-up: no 4 in R1C3, no 9 in R1C7, no 2 in R67C5, no 2 in R8C5
29a. Naked pair {18} in R67C5, locked for C5
29b. Naked pair {45} in R89C5, locked for C5 and N8, clean-up: no 6 in R7C7, no 6 in R8C7, no 6 in R9C3
29c. Naked pair {48} in R1C67, locked for R1, clean-up: no 3 in R1C3
29d. 8 in N3 only in R12C7, locked for C7, clean-up: no 2 in R5C6, no 2 in R7C6, no 3 in R8C6
29e. 3 in N8 only in R789C4, locked for C4

30. R13C4 = {15} (hidden pair in N2), locked for C4, clean-up: no 1,5 in R1C3, no 8 in R5C3, no 9 in R6C3
30a. Naked pair {68} in R6C34, locked for R6 -> R67C5 = [18], clean-up: no 6 in R4C7, no 9 in R5C7
30b. 9 in C7 only in R78C7, locked for N9

31. R46C6 = {35} (hidden pair in C6) -> R46C7 = {24}, locked for C7 and N9 -> R1C67 = [48], R2C67 = [86], R5C7 = 3, R5C6 = 7, R7C7 = 9, placed for D\, R7C6 = 1, R8C7 = 5, R8C6 = 6, R9C67 = [21], clean-up: no 6 in R5C34, no 4 in R9C3

32. Naked pair {37} in R9C34, locked for R9
32a. R8C4 = 9 (hidden single in N8)

33. R46C7 = {24} = 6, 45 rule on N6 2 remaining singles R4C9 + R6C8 = 17 = [89], R1C8 = 5, R1C9 = 9, placed for D/, R2C9 = 3 (cage sum), R1C4 = 1, R1C3 = 6

34. R6C9 = 5, R6C6 = 3, placed for D\, R6C7 = 4, R6C3 = 8, R6C4 = 6, placed for D/, R5C5 = 2, placed for both diagonals, R4C6 = 5, placed for D/, R1C1 = 7, R4C4 = 4, both placed for D\
34a. R8C8 = 8, R9C9 = 6, R9C8 = 4
34b. R4C4 = 4, R3C34 = [15], R4C3 = 3 (cage sum)

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

I'll rate my walkthrough at 1.5. I used some implied chains but no real chains; also some steps were hard to spot.

3x3::k:7168:7168:7168:2563:4868:4868:7686:7686:2568:7168:2314:2314:2563:4868:1806:1806:7686:2568:7168:3603:3604:2563:3606:3606:3606:7686:7686:3603:3603:3604:3604:3103:3103:3103:5154:2339:2852:1829:1829:5159:5159:5159:5154:5154:2339:2852:1829:4399:4399:4399:4146:4146:3124:3124:6454:6454:4664:4664:4664:3131:4146:3124:6974:1599:6454:2369:2369:5187:3131:2373:2373:6974:1599:6454:6454:5187:5187:3131:6974:6974:6974:

I seem to have done it the hard way!

In steps 1-16 candidates are entered on to grid:

1. 7(3) N4 = {124} not elsewhere in N4
-> 11(2) N4 = {38/56}

2. Innies N1: r3c23 = 8 = {17/26/35}
-> r4c1234 = 20
-> r4c89 = 13 = [94]/{58/67} -> r5c789 = 16
-> r5c9 = (12345)
-> 28(5) in N1 = {9…}

3. Innies N9: r7c78 = 9 {18/27/36/45}
-> r6c6789 = 19
-> r6c12 = 9 [81/54] -> r6c1 <> 3,6; r5c1 <> 5,8; r6c2 <> 2 -> 11(2) N4 = [38/65]
-> r5c123 = 9 = [324/342/612/621]
-> 2 locked to r5c23 not elsewhere in r5 -> r4c9 <> 7 -> r4c8 <> 6
-> 27(5) in N9 = {9…}

4. Innies N3: r23c7 = 5 = {14/23} -> r2c6+r3c56 = 16
-> 7(2) r2c67 = [61/52]/{34}
-> 30(5) in N3 must have 5

5. Innies N7: r78c3 = 14 = [95]/{68} -> r8c4 = (134)

6. 6(2) r89c1 = {15/24} -> 25(5) in N7 = {37…}

7. Outies - Innies N4: r6c3 - r4c4 = 7
-> r4c4 = (12), r6c3 = (89)
-> r3c3 <> 1,2; r3c2 <> 6,7

8. O-I N6: r6c6 - r4c7 = 3
-> r6c6 = (4…9), r4c7 = (1…6)

9. Split 14(2) r78c3 and r6c3 form Killer Pair -> 8,9 not elsewhere in c3
-> 14(3) r3c3 + r4c34 = [761/752/671/572] -> r4c3 = (567), r3c3 <> 3 -> r3c2 <> 5

10. 7 locked to r4c123, not elsewhere in r4.
-> r4c9 <> 6 -> r5c9 <> 3
-> 12(3) r4c567 = {138/156/246/345} ({129} blocked by r4c4)
-> r4c45 = (1234568)

11. 10(3) r123c4 = {136/145/235} ({127} blocked by r4c4) -> r123c4 = (1…6)

12. r34678c3 Naked Quin {56789} -> r129c3 = (1234) -> r2c2 = (5678)

13. Innies N69: r467c7 = 13
-> r1589c7 = 27 must have 9 in r159c7, not elsewhere in c7
-> 27(4) r1589c7 = {3789/4689/5679} -> r159c7 = (3…9), r8c7 = (3…8) -> r8c8 = (1…6)
-> split 13(3) r467c7 = {148/157/238/256} ({247/346} blocked by split 5(2) r23c7) -> r6c7 = (1…8)

14. 14(3) r3c2 + r4c12 = [167/176/239/293/257/275/356/365] -> r4c12 = (35679)
-> 8 locked to r6c13, not elsewhere in r6 -> r4c7 <> 5 (step 8)

15. 14(3) r3c2 + r4c12:
a) if 1{67} -> 14(3) r3c3 + r4c34 = [752]
b) if 2{39} -> 14(3) r3c3 + r4c34 = [671]
c) if 2{57} -> no options for 14(3) r3c3 + r4c34
d) if 3{56} -> 14(3) r3c3 + r4c34 = [572]
-> 14(3) r3c2 + r4c12 can’t be 2{57}
-> r4c3 <> 6

16. r4c1234 = 20 = {39}[71]/{567}2

Enter available candidates in all remaining cells for elimination
r123c1+r1c2 = (1…9); 19(3) in N2 = (2…9); r3c56 = (1…9); 10(2) in N3 = (1…4, 6…9);
r123c8+r3c9 = (1…9); 20(3) in N5 = (3…9); r6c45 = (1…7); r5c8 = (3…9); r6c89 = (1…7,9);
r7c1+r789c2 = (1…9); r7c45 = (1…9); 20(3) in N8 = (3…9); 12(3) in N8 = (1…9);
r9c8+r789c9 = (1…9)

17. 16(3) r6c67 + r7c7 = {178/259/268/349/358/367/457}
({169} blocked by split 13(3) r467c7)
Analysis: r7c7 <> 1 -> r7c8 <> 8

18. Outies r12: r3c1489 = 23

19. Outies r89: r7c1269 = 18

20. 10(3) r123c4 = {136/145/235}. Forms KP with r4c4 -> 1,2 not elsewhere in c4
-> r8c3 <> 8, r7c3 <> 6 -> NP {89} r67c3; NT {567} r348c3

21. 18(3) r7c345 = [891/981/972]/8{37}/8{46}/9{36}/9{45}
-> r7c5 <> 8,9

22. 10(3) r123c4 forms KP with r8c4 -> r5679c4 <> 3,4
-> 18(3) r7c345 = [891/981/972/873/864/963/954] -> r7c5 <> 5,6,7

23. 17(3) r6c345 = [872/863/971/962/953] ([854] blocked by r6c1 -> r6c5 <> 4,5,6,7

24. Split 13(2) r4c89 = [94]/{58}, Split 20(4) r4c1234 = {1379/2567} -> Forms KP on 5 and 9 -> r4c56 <> 5 -> 12(3) r4c567 = {138/246}

25. Innies N8: r7c45 + r8c4 = 13
Max from r7c5 + r8c4 = 7 -> r7c4 <> 5

26. 25(5) in N7 = {13579/13678/23479} - all other combos blocked by must have both 3 and 7 and can’t have both 89/56/14/25/45

27. 12(3) r789c6 = {129/138/147/156/237/246} ({345} blocked by r8c4)

28. Outies N14: r46c4 + r6c5 = 10 = {127/136/235}
-> r4c56 + r6c6 = 15 = {168/249/258/267/348/456}

29. 20(3) in N6 = {389/479/569} ({578} blocked by remaining options for 9(2) r45c9)
-> r6c89 <> 9

30. Innies N14: r346c3 = 21 = {579/678}

31. Split 16(3) r5c789 = {169/349/358}
{178} blocked by options for 20(3); {457} blocked by split 13(2) r4c89
-> r5c78 = (3689)
-> split 16(3) forms KP with r5c1 -> 20(3) r5c456 <> 3,6 -> options {479/578}
-> r6c46 <> 7 -> r4c7 <> 4

32. 10(3) r123c4 forms KP with r6c4 -> r579c4 <> 5,6 (NT {789} r579c4) -> r7c5 <> 4

33. 17(3) r6c345 = [863/953/962] -> r6c5 <> 1
-> 1 locked to r4c456 -> r4c7 <> 1 -> r6c6 <> 4
-> NQ {5689} r6c1346 -> r6c789 <> 5,6
-> r4c56 <> 6 -> r4c7 <> 2 -> r6c6 <> 5

34. 12(3) r4c567 = {18}3/{24}6 -> r4c56 <> 3 -> HS r6c5 = 3

35. 18(3) r7c345 = [972]/{89}1 -> 9 not elsewhere in r7

36. From step 8 options for O-I N6 now:
r6c6 = 6, r4c7 = 3 OR r6c9 = 9, r4c7 = 6
Either case r7c7 <> 6 -> r7c8 <> 3

37. Colouring 9s: [r6c6] =9= [r6c3] =9= [r7c3] =9= [r7c4]
r5c4 and r89c6 see both r6c6 and r7c4 -> r5c4, r89c6 <> 9
-> 9 locked to r79c4 <> r89c5 <> 9

38. Innies c4: r8c4 = 4 -> r8c3 = 5, r7c3 = 9, r6c3 = 8, r4c3 = 7, r3c3 = 6, r4c4 = 1, r6c4 = 6, r6c1 = 5, r5c1 = 6, r6c6 = 9, r4c7 = 6 …

Straightforward combinations and singles from here.

As promised in my previous message, I've now looked at the walkthroughs posted by J-C and Cathy and then checked my own one. Sorry for the delay; I only found time to look at them this evening.

J-C's step 2 was very powerful! I assume this must be what Mike called the Achilles Heel. I also liked his step 18c.

My walkthrough was more like Cathy's although there are quite a lot of differences. Here is mine.

1. R12C9 = {19/28/37/46}, no 5

2. R2C23 = {18/27/36/45}, no 9

3. R2C67 = {16/25/34}, no 7,8,9

4. R45C9 = {18/27/36/45}, no 9

5. R56C1 = {29/38/47/56}, no 1

6. R89C1 = {15/24}

7. R8C34 = {18/27/36/45}, no 9

8. R8C78 = {18/27/36/45}, no 9

9. R123C4 = {127/136/145/235}, no 8,9

10. 19(3) cage at R1C5 = {289/379/469/478/568}, no 1

11. 7(3) cage at R5C2 = {124}, locked for N4, clean-up: no 7,9 in R56C1

12. 20(3) cage at R4C8 = {389/479/569/578}, no 1,2

13. R5C456 = {389/479/569/578}, no 1,2

14. 20(3) cage at R8C5 = {389/479/569/578}, no 1,2

15. 45 rule on N1 2 innies R3C23 = 8 = {17/26/35}, no 4,8,9
15a. 9 in N1 locked in 28(5) cage

16. 45 rule on N3 2 innies R23C7 = 5 = {14/23}, clean-up: no 1,2 in R2C6
16a. 5 in N3 locked in 30(5) cage

17. 45 rule on N7 2 innies R78C3 = 14 = {68}/[95], clean-up: R8C4 = {134}

18. 45 rule on N9 2 innies R7C78 = 9 = {18/27/36/45}, no 9
18a. 9 in N9 locked in 27(5) cage

19. 45 rule on R1234 2 innies R4C89 = 13 = {58/67}/[94], no 1,2,3, clean-up: no 6,7,8 in R5C9

20. 45 rule on R6789 2 innies R6C12 = 9 = [54/81], clean-up: R5C1 = {36}

21. 2 in N4 locked in R5C23, locked for R5, clean-up: no 7 in R4C9, no 6 in R4C8 (step 19)

22. 45 rule on C123 3 innies R678C3 – 21 = 1 outie R4C4
22a. Max R678C3 = 24 -> max R4C4 = 3
22b. Min R678C3 = 22, no 3 in R6C3
22c. R678C3 = {589/679/689/789} = 9{58/67/68/78}
22d. 9 locked in R67C3, locked for C3

23. Killer quad 1/2/3/4 locked in R123C4 (contains two of 1,2,3,4, step 9), R4C4 and R8C4, locked for C4

24. Max R4C4 = 3 -> min R34C3 = 11, no 1,2, clean-up: no 6,7 in R3C2 (step 15)
24a. Max R3C3 + R4C4 = 10, no 3 in R4C3
24b. Max R4C34 = [83] = 11 but that would make 14(3) cage [383] which isn’t allowed
24c. Max R4C34 = 10 -> no 3 in R3C3 clean-up: no 5 in R3C2 (step 15)

25. Naked quint {56789} in R34678C3, locked for C3, clean-up: R2C2 = {5678}
[Later I spotted 45 rule on C12 4 outies R1259C3 = 10 = {1234}, locked for C3, which could have been done immediately after the preliminary steps.]

26. 45 rule on N4 3 outies R3C23 + R4C4 – 1 = 1 innie R6C3
26a. R3C23 = 8 (step 15) -> R6C3 – 7 = R4C4 -> R4C4 = {12}, R6C3 = {89}
26b. R678C3 (step 22c) = 9{58/68}, 8 locked for C3

27. R6C345 = {179/269/278/359/368} (cannot be {458} which clashes with R6C1, cannot be {467} because no 4,6,7 in R6C3) -> R6C4 = {567}, R6C5 = {123}

28. 45 rule on N6 3 outies R6C6 + R7C78 – 12 = 1 innie R4C7
28a. R7C78 = 9 (step18) -> R6C6 – 3 = R4C7, no 1,2,3 in R6C6, no 7,8,9 in R4C7

29. 7 in N4 locked in R4C123, locked for R4, clean-up: no 6 in R4C9 (step 19), no 3 in R5C9

30. R4C4 = {12} -> R4C567 must contain 1 or 2 = {138/156/246} (cannot be {129/345}), no 9

31. R123C4 (step 9) = {136/145/235} (cannot be {127} which clashes with R4C4), no 7
31a. Killer pair 1/2 in R123C4 and R4C4, locked for C4, clean-up: no 8 in R8C3, no 6 in R7C3 (step 17)

32. R789C6 = {129/138/147/156/237/246} (cannot be {345} which clashes with R8C4)

33. 20(3) cage at R4C8 (step 13) = {389/479/569} (cannot be {578} which clashes with R45C9) = 9{38/47/56}, 9 locked for N6
33a. 12(3) cage at R6C8 cannot be {129}

34. R48C4 = 5 = {14/23}, here’s how
34a. If R4C4 = 1 => R34C3 = {67} => R8C3 = 5 => R8C4 = 4
34b. If R4C4 = 2 => R34C3 = {57} => R8C3 = 6 => R8C3 = 3
[Later I spotted the more direct 14(3)cage at R3C3 + R8C34 = 23, R348C3 = {567} = 18 -> R48C4 = 5.]
34c. R123C4 (step 31) = {145/235} (cannot be {136} which clashes with R48C4) = 5{14/23}, no 6, 5 locked for C4 and N2, clean-up: no 2 in R2C7, no 3 in R3C7 (step 16)
34d. R6C345 (step 27) = {179/269/278/368}

35. R7C345 = {189/279/369/378/468} (cannot be {459} because no 4,5 in R7C3, cannot be {567} because no 5,6,7 in R7C3) -> R7C5 = {1234}

36. R789C6 (step 32) = {129/138/147/156/237/246}.
36a. If {147} => R8C4 = 3 => R7C5 = 2 => R7C4 = 7 (step 35) clashes with R789C6
36b. If {156} and R8C4 = 3 => 20(3) cage = {479} => R7C5 = 2 => R7C4 = 7 (step 35) clashes with 20(3) cage
36c. R789C6 = {129/138/156/237/246} and {156} combination cannot be with R8C3 = 3

37. 25(5) cage in N7 contains 3,7 and must have 1/2 and 8/9, valid combinations 37{159/168/249} (cannot be {23578} which clashes with R89C1)
37a. If {13678} => R78C3 = [95] => R8C4 = 4 => R89C1 = [24]
[If {13579} R89C1 is {24}; step 37a is a restriction, not an elimination.]

38. 45 rule on C89 4 outies R1589C7 = 27 = 9{378/468/567}, no 1,2, clean-up: no 7,8 in R8C8

39. 14(3) at R3C2 = {158/167/239/257/356}
39a. If {158} => R3C2 = 1 => R4C12 = {58} clashes with R6C1)
If {167} => R3C2 = 1 => R4C12 = {67} => R56C1 = [38] => R6C2 = 1 (step 20) clashes with R3C2
If {257} => R3C2 = 2, R3C3 = 6 (step 15), R4C12 = {57} => R4C3 = 6 clashes with R3C3
39b. 14(3) = {239/356}, no 1,7,8, clean-up: no 7 in R3C3
39c. 8 in N4 locked in R6C13, locked for R6, clean-up: no 5 in R4C7 (step 28a)

40. R4C3 = 7 (hidden single in R4/C3/N4)

41. Consider the remaining combinations for 14(3) at R3C2 = {239/356} (step 39b)
41a. If {239} => R3C2 = 2 => R4C12 = {39} => R56C1 = [65] => R6C2 = 4 (step 20) => R5C2 = 1
If {356} => R3C2 = 3 => R4C12 = {56} => R56C1 = [38] => R6C2 = 1 (step 20)
41b. 1 in N4 locked in R56C2, locked for C2

42. Killer pair 5/9 in R4C12 and R4C89, locked for R4
42a. R4C567 (step 30) = {138/246}

43. R3C567 = {149/167/248/347} (cannot be {239} which clashes with R3C2)
43a. If {149} and R3C7 = 1 => R3C56 = {49} => R2C67 = [34], R2C6 and R3C56 clash with R123C4
If {248} and R3C7 = 2 => R3C56 = {48} => R2C7 = 3 => R2C6 = 4 clashes with R3C56
43b. No 2 in R3C7, clean-up: no 3 in R2C7 (step 16), no 4 in R2C6

44. Naked pair {14} in R23C7, locked for C7 and N3, clean-up: no 6,9 in R12C9, no 4,7 in R6C6 (step 28a), no 5,8 in R7C8 (step 18)

45. 45 rule on C789 5 innies R23467C7 = 18, must contain 1,4 in R23C7 = 124{38/56}, no 7, clean-up: no 2 in R7C8 (step 18)

46. 16(3) cage at R6C6 = {259/268/358}
46a. For {268} R6C6 = 6 -> no 6 in R67C7, clean-up: no 3 in R7C8 (step 18)
46b. For {358} R6C6 = 5, R67C7 = [38] -> no 3 in R7C7, clean-up: no 6 in R7C8 (step 18)

47. R4C567 (step 42a) = {138/246}
47a. If {138} R4C7 = 3 -> no 3 in R4C56

48. 12(3) at R6C8 = {147/156/237/246/345}
48a. If {147} R7C8 cannot be 7 because R6C89 = {14} clashes with R6C2
If {156} R7C8 = 1 => R6C89 = {56} => R7C7 = 8 => R6C67 = [53/62], R6C6 clashes with R6C89
If {237} R7C8 = 7 => R6C89 = {23}, R7C7 = 2, R6C67 = [95] clashes with R6C13 which cannot both be 8
If {246} R7C8 = 4 => R6C89 = {26}, R7C7 = 5, R6C67 = [92] clashes with R6C89
If {345} R7C8 = 4 => R6C89 = {35}, R7C7 = 5, R6C67 = [92] => R6C3 = 8 => R6C1 = 5 clashes with R6C89
48b. 12(3) at R6C8 = {147}, R7C8 = {14}, R6C89 = {147} -> no 4 in R5C8, clean-up: no 2 in R7C7 (step 18)
48c. 7 locked in R6C89, locked for R6 and N6 -> R6C4 = 6 -> R6C35 = [83/92], no 1

49. Naked triple {589} in R6C136, locked for R6

50. R4C567 (step 42a) = {138/246}
50a. 6 only in R4C7 -> no 2 in R4C7

51. R6C7 = 2 (hidden single in C7) -> R6C5 = 3, R6C3 = 8, R6C1 = 5, R6C6 = 9, R7C7 = 5, R5C1 = 6, R7C8 = 4 (step 18), R4C7 = 6 (step 28a), R7C3 = 9, clean-up: no 3 in R8C8, no 1 in R89C1

52. Naked pair {17} in R6C89, locked for R6 and N6 -> R6C2 = 4, R5C23 = [12], clean-up: no 7 in R2C2, no 8 in R4C9

53. R9C4 = 9 (hidden single in C4)

54. R8C9 = 9 (hidden single in N9)

55. R4C7 = 6 -> R4C56 = {24} (step 50), locked for R4 and N5 -> R4C4 = 1, R3C3 = 6, R8C3 = 5, R8C4 = 4, R89C1 = [24], R45C9 = [54], R4C8 = 8 (hidden single in R4), clean-up: no 7 in R8C7

56. R4C12 = {39} -> R3C2 = 2

57. R123C4 = {235}, locked for N2 -> R2C67 = [61], R3C7 = 4, clean-up: no 8 in R2C2, no 3 in R2C3 -> R2C23 = [54]

58. R3C7 = 4 -> R3C56 = [91]

59. R1C8 = 6 (hidden single in R1) -> R8C78 = [81], R6C89 = [71]

60. Naked pair {35} in R3C48, locked for R3

61. Killer pair 7/8 in R12C9 and R3C9, locked for C9 and N3

62. R9C7 = 7 (hidden single in C7)

63. R3C8 = 5 (hidden single in C8) -> R3C4 = 3, R12C4 = [52]

64. R1C9 = 2 (hidden single in R1), R23C9 = [87], R3C1 = 8, R2C5 = 7

65. R9C8 = 2 (hidden single in C8)

66. R7C5 = 1 (hidden single in C5), R7C4 = 8, R5C4 = 7

67. R9C3 = 1 (hidden single in R9)

and the rest is naked singles


Now to try to find time to look at Mike's v1.5