Assassin 54

[CathyW]

Excellent puzzle

Will post walkthrough later but a UR move cracked it for me.

[mhparker]

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

[mhparker]

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...

[Para]

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

[CathyW]

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

[mhparker]

Hi all,

Since we've seen off the V1, here's an A54V2 I created. It should be harder than the original, but is definitely still solvable via logic.

Enjoy!



3x3::k:4864:4864:4864:4867:4867:4869:4869:48694105:4105:4867:4867:5901:5901486941052067:5901:59013087:61694105:61722845:69434129:6169:6169:4644:6172:6172:6943:6943:6943:4129:4129:6169:4644:4644:617269432866:4129:5941:4662:4644209638982364:5941:4662:646438985444:5444:5941:5941:4662:6464:6464:6464:5444:54443150

[sudokuEd]

here's an A54V2 I created.

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.

[Andrew]

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.

This Assassin needs a lot of advanced moves to be solved.

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