Assassin 32

[Nasenbaer]

After a hard days work on Ruud's "Last Killer 2006" with sudokued, rcbroughton and Para (I had a lot of fun there) Assassin 32 was almost relaxing.

I could have been faster, step 33 should have come a lot earlier. Oh well. The walkthrough is the way I solved it, so it's the long(er) way. Want a condensed version? Feel free to do it!

One other thing: for multiple solved cells I used square brackets because I'm sometimes lazy. (example: r34c2 = [28] would mean r3c2 = 2, r4c2 = 8)

Edit: Corrections included, thanks to Andrew.

Edit 2: Finally got around to do the rest of the corrections, sorry for the delay. Thanks again to Andrew for the input.

Walkthrough Assassin 32

1. N1 : 15(2) = {69|78}, N7 : 15(2) = {69|78} -> 6,7,8,9 locked in c1
2. N1 : 8(2) = {17|26|35} -> no 4,8,9
3. N1 : 11(2) = {29|38|47|56} -> no 1
4. N3 : 7(2) = {16|25|34} -> no 7,8,9
5. N3 : 12(2) = {39|48|57} -> no 1,2,6
6. N3 : 9(2) = {18|27|36|45} -> no 9
7. N7 : 10(2) = {19|28|37|46} -> no 5
8. N7 : 6(2) = {15|24} -> no 3,6,7,8,9
9. N9 : c7 : 11(2) = {29|38|47|56} -> no 1
10. N9 : c9 : 11(2) = {29|38|47|56} -> no 1
11. N9 : 9(2) = {18|27|36|45} -> no 9
12. r5 : 8(2) = {17|26|35} -> no 4,8,9
13. r5 : 12(2) = {39|48|57} -> no 1,2,6
14. N6 : 11(3) : no 9
15. 45 on N1 : r12c4 = 16(2) = {79} -> 7,9 locked for 27(5), N2 and c4
16. -> r1c123 11(3) = {128|146|236|245}
17. 45 on N3 : r12c6 = 6(2) = {15|24} -> no 3,6,8 -> r1c789 = 17(3)
18. 45 on N2 : r3c46 = 14(2) = {68} -> 6,8 locked for N2 and r3
19. Clean-up -> no 7,9 in r2c1, no 2 in in r2c2, no 3,5 in r2c3, no 1 in r2c7, no 4 in r2c8, no 1,3 in r2c9, no 1 in r5c3
20. N2 : 9(3) = 3{15|24} -> 3 locked for N2 and c5
21. N4 : 13(3) : r45c1 = {12345} -> r4c2 = {6789}
22. 45 on N7 : r89c4 = 6(2) = {15|24}
23. 45 on N9 : r89c6 = 13(2) = {49|58|67} -> r9c789 = 14(3)
24. 45 on N8 : r7c46 = 5(2) = {23} ({14} blocked by 6(2) in r78c4) -> 2,3 locked for N8 and r7
25. -> N8 : 6(2) = {15} -> 1,5 locked for 20(5), N8 and c4
25a. -> r9c123 = 14(3) = 15{239|248|347} -> N7 : 6(2) = {15} -> 1,5 locked for N7 and c3
26. Clean-up -> no 8 in r89c6, no 7,8,9 in r8c2, no 9 in r7c2, no 8,9 in r8c7, no 6,7 in r8c8, no 8,9 in r8c9, no 3,7 in r5c3, no 3 in r5c4
27. r5 : 8(2) = {26} -> 2,6 locked for r5
28. N8 : 21(3) = 8{49|67} -> 8 locked for N8 and c5
29. 4 locked in r46c4 for c4 and N5 -> no 8 in r5c7
30. N5 : 15(3) = {159|267}
31. N245 : 17(4) = {269|278|368|467} -> 2,4 not possible in r4c3
32. N256 : 14(3) = {158|167|248|356} -> no 6,8,9 in r4c67, no 2 in r4c7
33. 45 on r1 : r1c456 = 17(3) = [935] -> r2c46 = [71] -> r23c5 = {24} -> 2,4 locked for c5
34. N8 : 21(3) = {678} -> 6,7,8 locked for N8 and c5
35. N8 : 13(2) = {49} -> 4,9 locked for 27(5), N8 and c6 -> r9c789 = 14(3) = {158|167|257|356}
36. Clean-üp -> no 1,7 in r3c2, no 6 in r2c3, no 4 in r3c3, no 5 in r3c8, no 2 in r3c9, no 3,7 in r5c7
37. N256 : 14(3) : {158} not possible
37a. not connected: no 3,7 in r4c7 possible
38. 5 locked in r23c2 for for N1 and c2 -> 8(2) = {35} -> 3,5 locked for N1 and c2 -> no 8 in r2c3, no 7 in r7c2
new 39. N7 : 14(3) : 3 locked in r9c13 for N7 and r9, 14(3) = 3{29|47}, no 6,8 (same result as old version but clearer)
(old 39. N7 : 14(3) : {248} not possible, blocked by 10(2) -> no 6, 8 in 14(3) -> 14(3) = 3{29|47} -> 3 locked for N7 and r9)
40. 45 on r9 : r9c456 = 17(3) = [179]|[584] -> no 6 in r9c5
41. 6 locked in r9c789 for N9, r9 and 27(5) -> r9c789 = {167}, locked for r9 and N9
42. r9c456 = [584] -> r8c46 = [19] -> r78c3 = [15]
43. N9 : 14(3) = {239} -> 10(2) = {46} -> 4,6 locked for c2 -> 15(2) = {78} -> r23c1 = [69] -> r23c3 = [47] -> r23c5 = [24] -> r23c8 = [93] -> r23c7 = [52] -> r23c9 = [81] -> r23c2 = [35]
44. Clean-Up -> no 7 in r5c6, no 3 in r4c6, no 9 in r7c7, no 6 in r8c7, no 8 in r8c8, no 4,8 in r7c8, no 3 in r8c9, no 5 in r7c9
45. 9 single in N9 -> r78c9 = [92] -> r78c8 = [54] -> r78c2 = [46] -> r78c5 = [67] -> r78c7 = [83] -> r78c1 = [78]
46. N568 : 16(3) = 6{28|37} ({349} not possible) -> 6 locked for r6 -> r6c6 = {678}, r6c7 = {67}
47. N458 : 16(3) = [943] -> N568 : 16(3) = [862] -> N256 : 14(3) = [671] -> N245 : 17(3) = [836]

The rest is simple clean-up

Have fun.
Peter

[Para]

Did 32 to relax after not getting any progress in the "Last Killer 2006". Next time i'll keep a walkthrough. Just started doing this forum thing. I do need some practise in reading the clues. Takes me a while sometimes to get the explanations given.
Well let's just see what Ruud gives us next to tag on.

Para

[frank]

I used the same solution, rotated the cages 90 degrees,
and after a minor adjustment I came up with this:



3x3::k:7424:23094876:487623075922:74241796:487620613846:5922:74243091:437336071304:5922:7424:7424:4373:437336075922:5922362025992599:5418:5418:5418:6446:6446:4153:41534155:4155:6992:6992:64461335:415341553388:6992:644618565187:51871799:6992:644618563650:51872117:6992:

I found it a very slow start, and then once I got the first
few numbers the rest were easy. Surprisingly SumoCue
gets stuck early on and needs a prod.

More puzzles on my home page http://members.shaw.ca/fdkr2/

[sudokuEd]

Fun puzzle Frank - does finish surprisingly quick, but did take a while to get to there.

You've inspired me with this type of variation.

I used a slightly different solution, rotated the cages 90 degrees, and after a minor adjustment I came up with this:

3x3::k:66611553361828786980:66613088388331336980:66612575311321086980:6661:6661360733786980:698012843329:6943:6943:6943:6943:6943:6144:614436054143:4143:5954:5954:614428272086:41432360:5954:614428264389:43892615:5954:614438492587:43893126:5954:

Faster start than Frank's, but then makes you work a bit harder in the middle. Unsurprisingly, got no idea how SumoCue handles it

[frank]

Very nice sudokuEd. SumoCue managed this without much trouble.
I on the other hand, got off to a fast start on the left stack, but then
had to work hard as you predicted. Eventully I found a key to unlock
the brute, and then sailed through the rest.

Great fun!

Thanx - Frank

[Andrew]

Not my best walkthrough but I decided to post it anyway. It has a couple of interesting moves which I wouldn't have found if I had followed a more direct solving path. I've included some comments on moves that I missed first time.

I liked the way that Peter used clean-up steps so I've done something similar in my walkthrough.

Thanks Ed for a couple of comments, which I have added, and the correction of a typo; I also found another typo while going through Ed's comments.

Clean-up is used in various steps, using the combinations in steps 1 to 14 for further eliminations from these two cell cages; it is also used for the two cell sub-cages that are produced by applying the 45 rule to N1, N3, N7 and N9. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R23C1 = {69/78}

2. R23C2 = {17/26/35}, no 4,8,9

3. R23C3 = {29/38/47/56}, no 1 [Ed has pointed out that {56} is blocked. It would make R23C1 = {78} and then 5,6,7 would all be blocked from R23C2 which is impossible. Nice one Ed! I must admit that I was making the first 17 steps fairly mechanically, not looking for any interactions at that stage, before I started thinking.]

4. R23C7 = {16/25/34}, no 7,8,9

5. R23C8 = {39/48/57}, no 1,2,6

6. R23C9 = {18/27/36/45}, no 9

7. R5C34 = {17/26/35}, no 4,8,9

8. R5C67 = {39/48/57}, no 1,2,6

9 R78C1 = {69/78}, 6,7,8,9 locked in R2378C1 for C1

10. R78C2 = {19/28/37/46}, no 5

11. R78C3 = {15/24}

12. R78C7 = {29/38/47/56}, no 1

13. R78C8 = {18/27/36/45}, no 9

14. R78C9 = {29/38/47/56}, no 1

15. R123C5 = {126/135/234}, no 7,8,9

16. 11(3) cage in N6, no 9

17. R789C5 = {489/579/678}, no 1,2,3

18. 13(3) cage in N4, max R45C1 = {45} = 9, min R4C2 = 6 (4,5 already being used)

19. 45 rule on N1 3 innies R1C123 = 11 -> R12C4 = 16 = {79}, locked for C4 and N2, no 7,9 in R123C1, clean-up: no 1 in R5C3

20. 17(3) cage in N254, max R34C4 = 14 -> min R4C3 = 3

21. 45 rule on N3 3 innies R1C789 = 17 -> R12C6 = 6 = {15/24} (when I was checking this walkthrough before posting it, I noticed that this eliminates {126} from R123C5 but haven’t changed the walkthrough; that combination is eliminated in step 23)

22. 45 rule on N2 2 remaining innies R3C46 = {68}, locked for R3 and N2, clean-up: no 7,9 in R2C1, no 2 in R2C2, no 3,5 in R2C3, no 1 in R2C7, no 4 in R2C8, no 1,3 in R2C9

23. R123C5 = 3{15/24}, 3 locked for C5 [Ed has pointed out that this eliminates {258/456} from R456C5]

24. 17(3) cage in N254, R3C4 = {68}, no 1 in R4C4 (cannot have {188})

25. 14(3) cage in N256, R3C6 = {68}, min R34C6 = 7, max R4C7 = 7, similarly min R3C6 + R4C7 = 7, max R4C6 = 7

26. 45 rule on N7 3 innies R9C123 = 14 -> R89C4 = 6 = {15/24}

27. 45 rule on N9 3 innies R9C789 = 14 -> R89C6 = 13 = {49/58/67}, no 1,2,3

28. 45 rule on N8 2 remaining innies R7C46 = 5 = {23} (the only remaining 3s in N8), locked for R7 and N8, clean-up: no 7,8 in R8C2, no 4 in R8C3, no 8,9 in R8C7 and R8C9, no 6,7 in R8C8, no 4 in R89C4

29. R89C4 = {15}, locked for C4 and N8, no 1,5 in R9C123, clean-up: no 3,7 in R5C3, no 8 in R89C6

30. 8 in N8 locked in R789C5 = 8{49/67}, 8 locked for C5 [Corrected]

31. 16(3) cage in N458, R7C4 = {23}, valid combinations {268/349/358/367} ({259} not possible because 5,9 are in same cell), no other 2,3 in this cage, no 1,4 in R6C3 -> R6C3 = {56789}, R6C4 = {468}

32. 4 in C4 locked in R46C4, locked for N5, clean-up: no 8 in R5C7

33. 16(3) cage in N658, R7C6 = {23}, valid combinations {259/268/349/358/367}, no other 2,3 in this cage, no 1 in R6C67 -> R6C6 = {56789}, R6C7 = {456789}

34. 17(3) cage in N254, R3C4 = {68}, valid combinations {269/278/368/458/467}, no 4 in R4C3 -> R4C3 = {356789}, R4C4 = {23468}

35. 14(3) cage in N256, R3C6 = {68}, valid combinations {158/167/248/356}, no 6 in R4C6, no 2,6 in R4C7 -> R4C6 = {12357}, R4C7 = {13457}

36. 14(3) sub-cage R9C123 in N9, valid combinations {239/248/347}, no 6

37. R78C3 = {15} (the only remaining 5s in N7 - I should have spotted this earlier), locked for C3 and N9, clean-up: no 6 in R2C3, no 9 in R78C2, no 3 in R5C4 -> R5C34 = {26}, locked for R5

38. 45 rule on N4 3 innies R456C3 = 18, R5C3 = {26} (step 37), valid combinations {279/369} = 9{27/36}, no 6,8 in R46C3, R4C3 = {379}, R6C3 = {79}, 9 locked for C3 and N4, 8 in N4 in R456C2, locked for C2, clean-up: no 2 in R23C3, no 2 in R8C2 [Some of this was added when I was checking my walkthrough]

39. 2 in N7 locked in R9C123, locked for R9, R9C123 = 2{39/48}, no 7

40. 15(3) cage in N5 = {159} ({267} would clash with R5C4), locked for C5 and N5 -> R123C5 = {234}, locked for C5 and N2, R789C5 = {678}, locked for N8, R12C6 = {15}, no 1,5 in R1C789, R89C6 = {49}, no 4,9 in R9C789, clean-up: no 3,7 in R5C7

41. 1 in N3 locked in R3C79, clean-up: no 7 in R2C2
41a. 1 in R3C7 -> R2C7 = 6
41b. 1 in R3C9 -> R2C9 = 8
41c. {68} in R2C79, {68} in R2C1, killer pair 6/8, no other 6,8 in R2 including no 6 in R2C9, clean-up: no 2 in R3C2, no 3 in R3C3, no 4 in R3C8, no 3 in R3C9
[At this stage the obvious move is to use R23C3 = {47} to fix R23C1 but I didn’t see that first time through because I hadn’t seen that 9 could be fixed in R46C3 (step 38) when I first solved this puzzle]
41d. {368} not valid in R1C789 (it would clash with R23C7 if that was {16/34} or would clash with R23C8 if R23C7 = {25}), no 3 in R1C789

42. 9 in N9 locked in R7C79, locked for R7, clean-up: no 6 in R8C1, 2 in N9 locked in R8C79 (one of the 11(2) cages must be {29}), no 7 in R7C8

43. R23C3 = {47}, locked for C3 and N1, R3C1 = 9 (naked single), R2C1 = 6, clean-up: no 1 in R2C2 -> R23C2 = {35}, locked for C2 and N1, no 7 in R7C2 -> R78C2 = {46}, locked for C2 and N7, R78C1 = {78}, locked for N7, R9C2 = 9 (hidden single), R9C13 = {23}, locked for R9, more clean-up: no 1 in R3C7, no 3 in R2C8

44. R3C9 = 1 (hidden single in N3), R2C9 = 8, clean-up: no 3 in R8C9 [Corrected]

45. R6C3 = 9 (naked single), R4C3 = 3 (naked single), R9C123 = [392], R9C6 = 4, R8C6 = 9

46. R1C3 = 8, R1C12 = {12}, locked for R1, R1C6 = 5, R2C6 = 1

47. R1C789 = {467} (subtraction combo), locked for R1 and N3, R1C5 = 3 (naked single in R1), clean-up: no 3 in R23C7 -> R23C7 = {25}, locked for C7 and N3, more clean-up: no 7 in R5C6, no 6 in R78C7, no 9 in R7C7, R23C8 = [93], R2C4 = 7, R1C4 = 9, R3C2 = 5, R2C2 = 3, R3C7 = 2, R2C7 = 5, R3C5 = 4, R2C5 = 2, R2C3 = 4, R3C3 = 7, clean-up: no 6 in R7C8

48. R5C3 = 6, R5C4 = 2, R7C4 = 3, R6C4 = 4, R34C4 = {68}, R7C6 = 2, no 7 in R6C67 (16(3) cage cannot be {277}) -> R6C67 = {68}, locked for R6

49. 4 in N4 locked in 13(3) cage = 4{18/27}, no 5 -> R6C1 = 5 (hidden single in N4), rest of 14(3) cage = {18/27}, R6C5 = 1 (naked single), no 8 in R5C2 -> R56C2 = [72]

50. R4C2 = 8, R45C1 = {14}, R1C12 = [21]

51. R6C89 = [73], R5C8 = 8, clean-up: no 1 in R78C8, R6C7 = 6, R6C6 = 8, R3C6 = 6, R3C4 = 8, R4C4 = 6, R5C6 = 3, R5C7 = 9

and the rest is naked and hidden singles, simple elimination and cage sums

[Andrew]

Many thanks to Frank and Ed for their variants on Assassin 32. Both definitely harder than Ruud's original puzzle.

I agree with the comments already posted about them except that I'm surprised SumoCue needed a prod early on. I found the difficult bit was near the end. After getting stuck and solving Ed's one first, I came back to Frank's version a couple of days ago and managed to break it open with a contradiction move, something I hadn't been looking for earlier.

For that reason, I found it the harder of the two variants. Ed's one came out by working on the L-shaped 3-cell cages which weren't as helpful on Franks's one.

I'm posting my walkthroughs in the next two messages since nobody else has posted walkthroughs for them.

[Andrew]

Frank’s Version

Clean-up is used in various steps, using the combinations in steps 1 to 14 for further eliminations from these two cell cages; it is also used for the two cell sub-cages that are produced by applying the 45 rule. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R1C23 = {18/27/36/45}, no 9

2. R12C7 = {18/27/36/45}, no 9

3. R12C8 = {69/78}

4. R2C23 = {16/25/34}, no 7,8,9

5. R3C23 = {39/48/57}, no 1,2,6

6. R3C78 = {14/23}

7. R34C5 = {49/58/67}, no 1,2,3

8. R67C5 = {16/25/34}, no 7,8,9

9. R7C23 = {14/23}

10. R7C78 = {49/58/67}, no 1,2,3

11. R89C2 = {69/78}

12. R89C3 = {16/25/34}, no 7,8,9

13. R8C78 = {16/25/34}, no 7,8,9

14. R9C78 = {17/26/35}, no 4,8,9

15. 8(3) cage in N2 = 1{25/34}, 1 locked for N2

16. 10(3) cage in N5 = {127/136/145/235}, no 8,9

17. 21(3) cage in N6 = {489/579/678}, no 1,2,3

18. 20(3) cage in N8 = {389/479/569/578}, no 1,2

19. 45 rule on N1 3 innies R123C1 = 17 -> R4C12 = 12 = {39/48/57}, no 1,2,6

20. 45 rule on N3 3 innies R123C9 = 16 -> R4C89 = 7 = {16/25/34}, no 7,8,9

21. 45 rule on N7 3 innies R789C1 = 18-> R6C12 = 7 = {16/25/34}, no 7,8,9

22. 45 rule on N9 3 innies R789C9 = 17 -> R6C89 = 10, no 5

23. 45 rule on N4 2 innies R46C3 = 12 = {39/48/57}, no 1,2,6

24. 45 rule on N6 2 innies R46C7 = 7 = {16/25/34}, no 7,8,9

25. 45 rule on C1 3 innies R456C1 = 10 = {127/136/145/235}, no 8,9, clean-up: no 3,4 in R4C2

26. 45 rule on N8 3 innies R7C456 = 11 = {146/236} (cannot be {128/137/245} which would clash with R7C23) = 6{14/23}, no 5,7,8,9, 6 locked for R7 and N8, 1,2,3,4 locked for R7C23456, clean-up: no 2 in R6C5, no 7,9 in R7C78 -> R7C78 = {58}, locked for R7 and N9, R7C19 = {79}, 9 locked in R789C9 for C9, clean-up: no 1 in R6C8, no 2 in R8C78, no 3 in R9C78

27. R12C8 = {69} (only remaining 9s in N3), locked for C8 and N3, clean-up: no 3 in R12C7, no 1 in R4C9, no 1,4 in R6C9, no 1 in R8C7, no 2 in R9C7

28. R5C7 = 9 (hidden single in N6), R5C89 = 12 = {48/57}, no 6

29. R12C7 must contain one of 5,7,8 -> 16(3) sub-cage R123C9 must contain two of 5,7,8 = {178/358/457}, no 2

30. 16(3) cage in N658, R6C7 = {123456}, R7C6 = {12346}, only valid combinations {169/259/268/349/358/367/457} -> R6C6 = {789}

31. Only valid combinations for 17(3) sub-cage R789C9 are 9{17/26} (9 was locked in step 26), no 3,4 -> R8C78 = {34}, locked for R8, clean-up: no 3,4 in R9C3
31a. R46C7 (step 24) cannot be {34} -> R46C7 {16/25}
31b. No {349} combination in 16(3) cage in N658

32. Only cells with 3,4 in N7 are R7C23 and R9C1. R7C23 = {14/23} -> R9C1 = {34} -> no 3,4 in R6C12

33. Only valid combinations for 18(3) sub-cage R789C1 with R7C1 = {79} and R9C1 = {34} are [783/963/954] -> R8C1 = {568}

Note. After step 31 could have used killer pair 1,2 in R7C23 and R89C3 but steps 32 and 33 are more powerful and eliminate 1,2 from the remaining cells in N7.

34. R6C12 must contain 1 or 2 -> R5C123 must contain 1 or 2, valid combinations are {158/167/248/257}, no 3 [must contain one of 4,5,6 and either 7 or 8]
34a. Either R4C12 or R46C3 must be {39}

35. 3 in R5 locked in R5C456, locked for N5, R5C456 = 3{16/25}, no 4,7, clean-up: no 4 in R7C5
35a. R5C89 = {48/57} [4/7] -> R5C123 = {167/248/257} (cannot be {158})

36. 1 in N6 must be either in the R46C7 7(2) sub-cage or in the R6C89 7(2) sub-cage so one of these sub-cages must be {16}, no other 6 in N6, clean-up: no 4 in R6C8 [this elimination could alternatively have been made because both R5C89 and R6C89 require 7 or 8 so must be {48/57} and {28/37} respectively]

37. 45 rule on N2 3 innies R3C456 = 18

38. 45 rule on R3 2 innies R3C19 = 10 = {19/28/37/46}, no 5, clean-up: no 1,4,8 in R3C1

39. 1 in R3 locked in R3C789, locked for N3, clean-up: no 8 in R12C7

40. R7C7 = 8 (hidden single in C7), R7C8 = 5, clean-up: no 2 in R4C9, no 7 in R5C9

41. 8 in N3 locked in R123C9, locked for C9, clean-up: no 4 in R5C8, no 2 in R6C8, R123C9 = 8{17/35}, no 4, no 7 in R3C9, clean-up: no 3,6 in R1C9

42. 4 in C9 locked in R45C9, locked for N6, clean-up: no 3 in R4C9

43. 1 in C9 only in R389C9 so either R123C9 or R789C9 must contain {17} -> no 7 in R6C9, clean-up: no 3 in R6C8, R56C8 = {78}, locked for C8, clean-up: no 1 in R9C7

44. 20(3) cage in N8 min R8C56 = 12 -> max R9C6 = 8

Looks like it now needs a contradiction move to break it open!

45. If R12C7 = {27} => R46C7 = {16} clashes with R9C7 = {67} -> R12C7 not {27}
45a. R12C7 = {45}, locked for C7 and N3, R8C78 = [34], R3C78 = [23], R46C7 = {16}, locked for C7 and N6, R9C7 = 7 (naked single), R9C8 = 1, clean-up: no 9 in R3C23, no 8 in R8C2, no 6 in R8C3
45b. R12C9 = {78}, R3C9 = 1
45c. R7C9 = 9 (naked single), R89C9 = {26}, locked for C9
45d. R6C9 = 3, R6C8 = 7, R5C8 = 8, R5C9 = 4, R4C9 = 5, R4C8 = 2 (naked singles), clean-up: no 7 in R4C12

Wow! That was an effective key.

46. R7C1 = 7 (naked single), R89C1 = [83] (step 33), clean-up: no 9 in R4C2, no 2 in R7C23
46a. R7C23 = {14}, locked for R7 and N7, R89C2 = {69}, locked for C2 and N7, R89C3 = {25}, locked for C3, clean-up: no 4,7 in R1C2, no 3 in R1C3, no 2,5 in R2C2, no 1 in R2C3, no 7 in R3C2, no 6 in R6C5

47. R4C12 = [48], R3C1 = 9 (naked singles), R12C1 = {26} (only valid combination), locked for C1 and N1, clean-up: no 3 in R1C2, no 7 in R1C3, no 1 in R2C2, no 1,5 in R6C2
47a. R2C23 = {34}, locked for R2 and N1, clean-up: no 5 in R1C2, no 5 in R1C7, no 8 in R3C23
47b. R12C7 = [45], R3C23 = [57] (naked singles)
47c. R2C56 = {12}, locked for R2 and N2, R1C6 = 5
47d. R3C456 = {468}, 19(3) cage in N2 = {379}

and the rest is naked and hidden singles, simple elimination and cage sums

[Andrew]

Ed’s Version

Clean-up is used in various steps, using the combinations in steps 1 to 16 for further eliminations from these two cell cages; it is also used for the two cell sub-cages that are produced by applying the 45 rule. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R1C23 = {15/24}

2. R1C78 = {29/38/47/56}, no 1

3. R2C23 = {39/48/57}, no 1,2,6

4. R2C78 = {39/48/57}, no 1,2,6

5. R3C23 = 10(2), no 5

6. R34C5 = {39/48/57}, no 1,2,6

7. R3C78 = {17/26/35}, no 4,8,9

8. R5C12 = {14/23}

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

10. R67C5 = {17/26/35}, no 4,8,9

11. R7C23 = {29/38/47/56}, no 1

12. R7C78 = {18/27/36/45}, no 9

13. R8C23 = {29/38/47/56}, no 1

14. R8C78 = 10(2), no 5

15. R9C23 = {69/78}

16. R9C78 = {39/48/57}, no 1,2,6

17. 10(3) cage in N8 = {127/136/145/235}, no 8,9

18. 45 rule on N1 3 innies R123C1 = 17 -> R4C12 = 9 = {18/27/36/45}, no 9

19. 45 rule on N3 3 innies R123C9 = 14 -> R4C89 = 13 = {49/58/67}

20. 45 rule on N7 3 innies R789C1 = 8 = 1{25/34}, 1 locked for C1, clean-up: no 8 in R4C2, no 4 in R5C2, R6C12 = 16 = {79}, locked for R6 and N4, clean-up: no 2 in R4C12, no 4,6 in R5C4, no 1 in R7C5

21. 45 rule on N9 3 innies R789C9 = 14 -> R6C89 = 9 = {18/36/45}, no 2

22. 45 rule on C1 3 innies R456C1 = 20, only valid combination = [839] -> R456C2 = [127], R456C3 = {456}, locked for C3, clean-up: no 8 in R2C2, no 4,6 in R3C2, no 3,8,9 in R3C3, no 4 in R3C5, no 5 in R4C89, no 5 in R5C4, no 5,6 in R7C2 and R8C2, no 9 in R7C3 and R8C3, no 9 in R9C2, no 8 in R9C3

23. R789C1 = {125}, locked for C1 and N7, clean-up: no 9 in R7C2 and R8C2

24. R9C2 = 6, R9C3 = 9 (hidden singles in N7), clean-up: no 3 in R2C2, no 3 in R9C78
24a. R9C78 = {48/57} -> no {45} in R7C78

25. R123C1 = {467}, locked for N1, clean-up: R1C2 = 5, R1C3 = 1, no 6 in R1C67, no 8 in R2C3, R2C23 = [93], R3C23 = [82], clean-up: no 6 in R3C78, no 4 in R4C5

26. 6 in N3 locked in R123C9, locked for C9, no 6 in R4C8, clean-up: no 7 in R4C89, no 3 in R6C8, R123C9 = 6{17/35}, no 2,4,8,9, R4C89 = {49}, locked for R4 and N6, clean-up: no 3 in R3C5, no 5 in R6C89

27. R123C9 = 6{17/35} (step 26) and R3C78 = {17/35}, 3,5,7 locked for N3 -> R2C78 = {48}, locked for R2 and N3 -> R1C78 = {29}, locked for R1

28. 45 rule on N2 3 innies R3C456 = 16, 9 locked in R3C456 = 9{16/34}, no 5,7 -> R3C5 = 9, R4C5 = 3, clean-up: no 5 in R67C5
28a. 5 in R3 locked in R3C789, locked for N3

29. R3C46 = {16/34}, R3C78 = {17/35}, killer pair 1,3 for R3
29a. Only 1 in R123C9 is in R2C9, no 7 in R2C9

30. 45 rule on N8 3 innies R7C456 = 18, valid combinations with R7C5 = {267} are {279/369/378/468/567}, no 1 (note that {279} can use either the 2 or the 7 in R7C5)

31. 2 in N6 locked in R46C7, locked for C7 -> R1C7 = 9, R1C8 = 2, clean-up: no 7 in R7C78, no 8 in R8C78, no 1 in R8C8

32. 2 in C9 locked in R789C9 = 2{39/48/57}, no 1

33. 45 rule on C9 3 innies R456C9 = 17, valid combinations are [458/953/971] -> R5C9 = {57}

34. R123C9 = 6{17/35} (step 26), if R123C9 is not {167} then R456C9 must be [971] -> no 7 in R789C9, clean-up: no 5 in R789C9

35. R9C78 = {57} (only remaining 5s in N9), locked for R9 and N9, clean-up: no 3 in R8C78

36. 10(3) cage in N8 = {136/145/235} (step 17), ({127} now not valid because it would clash with R9C1), no 7, 1,2,3,4 are locked in R9C45 for the 10(3) cage -> R8C4 = {56}
36a. R9C45 contain 1 or 2, R9C1 = {12}, killer pair 1,2 for R9

37. 17(3) cage in N8 valid combinations with R9C6 = {348} are {278/359/368/458/467}, no 1

38. 1 in N8 locked in R9C45, locked for R9, 10(3) cage = 1{36/45}, no 2

39. R9C1 = 2 (naked single in R9)

40. 14(3) cage in N254 valid combinations are {167/356} (cannot have {257} because 2,7 both in same cell) = 6{17/35}, no 2,4, clean-up: no 3 in R3C6

41. 2 in R4 locked in 13(3) cage in N256 = 2{47/56}, no 1, R3C6 = {46} -> R4C67 = {257}, clean-up: no 6 in R3C4

42. 6 in R4 locked in 14(3) cage in N254, R3C4 = {13} -> R4C34 = {567}

43. 14(3) and 15(3) cages in N2 must each have 7 or 8

44. Valid combinations for 14(3) cage in N2 {167/248/347/356} (cannot have {158} because 1,5 in the same cell, cannot have {257} because 2,5 in the same cell), no 6 in R2C4
44a. 14(3) cage must have 4 or 6, R3C6 = {46}, killer pair 4,6 for N2

45. Valid combinations for 15(3) cage in N2 {258/357} = 5{28/37}, no 7 in R1C6, no 1 in R2C56, 5 locked in R2C56, locked for R2
45a. {356} not now valid for 14(3) cage in N2

46. 1 in N2 locked in R23C4, locked for C4 -> R9C5 = 1 (hidden single in N8), clean-up: no 7 in R7C5, R67C5 = {26}, locked for C5
46a. From combinations for 14(3) cage in N2 (step 44), no 7 in R1C4
46b. From combinations for 17(3) cage in N8 (step 37), no 3,7 in R8C6

47. Combinations for 16(3) cage in N658 {169/178/259/268/349/358/367/457} -> no 4 in R7C6

48. Valid combinations for 14(3) cage in N458 with R6C3 = {456} are {248/257/356} (cannot be {347} because 3,7 in the same cell), no 9, no 4 in R6C4, no 4,5,6 in R7C4
48a. R5C4 = 9 (hidden single in C4), R4C4 = 4
48b. Valid combinations for 14(3) cage in N458 with R6C3 = {56} are {257/356} = 5{27/36}, no 8, no 2 in R7C4, 5 locked in R6C34, locked for R6

49. R1C4 = 8 (hidden single in C4) -> only valid combination for 14(3) cage in N2 = [842]

50. R1C6 = 3 (naked single), R2C56 = {57}, locked for R2, R3C4 = 1, R3C6 = 6, clean-up: no 7 in R3C78 = {35}, locked for R3

51. R2C1 = 6, R1C1 = 7, R3C1 = 4

52. R123C9 = [617], clean-up: no 8 in R6C8

53. R5C9 = 5, 6 in R5 locked in R5C78, locked for N6 -> R6C89 = [18], R5C78 = {67}, locked for R5 and N6 -> R5C56 = [81], clean-up: no 8 in R7C7, R4C7 = 2, R4C6 = 5, R6C7 = 3

and the rest is naked and hidden singles, simple elimination and cage sums