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Vortex Killer
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Andrew
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PostPosted: Sat Sep 08, 2007 9:09 pm    Post subject: Reply with quote

mhparker wrote:
Many people intuitively prefer this loop approach (even though they may be unaware that it's a Nice Loop), because they find it more natural to start on a true premise than on a false premise, as is the case with AICs.

I assume you mean that when people look for contradictions, which is what many of these advanced techniques appear to be, they prefer to start with "if cell/group equals" rather than "if cell/group doesn't equal".

At the current time AICs, ALSs, Nice Loops, Eureka notation etc. are just names to me. I can just about do very simple cases of "simple colouring". I was pleased that I managed to find an example of the simplest type of that yesterday in a jigsaw sudoku; one link one way and two links by the other path.

mhparker wrote:
It's well worth getting into the habit of trying to "normalize" contradiction moves in this way, if for no other reason than the fact that the WT sounds more impressive and less like T&E when using the correct lingo!

Hope this post has helped and not been totally confusing!

Not totally confusing. Even though I don't know these techniques I could understand part of what was being said until Eureka notation was used when I was totally lost.

I've got no problem with people using these techniques but they must also give the outcomes clearly so that those of us that don't yet know them can still understand what has been done and possibly try to work out the outcome using techniques that we understand.

Also my personal preference would be that if a particular stage of a puzzle can be solved using "normal" techniques that we all understand, then that should be done. Of course I'm not expecting that everyone will do that.

I do intend to find time to study advanced techniques but it's a matter of finding the time. I have to submit 3 daily and 3 weekly puzzles on www.sudoku.org.uk plus there are Assassins and forum variants and there is life outside sudokus.
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Andrew
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PostPosted: Mon Sep 10, 2007 5:57 am    Post subject: Reply with quote

mhparker wrote:
Vortex Lite Est. rating 1.25

and

Come on guys, the Vortex Lite is really no more difficult than a typical Assassin!

I'll agree with that although at one time I was struggling a bit. Then I had several restarts when I saw things that I ought to have spotted earlier and it became easier.

As Ruud said, a fun puzzle. Thanks Mike!

Here is my walkthrough for Vortex Lite. No "advanced techniques". Step 34 was combination crunching but not heavy stuff, the rest was normal killer moves.

Thanks Mike for the feedback on step 34. I've also done a bit of minor editing.

1. R1C45 = {17/26/35}, no 4,8,9

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

3. R56C1 = {16/25/34}, no 7,8,9

4. R9C56 = {69/78}

5. 20(3) cage in N2 = {389/479/569/578}, no 1,2

6. 11(3) cage at R4C7 = {128/137/146/236/245}, no 9

7. 21(3) cage at R5C3 = {489/579/678}, no 1,2,3

8. 19(3) cage at R6C5 = {289/379/469/478/568}, no 1

9. 22(3) cage at R7C7 = 9{58/67}, 9 locked for N9 and D\

10. 28(4) cage at R2C9 = {4789/5689} = 89{47/56}
10a. CPE no 8,9 in R12C8

11. 13(4) cage at R8C6 = {1237/1246/1345} = 1{237/246/345}, no 8,9
11a. CPE no 1 in R8C9

After the preliminary steps the first placement

12. 45 rule on D/ + D\, R5C5 counts toward both diagonals, total 89 -> R5C5 = 1, locked for D/ and D\, clean-up: no 7 in R1C4, no 6 in R6C1

13. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
13a.1 of {128/137} must be in R4C7 -> no 7,8 in R4C7

14. 45 rule on D/ 2 outies R4C4 + R6C6 = 7 = {25/34}

15. 45 rule on D\ 2 outies R4C6 + R6C4 = 15 = {69/78}

16. 12(3) cage in N3 = {237/246/345}, no 8,9

17. 15(3) cage in N1 = {258/267/348} (cannot be {357/456} which clash with R4C4 + R6C6)

18. 45 rule on N1 2 outies R2C4 + R4C1 = 10 = {19/28/37/46/55}

19. 45 rule on N3 2 outies R1C6 + R4C8 = 13 = {49/58/67}, no 1,2,3 in R1C6

20. R8C6 = 1 (hidden single in C6)
20a. 1 in N9 locked in R7C89, locked for R7 and 15(4) cage -> no 1 in R6C9
20b. R7C9 = 1 (hidden single in C9)
[Itís neat the way that fixing R8C6 leads directly to fixing R7C9!]

21. 45 rule on N7 2 outies R6C2 + R9C4 = 4 = [13/22]

22. 45 rule on N9 1 remaining outie R6C9 = 4, clean-up: no 9 in R1C6 (step 19), no 3 in R4C4 (step 14), no 7 in R45C9, no 3 in R5C1
22a. 15(4) cage at R6C9 = 14{28/37}, no 5,6
22b. 4 in N9 locked in 13(4) cage at R8C6 = 14{26/35}, no 7
[Alternatively killer pair 7,8 in R7C8 + R8C9 and 22(3) cage, locked for N9]

23. 28(4) cage at R2C9 (step 10) = {4789/5689}
23a. 4 of {4789} must be in R3C8 -> no 7 in R3C8

24. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
24a. 4 of {245} must be in R5C6 -> no 5 in R5C6

25. 17(3) cage in N7 = {269/278/359/458} (cannot be {368/467} which clash with R4C6 + R6C4)

26. 45 rule on R123 3 outies R4C158 = 20 = {389/479/569/578}, no 1,2, clean-up: no 8,9 in R2C4 (step 18)

27. 45 rule on R789 2 remaining outies R6C25 = 9, R6C2 = {12} -> R6C5 = {78}

28. 45 rule on C123 3 outies R259C4 = 11 = {128/137/236/245} (cannot be {146} because R9C4 only contains 2,3), no 9
28a. 7 of {137} must be in R5C4 -> no 7 in R2C4
28b. 6 of {236} must be in R5C4 -> no 6 in R2C4
28c. Clean-up: no 3,4 in R4C1 (step 18)

29. 45 rule on C789 2 remaining outies R15C6 = 9 = [54/63/72], clean-up: no 5,9 in R4C8 (step 19)

30. 28(4) cage at R2C9 (step 10) = 89{47/56}, 9 locked in R2C9 + R3C89 for N3
30a. 4/5 now in N3 -> 12(3) cage in N3 (step 16) = {237/246} (cannot be {345} which clashes with 28(4) cage) = 2{37/46}, no 5, 2 locked for N3 and D/

31. Killer pair 6,7 in 12(3) cage in N3 and R4C6 + R6C4, locked for D/

32. 5 on D/ locked in 17(3) cage, locked for N7
32a. 17(3) cage (step 25) = {359/458}
[Step 31 wasnít actually needed but I saw it before I checked for an earlier step with the 17(3) cage.]

33. 45 rule on C1234 6 remaining outies R1348C5 + R46C6 = 25, min R46C6 = 8 -> max R1348C5 = 17, no 9
33a. R1348C5 = 14, 15, 16 or 17 = {2345/2346/2347/2348/2356/2357/2456}, 2 locked for C5

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. If {1467} with R1C6 = 6 => R1C78 + R2C7 = {147} clashes with 12(3) cage
34c. If {1467} with R1C6 = 7 => R1C78 + R2C7 = {146} => R2C9 + R3C89 => {589} => R4C8 = 6 => no 6 in R1C8
34d. -> 18(4) cage at R1C6 = {1458/1467} = 14{58/67}, no 3, 4 locked for N3, no 6 in R1C6 (steps 34a and 34b) no 6 in R1C8 (step 34c), clean-up: no 6 in 12(3) cage (step 30a), no 7 in R4C8 (step 19) , no 3 in R5C6 (step 29)
See discussion of this after the walkthrough

35. Naked triple {237} in 12(3) cage in N3, locked for D/ and N3, clean-up: no 8 in R4C6 + R6C4 (step 15), no 9 in 17(3) cage in N7 (step 32a)

36. 11(3) cage at R4C7 (step 6) = {128/146/236/245} (cannot be {137} because R5C6 only contains 2,4), no 7
36a. 7 in N6 locked in 15(3) cage = 7{26/35}, no 1,8,9

37. R4C7 = 1 (hidden single in N6)
37a. R5C67 = 10 = [28/46]

38. R1C8 = 1 (hidden single in C8), clean-up: no 7 in R1C5
38a. R9C8 = 4 (hidden single in C8)

39. Naked pair {68} in R4C8 + R5C7, locked for N6, clean-up: no 3,5 in R45C9

40. Naked pair {29} in R45C9, locked for C9 and N6, clean-up: no 8 in R7C8 (step 22a)

41. R3C8 = 9 (hidden single in N3)

42. R7C7 = 9 (hidden single in N7)

43. 28(4) cage at R2C9 = {5689}
43a. 5 locked in R23C9, locked for C9 and N3, clean-up: no 8 in R8C8 (step 9)

44. R4C8 = 8 (hidden single in C8), R5C7 = 6, R5C6 = 4, R1C6 = 5 (step 29), clean-up: no 3 in R1C45, no 2 in R89C7 (step 22b), no 2 in R4C4, no 3 in R6C6 (both step 14) -> R4C4 = 5, R6C6 = 2, 2,5 locked for D\, R6C2 = 1, no 5 in R5C1, no 3 in R6C1 -> R56C1 = [25], R45C9 = [29] , clean-up: no 8 in R9C9 (step 9) [Clean-up edited]
44a. R5C8 = 5 (hidden single in R5)
44b. R9C3 = 1 (hidden single in R9)

45. R3C7 = 2, R6C7 = 7 (hidden singles in C7), R6C5 = 8, R6C8 = 3, clean-up: no 7 in R9C6

46. R2C8 = 7, R7C8 = 2, R8C8 = 6, R1C9 = 3, R89C9 = [87], 6,7 locked for D\, clean-up: no 8 in R9C6

47. R5C4 = 7, R5C23 = [38], R4C5 = 3 (hidden single in N5)

48. R4C5 = 3 -> R3C45 = 11 = [47]
48a. R7C6 = 7 (hidden single in C6), R7C5 = 4, R7C3 = 5, R8C2 = 4, R9C1 = 8
48b. R8C5 = 5 (hidden single in C5), R89C7 = [35]

49. R7C4 = 8 (hidden single in C4), R8C4 = 2

and the rest is naked singles and a cage sum

4 9 7 6 2 5 8 1 3
6 8 2 1 9 3 4 7 5
1 5 3 4 7 8 2 9 6
9 7 4 5 3 6 1 8 2
2 3 8 7 1 4 6 5 9
5 1 6 9 8 2 7 3 4
3 6 5 8 4 7 9 2 1
7 4 9 2 5 1 3 6 8
8 2 1 3 6 9 5 4 7


Step 34. Mike commented that the logic was too complicated and should be broken down into smaller steps, for example

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1458}, 5 must be in R1C6 -> no 5 in R1C78+R2C7
34b. -> 5 in N3 locked in 28(4) at R2C9 = {5689} (no 4,7) -> no 6 in R12C8.
34c. Hidden killer pair on {47} in N3, as follows. 12(3) at R1C9 contains 1 of {47}. Only other place for {47} in N3 = 18(4) at R1C6 = {(4/7)..} = {1458/1467} (no 3) -> 4 locked in R1C78+R2C7 for N3
34d. Clean-up: no 6 in R1C6 (step 19), no 6 in 12(3) cage (step 30a), no 3 in R5C6 (step 29)

There is a reason why I missed his step 34a, which I ought to have seen. After typing step 34 with "must have 1" I used Ruud's combination calculator and forgot to enter that so I got {1368/1458/1467/3456}. When I was part way through my sub-steps I realised why {3456} wasn't allowed but forgot to go back and look at {1458} again, having already accepted that it was valid.

Mike's step 34c is a difficult one to find. You either spot it immediately or have to work hard to find it.

One can combine thoughts from both sets of sub-steps and come up with

34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. 18(4) cage at R1C6 = {1458/1467}, no 3, 4 locked for N3, clean-up: no 6 in 12(3) cage (step 30a)
34c. 28(4) at R2C9 (step 10) = {5689} (only remaining combination), no 7 -> CPE no 6 in R1C8
34d. Clean-up: no 6 in R1C6 (step 19), no 3 in R5C6 (step 29)

I'll leave it to you to decide whether Mike's step 34c or my step 34a is the easier one to find.


Last edited by Andrew on Wed Sep 19, 2007 9:39 pm; edited 1 time in total
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CathyW
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PostPosted: Mon Sep 10, 2007 8:55 am    Post subject: Reply with quote

mhparker wrote:
Hope this post has helped and not been totally confusing!


Yes - thanks Mike. It all makes sense. Need some more puzzles to try putting it into practice!!
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mhparker
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PostPosted: Mon Sep 10, 2007 10:56 am    Post subject: Reply with quote

CathyW wrote:
Need some more puzzles to try putting it into practice!!

Although I have already managed to come up with an interesting (and still do-able!) A67 variant, I won't be posting it on the forum, because Para has indicated that he's got something up his sleeve for us this week. He told me last week via PM that he had created a promising one, but with only one problem: (even) he couldn't solve it!

Hopefully, he'll be able to tweak it for the better before publishing it, otherwise it looks like we're in for some real evil here! Twisted Evil
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Para
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PostPosted: Thu Sep 13, 2007 8:46 pm    Post subject: Reply with quote

Hi all

I have to admit, this isn't the nicest walk-through i have ever written, but couldn't find the break-through move i used before. But here's a walk-through, i hope to improve on it though. Someone else have any thought, feel free to share.

Walk-through Vortex Killer

1. 12(4) at R1C2 = {1236/1245}: no 7,8,9; 1,2 locked within cage: R2C12: no 1,2

2. R1C45 = {79} -->> locked for R1 and N2

3. 20(3) at R1C9 = {389/479/569/578}: no 1,2

4. 28(4) at R2C1 = {4789/5689}: no 1,2,3; 8 locked within cage: R1C1: no 8

5. 11(3) at R2C5 = {128/146/236/245}

6. 9(3) at R3C4 = {126/135/234}: no 7,8,9

7. R45C9 = {59/68}: no 1,2,3,4,7

8. R56C1 = {12} -->> locked for C1 and N4

9. 15(3) at R4C2 = {348/357/456}: no 9

10. 19(3) at R5C8 = {379/478}: {289/469/568} blocked by R45C9: no 1,2,5,6; 7 locked for N6

11. 11(3) at R7C7 = {128/137/146/236/245}: no 9

12. R9C56 = {29/38/47/56}: no 1

13. 45-overlap on both diagonals: Total sum 5 cages = 86 -->> R5C5 = 90-86 = 4
13a. 11(3) at R7C7 = {128/137/236}: no 5
13b. Clean up: R9C6: no 7

14. 45 on D/: 2 outies: R4C4 + R6C6 = 14 = {59/68}: no 1,2,3,7

15. 45 on D\: 2 outies: R4C6 + R6C4 = 9 = {18/27/36}: no 5,9

16. 45 on N1: 2 outies: R2C4 + R4C1 = 11 = [29/38/47/56/65]: R2C4: no 1; R4C1: no 4
16a. 1 in 12(4) cage at R1C2 locked within R1C23 + R2C3 -->> locked for N1

17. 45 on N3: 2 outies: R1C6 + R4C8 = 11 = [29]/{38/56}: no 1,4 R4C8: no 2

18. 45 on N7: 2 outies: R6C2 + R9C4 = 11 = [92]/{38/47/56}: R9C4: no 1,9

19. 45 on N9: 2 outies: R6C9 + R8C6 = 11 = {29/38/56}/[47]: no 1; R8C6: no 4

20. 1 in D\ locked within 11(3) cage at R7C7: 11(3) = {128/137}: no 6; 1 locked for N9

21. 45 on R123: 3 outies: R4C158 = 13 = [526/625/715/823/913]: R4C5: no 3,5,6; R4C8: no 8,9
21a. Clean up: R1C6: no 2,3(step 17)

22. 45 on C123: R259C4 = 10 = [217/316/415/514/613]/{235}: R5C5: no 6,7,8,9; R9C4: no 8
22a. Clean up: R6C2: no 3(step 18)

Missed earlier
23. Killer Pair {89} in 14(2) at R4C9 and 19(3) at R5C8 -->> locked for N6
23a. Clean up: R8C6: no 2,3

24. 45 on C789: 3 outies: R158C6 = 22 = {589/679}: no 1,2,3,; 9 locked for C6: When {679}, R1C6 = 6: R58C6: no 6
24a. Clean up: R4C4: no 5(step 14), R6C9: no 5(step 19)

25. 1 locked in N6 for C7 and 12(3) cage at R4C7
25a. 12(3) = [9]{12}/[8]{13}/[7][41]/[5]{16}: R45C7: no 5

26. R158C6 = {589}/[6]{79} -->> R6C6 = [6]/{58} -->> 6 locked in R16C6 for C6; R1C1: no 6
26a. Clean up: R6C4: no 3; R9C5: no 2,5

27. 16(3) at R1C1 = [592]/[3]{67}: {358} blocked by R4C4 + R6C6(step 14): R2C2: no 3,5,8; R3C3: no 3,5,8,9

28. 12(4) at R1C2 = {1236/1245}, canít have both {26} within N1 because of 16(3) cage at R1C1 -->> R2C4: no 3
28a. Clean up: R4C1: no 8

29. R259C4 = 10 = [217/613/415/514]/{235}: R9C4: no 6
29a. Clean up: R6C2: no 5

30. R259C4 = 10 = [217/613/415/514]/{235}: When R5C4 = 1 -->> R4C5 = 2, so either R4C5 = 2 or R259C4 = {235}: R3C4 + R6C4: no 2(sees all 4 cells)
30a. Clean up: R4C6: no 7

31. 7 in R4 locked for N4
31a. Clean up: R9C4: no 4

32. 7 in R4 locked within R4C123: R4C1 = 7 or 15(3) at R4C2 = {357} -->> R4C1: no 5
32a. Clean up: R4C2: no 6

33. 15(3) at R5C3 = [1]{59}/[1]{68}/[2][94]/[3][84]/[5][64]: [2]{58} blocked by 15(3) cage at R4C2: R56C3: no 3

34. 3 in N4 locked within 15(3) cage at R4C2: 15(3) = {348/357}: no 6

35. R4C158 = 13 = [625/715/913]: R4C8: no 6; Clean up: R1C6: no 5

36. Killer Triple {358} in R4C8 + R45C9 + 19(3) at R5C8 -->> locked for N6
36a. Clean up: R5C6: no 8; R8C6: no 8

37. 45 on N5: 4 innies: R46C5 + R5C46 = 18: Max R5C46 = 14: Min R46C5 = 4: R6C5: no 1,2

I know this is not nice, I will find something else later.
38. 45 on R6789: 2 outies and 3 innies: R5C18 + 6 = R6C346: Analysis: R6C4 = 7 -->> R6C2 = 4 -->> R9C4 = 7: R6C4: no 7
38a. Clean up: R4C6: no 2

39. 27(5) at R4C4 = 4{1589/3569} -->>R4C4 = 9; R6C6 = 5
39a. R1C45 = [79]; R1C1 = 3; R5C6 = 7; R8C6 = 9; R1C6 = 6(step 24)

40. R45C7 = [41](last combo)
40a. R56C1 = [21]; R5C4 = 3; R4C5 = 2(hidden); R4C6 = 1(hidden)
40b. R6C45 = [86]; R6C9 = 2; R1C9 = 5; R4C8 = 5(hidden)

41. R56C3 = [84](last combo in 15(3) at R5C3)

And the rest is all hidden and naked singles.

greetings

Para


Last edited by Para on Thu Nov 01, 2007 11:44 pm; edited 1 time in total
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Andrew
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PostPosted: Tue Sep 25, 2007 11:47 pm    Post subject: Reply with quote

In the past few weeks I've been working on Assassin 66V1.5 and Vortex Killer as "background" jobs.

Vortex Killer was definitely the easier and more enjoyable of these two puzzles. Maybe it was easier for those of us who solved SKX4 earlier this year.

I would rate Vortex Killer between 1.5 and 1.75. Ed currently has Vortex Killer as 1.75 and A66V.15 as 1.5. Maybe he should consider swapping those ratings?

Thanks Mike for some excellent comments.

Here is my walkthrough for Vortex Killer

1. R1C45 = {79}, locked for R1 and N2

2. R45C9 = {59/68}

3. R56C1 = {12}, locked for C1 and N4

4. R9C56 = {29/38/47/56}, no 1

5. 9(3) cage at R3C4 = {126/135/234}, no 7,8,9

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

7. 15(3) cage at R4C2 = {348/357/456}, no 9

8. 19(3) cage in N6 = {379/469/478/568} (cannot be {289} which clashes with R45C9), no 1,2
[Para also had {469/568} clash with R45C9, giving 7 locked for N6. Missed them! Fortunately it didnít have a significant effect on the solving path.]
8a. Killer pair 8,9 in R45C9 and 19(3) cage, locked for N6
[This shouldnít really be here, because itís not a preliminary step, but Iíve moved it after Mike pointed out that I had two step 15s.]


9. 11(3) cage in N9 = {128/137/146/236/245}, no 9

10. 12(4) cage at R1C2 = 12{36/45}
10a. CPE no 1,2 in R2C2

11. 28(4) cage at R2C1 = {4789/5689} = 89{47/56}
11a. CPE no 8 in R1C1

After the preliminary steps the first placement

12. 45 rule on D/ + D\, R5C5 counts toward both diagonals, total 86 -> R5C5 = 4, locked for D/ and D\, clean-up: no 7 in R9C6

13. 45 rule on D/ 2 outies R4C4 + R6C6 = 14 = {59/68}

14. 45 rule on D\ 2 outies R4C6 + R6C4 = 9 = {18/27/36}, no 5,9

15. 16(3) cage in N1 = {259/268/358/367} (cannot be {169} which clashes with R4C4 + R6C6, cannot be {178} because R1C1 only contains 3,5,6), no 1
15a. 2 of {259} must be in R3C3 -> no 9 in R3C3
15b. 1 in N1 locked in R1C23 + R2C3 of 12(4) cage -> no 1 in R2C4

16. 1 in D\ locked for 11(3) cage = 1{28/37}, no 5,6, 1 locked for N9

17. 16(3) cage in N1 (step 15) = {259/268/367} (cannot be {358} which clashes with 11(3) cage in N9)
17a. 2 of {268} must be in R3C3 -> no 8 in R3C3
17b. 5 of {259} must be in R1C1 -> no 5 in R2C2 + R3C3

18. 12(3) cage in N7 = {129/138/156/237}
18a. 9 in {129} must be in R9C1 -> no 9 in R8C2 + R7C3

19. 45 rule on N1 2 outies R2C4 + R4C1 = 11 = [29/38/47/56/65], no 4 in R4C1

20. 45 rule on N3 2 outies R1C6 + R4C8 = 11 = [47/56/65/83], no 1,2,3 in R1C6, no 1,2,4 in R4C8

21. 45 rule on N7 2 outies R6C2 + R9C4 = 11 = {38/47/56}/[92], no 1,9 in R9C4

22. 45 rule on N9 2 outies R6C9 + R8C6 = 11 = [29/38]/{47/56}, no 1 in R6C9, no 1,2,3 in R8C6

23. 1 in N6 locked in R45C7, locked for C7 and 12(3) cage at R4C7 -> no 1 in R5C6
23a. 12(3) cage at R4C7 = 1{29/38/47/56}
23b. 7 of {147} must be in R5C6 -> no 7 in R45C7
23c. 8,9 only in R5C6 -> no 2,3 in R5C6

24. 45 rule on R123 3 outies R4C158 = 13 = {139/157/238/256}
24a. 1,2 only in R4C5 -> R4C5 = {12}

25. 45 rule on C123 3 outies R259C4 = 10 = {127/136/145/235}, no 8,9, clean-up: no 3 in R6C2 (step 21)
25a. 1 of {127/136} must be in R5C4 -> no 6,7 in R5C4

26. 45 rule on C789 3 outies R158C6 = 22 = 9{58/67}, no 4, 9 locked for C6, clean-up no 5 in R4C4 (step 13), no 7 in R4C8 (step 20), no 7 in R6C9 (step 22), no 2 in R9C5
26a. 6 of {679} must be in R1C6 -> no 6 in R58C6, clean-up: no 5 in R45C7 (step 23a), no 5 in R6C9 (step 22)

27. CPE 5 in D\ must be in R1C1 or R6C6 -> no 5 in R1C6, clean-up: no 6 in R4C8 (step 20)

28. R158C6 (step 26) = 9{58/67}
28a. 8 of {589} must be in R1C6 -> no 8 in R58C6, clean-up: no 3 in R45C7 (step 23a), no 3 in R6C9 (step 22)

29. R4C158 (step 24) = {139/157/238/256}
29a. 5 of {256} must be in R4C8 -> no 5 in R4C1, clean-up: no 6 in R2C4 (step 19)

30. 45 rule on N2 4 innies R1C6 + R2C4 + R3C45 = 18 = {1368/2358/3456} (cannot be {1458} because R3C45 cannot be {14/15}), 3 locked for N2
30a. 3,5 of {2358} must be in R3C45 -> no 2 in R3C45

31. 45 rule on N6 4 innies R4C78 + R5C7 + R6C9 = 12 = 12{36/45}
31a. Killer pair 5,6 in R4C78 + R5C7 + R6C9 and R45C9, locked for N6

32. 19(4) cage at R1C6 = {1369/1378/1468/1567/2368/2458/2467} (cannot be {1279/1459/2359/3457} because R1C6 only contains 6,8) [1/2]
33. 17(4) cage at R2C9 must contain 1/2 = {1349/1358/1367/1457/2348/2357/2456} (cannot be {1259/1268} which contain both 1 and 2)
[Steps 32 and 33 are included for completeness even though they didnít lead to anything. I spent some time examining them because the 4-cell cage at R1C6 had proved to be a breakthrough area for Vortex Lite.]

34. 16(3) cage in N1 (step 17) = {259/268/367}
34a. If R3C3 = 2 => R2C4 = 2 => R4C1 = 9 (step 19) => R2C2 = 9 -> 16(3) cage in N1 cannot be {268} = {259/367}, no 8
34b. 8 in N1 locked in R2C1 + R3C12 -> no 8 in R4C1, clean-up: no 3 in R2C4 (step 19)
[Mike commented "BTW, another (alternative, not better) way of getting the same result is:
34. from step 26, R158C6 = 22 = 9{58/67}
34a. if {679}, R1C6 = 6
34b. if {589}, R6C6 = 6
34c. -> 6 locked in R16C6
34d. -> CPE: no 6 in R1C1
34e. -> {268} combo blocked for 16(3) cage in N1 (because none of these digits now available in R1C1)"
Then need to do my step 34b.
In one way Mike's version of this step is better because his step 34c can be extended as 34c. -> 6 locked in R16C6, not elsewhere in C6. Para had the extended version of Mike's step in his walkthrough.]


35. R259C4 (step 25) = {127/145/235} (cannot be {136} because R2C4 only contains 2,4,5), no 6, clean-up: no 5 in R6C2 (step 21)
[You may want to save this position to look at Mikeís comment after step 44.]

36. R1C6 + R2C4 + R3C45 (step 30) = {2358/3456} (cannot be {1368} because R2C4 only contains 2,4,5) = 35{28/46}, no 1, 5 locked for N2
36a. 11(3) cage in N2 = 1{28/46}
36b. Killer pair 6,8 in R1C6 and 11(3) cage, locked for N2
36c. 3 in N2 locked in R3C45, locked for R3

37. 45 rule on N8 2 innies R8C6 + R9C4 Ė 5 = 1 outie R6C5, min R8C6 + R9C4 = 7 -> min R6C5 = 2

38. 45 rule on N5 4 innies R4C5 + R5C46 + R6C5 = 18 = {1278/1359/2367} (cannot be {1269/2358} which clash with R4C4 + R6C6, cannot be {1368} because R5C6 only contains 5,7,9)
38a. 1,2 of {1278} must be in R4C5 + R5C4
38b. 2 of {2367} must be in R4C5
38c. Taking steps 38a and 38b together -> no 2 in R6C5
38d. 7 of {1278/2367} must be in R5C6 -> no 7 in R6C5

39. 45 rule on N124 2 innies R1C6 + R6C2 Ė 10 = 2 outies R4C5 + R5C4
39a. Min R4C5 +R5C4 = 3 -> min R1C6 + R6C2 = 13, no 4 in R6C2, clean-up: no 7 in R9C4 (step 21)

40. R259C4 (step 35) = {145/235} = 5{14/23}, 5 locked for C4
[Mike commented "This leaves R3C4 = {34} -> R3C4 and R259C4 form killer pair on {34} in C4. This would have been very useful (see comments after step 44), because it knocks out the 3 in R6C4."
I should have spotted that killer pair. I must have done that step fairly automatically having already eliminated the 6 from R259C4 in step 35.
BTW The 3 would have been eliminated from R6C4 by using the extended form of Mikeís step 34c.]


41. 45 rule on N4 2 innies R4C1 + R6C2 Ė 12 = 1 outie R5C4
41a. R4C1 + R6C2 + R5C4 = {67}1/[68]2/{69}3/[78]3/[98]5

42. 15(3) cage at R5C3 = {159/249/348/357/456} (cannot be {168/267} which clash with R4C1 + R6C2 + R5C4 (step 41a), cannot be {258} which clashes with 15(3) cage at R4C2 Ė step 7)
42a. 4 of {348/456} must be in R6C3 -> no 6,8 in R6C3

43. 45 rule on R789 3 outies R6C259 = 17 = {269/278/368/458/467} (cannot be {359} because R6C9 only contains 2,4,6)
43a. 3 of {368} must be in R6C5 => R6C6 = 5 when R6C56 = [35] clash with split-cage R4C5 + R5C46 + R6C5 = {1359}
43b. 3 of split-cage R4C5 + R5C46 + R6C5 = {2367} must be in R5C4
43c. Taking steps 43a and 43b together -> no 3 in R6C5
43d. R6C259 = {269/278/458/467}
43e. 2,4 only in R6C9 -> no 6 in R6C9, clean-up: no 5 in R8C6 (step 22)

44. Killer quint {56789} in R4C4 + R5C6 + R6C56 + (R4C6 + R6C4 which need one of 6,7,8) -> no 5 in R5C4
[Alternatively hidden killer triple {123} in R4C5 + R5C4 + (R4C6 + R6C4 which need one of 1,2,3) but I saw the quint first.]
[Mike commented "This candidate could have been eliminated much earlier (after step 35), as follows:
35a. CPE: R5C4 sees all 5's in R6 -> no 5 in R5C4
This could be followed up by:
35b. -> max. R5C4 = 3 -> no 3 in R56C3 (otherwise 15(3) cage sum unreachable)

Note also that the N5 innies (step 38) can only contain 2 of {123}, which (after step 35b above is applied), must go in R4C5 and R5C4 -> no 1,2,3 in R6C5.

Note that, after doing all this (including Mikeís addition to step 40), the 3 in R6 has been locked into R6C78 -> not elsewhere in N6 -> R4C8 = 5, R1C6 = 6."

Mike and Cathy are much better at spotting CPEs than I am.

I saw Mike's elimination of 3 from R56C3, after removing 5 from R5C4, but didn't include it in my walkthrough as it didn't immediately lead to anything; it's so easy to just move on to other moves that make more progress.]


45. R4C5 + R5C46 + R6C5 (step 38) = {1278/1359/2367}
45a. If {1278} => R5C4 = {12}, R5C6 = 7, R45C7 = [41] (step 23a) -> R5C4 + R5C7 clash with R5C1 -> cannot be {1278}
[This was there at step 38 but I didnít see it then.]
45b. If {1359} => R4C5 = 1, R5C4 = 3, R5C6 + R6C5 = {59}, R45C7 = [61/21]
45c. If {2367} => R4C5 = 2, R5C4 = 3, R5C6 = 7, R6C5 = 6, R45C7 = [41]
45d. -> R5C4 = 3, R5C7 = 1, R6C5 = {569}, R56C1 = [21], clean-up: no 6,8 in R4C6, no 6 in R6C4 (both step 14), no 8 in R6C2 (step 21)
45e. R4C5 + R5C46 + R6C5 = {1359/2367} = 3{159/267}

46. R3C4 = 4 (naked single) -> R3C5 = 3 (hidden single in R3), R4C5 = 2, R5C6 = 7, R6C5 = 6 (both step 45e), R4C6 = 1, R6C4 = 8 (both locked for D/), R4C4 = 9, R6C6 = 5 (both locked for D\), R1C45 = [79], R8C6 = 9, R1C6 = 6 (step 26), [I forgot about R4C8 here, itís in step 47], clean-up: no 5 in R5C9, no 5 in R9C5, no 2,8 in R9C6, no 2 in R2C4, no 7 in R4C1 (both step 19) -> R2C4 = 5, R4C1 = 6, R4C7 = 4, R6C9 = 2, R9C4 = 2, clean-up: no 3,6 in R1C23 + R2C3 (step 10), no 4,7 in R2C1 + R3C12 (step 11), R6C2 = 9 (step 21)
46a. R1C1 = 3 (naked single, locked for D\), R2C2 + R3C3 = {67} (step 34a, locked for D\)
46b. R1C9 = 5 (naked single, locked for D/), R2C8 + R3C7 = {69} (only remaining combination, locked for N3 and D/), clean-up: no 9 in R5C9
46c. R9C1 = 7 (naked single), R8C2 + R7C3 = {23}, locked for N7, R9C56 = [83], R2C5 = 1, R9C9 = 1, R7C7 + R8C8 = {28}, locked for N9

47. Naked pair {58} in R5C23, locked for R5 and N4 -> R5C8 = 9, R45C9 = [86], R2C8 = 6, R3C7 = 9, R2C2 = 7, R3C3 = 6, R3C9 = 7, R4C23 = [37], R4C8 = 5, R6C3 = 4, R5C3 = 8, R5C2 = 5

and the rest is naked singles

3 4 1 7 9 6 8 2 5
9 7 2 5 1 8 3 6 4
5 8 6 4 3 2 9 1 7
6 3 7 9 2 1 4 5 8
2 5 8 3 4 7 1 9 6
1 9 4 8 6 5 7 3 2
8 1 3 6 5 4 2 7 9
4 2 5 1 7 9 6 8 3
7 6 9 2 8 3 5 4 1
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