Chevron Killer

[Ruud]

Since it would be confusing to include zero-killers in the Assassin series, this puzzle has to go here on the forum.

It's kinda difficult, but the regulars here will probably find some weak spots.



3x3::k:30723073:352566:46163073:11:43642318:154616194364:223864:25:4616:272836:30323618:353868:383359422348461575:46573379:52542351:4657:4657:46573901:622615185721163901339123773395:4173:4173:4173

For those not familiar with zero-killer: The uncaged cells can contain any number that does not violate the Sudoku rules.

Enjoy,
Ruud

[Para]

Hi

Very nicely crafted puzzle. But it goes pretty smoothly when you get it to it's bare essentials.

Para

[edit]

ok here is a walkthrough. I am sorry about not cleaning up the grid properly, but i made this walkthrough quickly and following these steps will get you to the correct solution without cleaning up much.

Walkthrough Chevron Killer

1. R34C7, R45C2, R56C6 and R78C8 = {69/78}
2. R45C5, R67C7, R89C1 and R89C5 = {49/58/67} : no 1,2,3
3. R12C5, R23C6, R56C9 and R67C3 ={18/27/36/45}: no 9
4. R34C3 = {29/38/47/56}: no 1
5. R8C34 and R89C9 = {16/25/34}: no 7,8,9
6. R8C67 = {17/26/35}: no 4,8,9
7. 10(3) in R1C7: no 8,9
8. 9(3) in R9C2: no 7,8,9
9. R23C4 = {89} -->> locked for N2 and C4
10. R45C8 = {59/68}-->> {59} : {68} clashes with R78C8 -->> 59 locked for C8 and N6
11. R7C8 = {78}-->> {78} locked for C8 and N9
12. R56C1 = {59/68} -->> {59} : {68} clashes with R45C2 -->> 59 locked for C1 and N4
13. R45C2 = {78} -->> locked for N4 and C2
14. Naked pair {59} in R5C18; no 5 or 9 anywhere else in R5
15. 45 on R89: 2 innies = 17 -->> R8C28 = [98]
16. R7C28 = [17]
17. 45 on C1: 2 innies = 6 -->> R47C1 = {24} -->> locked for C1
18. R89C1 = {67} -->> locked for C1 and N7
19. 12(3) in R1C1 = {138} -->> locked for N1
20. Hidden single 8 in R7C3 -->> R6C3 = 1
21. 9(3) in R9C2 = {135/234} : {126} not possible 1 and 6 only possible in 1 cell so not both possible in 9(3) -->> no 6 in R4C9; -->> 3 locked in R9
22. 16(3) in R9C6 = {169/259/268}-->> {178} not possible 7,8 only in one cell; {457} not possible, clashes with 9(3) in R9C2, no 7 in R9C6
23. Killer Pair {1,2} in 9(3) and 16 (3) in R9
24. Hidden Killer Pair in R9C5 and 16(3) -->> R9C5 = {89} -->> Clean up: R8C5 = {45}
25. 12(3) in R1C2 = {246} -->> locked for N1
26. Naked Single: R3C2 = 5
27. Hidden Single 9 in R1C9
28. R34C7 = {78} -->> locked for C7
29. R67C7 = [49]
30. Hidden Pair {89} in R9 and N8 -->> R9C56 = {89}
31. Killer Pair {89} in R569C6 -->> no {89} anywhere else in C6
32. Naked Pair {78} R4C27 -->> locked for R4
33. Hidden single 7 in R8C6
34. R56C6 = [69]
35. Clean up: No 1 in R12C5
there is a lot more clean up, but this is all you need to get all naked or hidden singles from now on

[Andrew]

A nice puzzle Ruud! I don't think I've done one like this before. It's interesting how one can sometimes make use of the blank cells, even though they don't form parts of cages. At other times one has to look for other ways to make progress.

It's kinda difficult, but the regulars here will probably find some weak spots.

It was fairly easy to find what I thought was the key move which was step 15 in my walkthrough, steps 13, 14 and 18 were also very useful.

I decided to post my walkthrough because, although I followed roughly the same solving path as Para, in several places we made moves for different reasons. I hope people find this walkthrough interesting.

Clean-up is used in various steps, using the combinations in preliminary steps 1 to 12 for further eliminations from these two cell cages. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. 9(2) cages R12C5, R23C6, R56C9 and R67C3 = {18/27/36/45}, no 9

2. R23C4 = {89}, locked for C4 and N2, clean-up: no 1 in R12C5 and R23C6

3. R56C4 = {15/24}

4. 7(2) cages R8C34 and R89C9 = {16/25/34}

5. 8(2) cage R8C67 = {17/26/35}

6. 10(2) cage R78C2, no 5

7. 11(2) cage R34C3, no 1

8. 13(2) cages R45C5, R67C7, R89C1 and R89C5 = {49/58/67}

9. 14(2) cages R45C8 and R56C1 = {59/68}

10. 15(2) cages R34C7, R45C2, R56C6 and R78C8 = {69/78}

11. R9C234 = {126/135/234}, no 7,8,9

12. 10(3) cage in N3 = {127/136/235}, no 8,9

13. 45 rule on R89 2 innies R8C28 = 17 = {89}, locked for R8, clean-up: R7C2 = {12}, R7C8 = {67}, no 4,5 in R9C1 and R9C5

14. 45 rule on N1 3 innies R2C3 + R3C23 = 21 = {489/579/678}, no 1,2,3, clean-up: no 8,9 in R4C3

15. 45 rule on N4 5 innies R4C13 + R5C3 + R6C23 = 16 = {12346}, locked for N4 -> R56C1 = {59}, locked for C1 and N4, R45C2 = {78}, locked for C2 -> R8C2 = 9, R7C2 = 1 -> R8C8 = 8, R7C8 = 7, clean-up: no 4,6 in R3C3, no 6 in R45C8, no 2 in R6C3, no 5,6 in R6C7, no 2,4 in R7C3, no 4 in R8C1, no 6 in R8C4, no 1 in R8C6, no 8 in R9C1

16. R89C1 = {67}, locked for C1 and N7, clean-up: no 3 in R6C3, no 1 in R8C4

17. R45C8 = {59}, locked for C8 and N6, clean-up: no 6 in R3C7, no 4 in R56C9, no 4 in R7C7

18. R3C2 = {456} -> R23C3 = {789} (from combinations in step 14)

19. Only valid combination for R123C1 = {138}, locked for C1 and N1 -> R23C3 = {79}, locked for C3, R3C2 = 5, clean-up: no 4 in R2C6, no 3,6 in R4C3

20. R7C3 = 8 (hidden single), R6C3 = 1, clean-up: no 5 in R5C4, no 8 in R5C9

21. R4C13 = {24} (naked pair), locked for R4 and N4, clean-up: no 9 in R5C5

22. 45 rule on N5 3 innies R4C46 + R6C5 = 11, no 9

23. 45 rule on C5 3 innies R367C5 = 10 = {127/136/235}, no 8,9

24. 8/9 in C5 locked in the two 13(2) cages, no 6,7 in these cages
24a. Killer pair 4/5 in the two 13(2) cages, locked for C5

25. 45 rule on R9 3 innies R9C159 = 20, minmax R9C15 = 14..16 -> R9C5 = {456}, clean-up: R8C5 = {123}

26. 1 in R8 locked in R8C79, locked for N9

27. 7 in N8 locked in R89C6, locked for C6, clean-up: no 2 in R23C6, no 8 in R56C6

28. R56C6 = {69}, locked for C6 and N5, clean-up: no 3 in R23C6, no 4 in R5C5, no 2 in R8C7

29. R23C6 = [54] (naked singles), clean-up: no 3 in R8C7

30. R45C5 = {58}, locked for C5 and N5 -> R89C5 = [49], clean-up: no 1 in R5C4, no 3 in R8C34

31. R56C4 = {24}, locked for C4 and N5 -> R8C34 = [25], clean-up: no 9 in R3C3, no 3 in R8C6, no 6 in R8C7, no 5 in R9C9 -> R34C3 = [74], R2C3 = 9, R47C1 = [24], R8C67 = [71], R89C1 = [67], R89C9 = [34], R9C234 = [351], R9C6 = 8 (hidden single), R9C78 = {26}, locked for N9, R1C3 = 6 (naked single), clean-up: no 3 in R2C5

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

BTW When I first started solving it, I reached an impossible position and then found that I was using the wrong diagram. I'd put a 14(2) cage as a 15(2) cage. It would have been interesting, and embarassing, if that had come out and I'd posted a walkthrough for the wrong puzzle.

[Para]

Just one quick question about my own walk-through. Is the term Hidden Killer Pair a proper term? Is it clear what i mean if write that down?

Para

[Andrew]

Just one quick question about my own walk-through. Is the term Hidden Killer Pair a proper term? Is it clear what i mean if write that down?

It was clear to me what you meant. However I'm not sure if it's possible to have a Hidden Killer Pair. No doubt someone will post an example if I'm wrong about that. If so, I'll be interested to see it.

What I think you should say is 24. 8,9 in R9 locked in R9C5 and 16(3) -->> R9C5 = {89} (16(3) cannot have both 8 and 9) -->> Clean up: R8C5 = {45}

Hope this helps

Andrew

[mhparker]

Just one quick question about my own walk-through. Is the term Hidden Killer Pair a proper term? Is it clear what i mean if write that down?

Para

After a lot of searching, I couldn't find a standard term for this either. However, I like the simplicity and fluency of the terms used here. That is:

• "killer pair" as shorthand for "killer naked pair" or "naked killer pair"
  "hidden killer pair" instead of "killer hidden pair"

BTW, I'm rather surprised that nobody seems to have written a comprehensive article about this anywhere on the web (at least I couldn't find anything, in contrast to the well-documented techniques for regular Sudoku).

[nd]

I wrote an article about it a few weeks back for the Sudopedia: Hidden Killer Subset. It still needs a "simple" example (I wanted to use a JC Godart puzzle I once solved but it seems to have been removed from his site).

I do encourage people to add to the Sudopedia to expand the sections on Killer techniques.

[Para]

Hi

That explanation is not really what i meant with my Hidden Killer Pair.
In my walk-through, where i used Hidden Killer Pair. The way i used this term i meant that R9C5 and the 16(3) cage in R9 were the only cells that contained the numbers {89} in R9. As the 16(3) cage can only contain either a 8 or a 9 (not both). Which means that R9C5 has to contain either an 8 or a 9. This way you can eliminate all other candidates from R5C5.

This comes to terms more with the normal Hidden Pair term because in a Hidden Pair you eliminate candidates from the cells that contain the Hidden Pair. You could also use this technique to eliminate certain combinations from cages.

Para

[nd]

Yes, that's the same technique described in the entry on Sudopedia; it's just a much simpler version (I haven't put in an example of that yet). There's a description of just the simpler version here:

http://www.ndorward.com/blog/?page_id=95

in the 2nd section (see the diagram).

[Para]

Well according to me it isn't really the same technique. It is really the opposite of what i meant.


It says. R145C7 has to have contains 3 digits of {56789} and the 18 (3) in R6C7 cage has to have 2 of these digits. Thus creating a subset {56789} in R145C7 and the 18(3) cage. Thus eliminating them from all other cells in C7.
Which i agree with .

But what i meant is the opposite of this technique actually.
R239C7 and the 18(3) in R6C7 cage are the only cells that can contain {1234}. But {1234} can only be used once in the 18(3) cage. So 3 of the digits {1234} has to be in R239C7, so eliminating {56789} from R239C7.
This would be a Hidden Killer Quad to me.

And you would call the technique using the opposite cells a Hidden Killer Quint.

So according your terms my elimination would be named a Hidden Killer Septet using the opposite cells i used.

I just found my explanation more appropriate to be called Hidden Killer Subset because normally in a hidden subset you would make the eliminations in the cells that are part of the subset and not in the peers of the subset.

So there must be made a difference between these techniques if you want proper terms for Killer Sudoku.

Hope it is clear what is the difference between your and my view.

Para

[mhparker]

Hi ND,

Although your logic is sound, I agree with Para here.

In your Sudopedia example (Assassin 34), the 18/3 cage in column 7 can (for this purpose) behave as if it contained 2 cells with the candidates {56789} and 1 cell with the candidates {1234}. Therefore, what you've found is actually (IMHO) a *Naked* Killer Quint, as if you had (in regular Sudoku) 5 uncaged cells with the values (reading down the column) {5689}, {5789}, {5789}, {56789} and {56789}. As you say, this allows these candidates {56789} to be eliminated from all cells outside the subset.

As always, for every naked subset, there's a corresponding hidden subset (which is what Para means). In this case, it just means turning the logic round, and saying that there are only 4 candidate positions in column 7 for the candidates {1234} (i.e., R239C7 and one effective cell in the 18/3 cage as mentioned above). This implies that all other candidates can be eliminated from the subset (i.e., R239C7 - no eliminations are possible in the 18/3 group because although we know that exactly one of the three cells must belong to the subset, we don't know which one it is). This is what I (and Para) would call a hidden Killer Quad. It's also directly analagous to hidden subsets in regular Sudoku and to the example in Para's walkthrough.

The basic rule of thumb I use is: if the result of applying the killer subset logic is that candidates *outside* of the subset are eliminated, it must by definition be a naked subset, not a hidden one.

[mhparker]

Arrghh - Para was quicker in responding than I was! But our simultaneously-composed responses show that Para and I do indeed share a common point of view, as I suspected.

So ND - now you're outvoted 2 to 1 (not that that necessarily means anything)!

[nd]

OK, I see we have a disagreement with terminology not the technique--that's what I wanted clarified. It's just two ways of looking at the same logic (the same way in regular sudoku there's always a corresponding [complementary] naked subset for every hidden subset & vice versa). I think much of the confusion stems from the fact that in a real sense both subset moves involve "hidden" subsets, i.e. they are buried within a cage.

Feel free to go into Sudopedia & rewrite things (I'm sure Ruud would welcome some non-spam edits at the moment!!).

I think the most important thing in the article is that both logics be listed as in some cases it's easier to spot such a move from one end ("hidden" or "naked") rather than the other, or some players may find one method more intuitively/visually obvious than the other.

[mhparker]

Hi ND,

I've added my suggestion to your Sudipedia discussion here:

http://www.sudopedia.org/wiki/Talk:Hidden_Killer_Subset