Name for this technique

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Name for this technique

Postby pochert » Fri Jan 06, 2006 11:53 pm

I wonder if the following situation has a name. It's come up for me a few times and relies on the fact that a unique solution should exist for each puzzle.
There were 4 cells at the corners of a rectangle and 3 had the same pair of possible numbers. The 4th cell had the same pair and one other number. I figured the other number had to be the answer to the 4th cell because otherwise there would be the same pair in the four cells and this would lead to 2 equally valid solutions.
Thanks for the input.
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Re: Name for this technique

Postby Red Ed » Sat Jan 07, 2006 12:06 am

How about uniqueness ?
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Postby PaulIQ164 » Sat Jan 07, 2006 2:00 am

This must surely be the single most often independently discovered tactic in the history of sudoku? Nevertheless, well done for finding it, but unlucky that you got to the game rather late.
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Postby Pi » Sat Jan 07, 2006 11:50 am

Is that a chain?
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uniqueness

Postby pochert » Sat Jan 07, 2006 5:44 pm

Thanks Red Ed for the input. Paul, I wasn't trying to claim credit for an original idea, it was just a question. If a post of mine seems to trivial or stupid to you please feel free to ignore it and don't
waste your time replying.
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Postby PaulIQ164 » Sat Jan 07, 2006 6:16 pm

Sorry, I wasn't trying to be hostile. Just pointing out that, for whatever reason, this tactic seems to get discovered a lot more than many others.
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Postby Pi » Sat Jan 07, 2006 6:20 pm

I am just guessing numbers here

If the candidates were
Code: Select all
|25------25|
|------------|
|256-----25|



why wouldn't

Code: Select all
|2------5|
|---------|
|5-----2|



be valid?[/code]
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Postby Moschopulus » Sat Jan 07, 2006 6:34 pm

Because the 2 and 5 could be interchanged in the solution grid to give another solution. Then the puzzle would have 2 solutions. You could have
2..5
....
5..2

or

5..2
....
2..5
(everything else the same).

We call a set of cells like this an "unavoidable" set, because the set of clues must contain one of these cells, it cannot be avoided.
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Postby Pi » Sat Jan 07, 2006 6:38 pm

so i asume that actually there wouldn't be any solutions if you didn't use the six
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Postby PaulIQ164 » Sat Jan 07, 2006 8:03 pm

Well, it depends. That's why this technique is a tiny bit controversial. If the puzzle is well-formed, and only has one unique solution, then the tactic can help you find it, by exploiting that knowledge. But if it was a bad puzzle to begin with, that had more than one solution, then this technique could get you in trouble.

What I'm not sure about, is whether you can get a 2-solutioned puzzle, use this technique (incorrectly, since it's the 'uniqueness' technique, and you don't have a unique solution to exploit) and end up in a position where you have no solutions. But this is a tangent, really. The technique is fine for proper sudoku puzzles.

PS: We've been hacked! How lovely!
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Postby Ocean » Sat Jan 07, 2006 9:01 pm

PaulIQ164 wrote:What I'm not sure about, is whether you can get a 2-solutioned puzzle, use this technique (incorrectly, since it's the 'uniqueness' technique, and you don't have a unique solution to exploit) and end up in a position where you have no solutions. But this is a tangent, really. The technique is fine for proper sudoku puzzles.


I think, that if you for a particular cell place a number that does not appear in that cell in any of the solution grids, you will end up in a contradiction sooner or later. (Example cases are easy to construct). But for proper sudokus the technique works fine.
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Postby Moschopulus » Sun Jan 08, 2006 2:38 am

PaulIQ164 wrote:What I'm not sure about, is whether you can get a 2-solutioned puzzle, use this technique (incorrectly, since it's the 'uniqueness' technique, and you don't have a unique solution to exploit) and end up in a position where you have no solutions.

Here is an example of this.

Consider this grid:
Code: Select all
5 6 2 3 8 9 4 7 1
8 4 9 7 1 6 2 5 3
1 3 7 4 2 5 8 9 6
3 5 8 1 9 4 6 2 7
9 7 4 2 6 3 1 8 5
2 1 6 8 5 7 3 4 9
6 9 1 5 4 2 7 3 8
7 2 5 6 3 8 9 1 4
4 8 3 9 7 1 5 6 2

Remove the digits in all the cells with 8 and 9, and one further cell (r1c1). Now consider what is left as a puzzle:
Code: Select all
. 6 2 3 . . 4 7 1
. 4 . 7 1 6 2 5 3
1 3 7 4 2 5 . . 6
3 5 . 1 . 4 6 2 7
. 7 4 2 6 3 1 . 5
2 1 6 . 5 7 3 4 .
6 . 1 5 4 2 7 3 .
7 2 5 6 3 . . 1 4
4 . 3 . 7 1 5 6 2

If you use the uniqueness rule, then you argue that 5 cannot go in r1c1, because if it did the puzzle has two solutions. But 5 cannot go in any other cell, because you either get two 5s in a row or a column. So the uniqueness rule implies no solution. Not using the uniqueness rule there are two solutions.

Edit: The puzzle was wrong when I first posted it, it got mangled somehow. It's now corrected, sorry.
Last edited by Moschopulus on Mon Jan 09, 2006 11:03 am, edited 1 time in total.
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Postby Moschopulus » Mon Jan 09, 2006 2:58 pm

Previous post corrected.
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Postby udosuk » Mon Jan 09, 2006 7:58 pm

I think what people fail to mention is that this technique not merely is restricted to "proper/valid" puzzles, it is restricted to "normal/ordinary" i.e. boring sudoku puzzles. Consider all the other variant forms: killer, diagonal (sudoku X), samurai, overlapping, etc. Almost all other normal sudoku techniques are appliable to these variants, but not this. I've seen numerous users complaint to djape on his X puzzles being with multiple solutions, failing to notice the "X" property. Okay, that's irrelevant... So I guess it's a "pure sudoku" technique...
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Postby PaulIQ164 » Mon Jan 09, 2006 11:03 pm

Yes, good example there, just what I was wondering.

In fact you can use uniqueness on Killer puzzles, as long as the four numbers being used are restricted to 2 cages (there may be more restrictions I can't fathom out in my head). Also in Samurai and X puzzles under similar limitations. But it's certainly true that it doesn't apply straigt away to variants like other rules do, because, I suppose, the extra restrictions that variants add mean that the underlying 'pure' sudoku involved could have more that one solution (eg, the individual puzzles in a Samurai will have multiple solutions, of which the extra restriction of fitting with the overlapping puzzles will restrics the solver to one).
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