Numerals on Grid

Everything about Sudoku that doesn't fit in one of the other sections

Postby Karyobin » Tue Aug 09, 2005 3:35 pm

Ooohh! A contradiction m'lud!

Obviously is [sic] a grid has two numbers missing from the clues, there'll always be at least two solutions.

Note the 'Obviously' and 'always'...followed later by:

As to whether you can get a puzzle with two numbers not present in the clues, but with only one possibility for which 9 of the 18 cells one of the numbers goes in, I think that's unresolved


I'm sure you've just disagreed with your first sentence there Paul.

I'd imagine it isn't true, myself. Essentially, you are conjecturing that a puzzle can be given that is missing two numbers from its original clue set but somewhere along the way it becomes apparent that, in fact only x can go here and y here.

I don't understand how you arrived at that conjecture. (I suspect it's just a gut feeling and I respect that, but I'm just being challenging.)
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Postby PaulIQ164 » Tue Aug 09, 2005 5:15 pm

What I was trying to say is that, while if there are two numbers not present in the clues, there'll always be at least two solutions, beause you can write the unclued numbers in the grid either way round, if you don't count that as different enough to be different, I'm not sure it's been resolved whether you can make a grid with only seven different numerals (I almost coined a new word: 'numberals', there) in the clues that does not have more than one different (under the new definition of 'different') solution. My suspicion that you can make such a grid is indeed nothing more than a gut feeling.

I hope that's clarified my position, though I suspect it probably hasn't.
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Postby Karyobin » Tue Aug 09, 2005 5:51 pm

Aaahh. The old different-but-not-different sleight of hand.

Position clarified.
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Postby PaulIQ164 » Tue Aug 09, 2005 7:47 pm

In fact, it's very easy to get a trival puzzle which satisfies this second definition. If you look at almost any completed sudoku grid, and remove any two numbers from throughtout the grid, you'll find that placing one back in leads necessarily to the other seventeen. I think it only fails if you have a construction like this:

*** xx* ***
*** *** ***
*** *** ***

*** *** ***
*** xx* ***
*** *** ***
*** *** ***

Where the xs are candidate places for the two missing numbers, when you can swap these pairs of cells round in isolation to the rest of the puzzle.
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Postby Karyobin » Tue Aug 09, 2005 8:03 pm

Understood. But if two original number clues were totally lacking, then you would never be able to complete the last eighteen boxes. Or if you could, I'm at a loss to see how. (Obviously just one further clue would solve the other seventeen, but in its absence - Brick Wall as far as I'm concerned.)

Are you saying that you can conceive of a puzzle with number clues that only include 7 of the 9 options, and that has a unique solution? I understood (perhaps erroneously) that originally you were, and triviality aside, I really don't see how this could be the case.
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Postby PaulIQ164 » Tue Aug 09, 2005 8:16 pm

No, obviously a grid with only seven different numbers in the clues will always have two different solutions - you can swap round the missing digits. What I meant was whether you can make a puzzle with seven different numbers in the clues where the only way any solutions differ is that all the 1a and 2s are swapped (presuming these are the missing numbers). In fact, you rather trivially can do this.
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Postby Karyobin » Wed Aug 10, 2005 8:21 am

Aaahh, on your wavelength now and I agree entirely. There will be vast amounts of arrangements, all integrally different from each other.
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Postby stuartn » Wed Aug 10, 2005 8:29 pm

has this thread degenerated to StarWars speak?.....
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Postby The Druid » Wed Aug 10, 2005 9:09 pm

stuartn wrote:has this thread degenerated to StarWars speak?.....


Sheeeeeeesh, Stuart, that wouldn't be "degenerating", that would be, uh, what's a biiiiig word for "going up a level"? <g>.

May the Force be with you!

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Postby Pat » Thu Aug 11, 2005 8:51 am

.

( just 2 days i was off and you've generated all this discussion?? )

sorry if my post wasn't clear, here are the references:

Hammerite (2005.Jun.21)
Hammerite (2005.Jun.21) wrote:Clearly if there are 7 clues in the grid, then at most 7 of the 9 symbols will appear in the initial grid. Then at least 2 of the symbols do not appear in the initial grid. Clearly in any solution of the grid, the grid produced by the transposition of the 2 non-appearing symbols is also a solution. However, can it really be called a different solution? The two symbols are, by virtue of not being in the grid, invented by us, the solvers, and so if all that changes is that we transpose them, while keeping their respective locations the same, we may decide that the two solutions are in fact equivalent.


gfroyle (2005.Jul.11)
gfroyle (2005.Jul.11) wrote:
Code: Select all
5 . 2  . . .  4 . . 
. . .  7 1 .  . . 3 
. . .  . . .  . . . 

. . .  . . 4  6 . . 
. 7 .  2 . .  . . . 
. 1 .  . . .  . . . 

6 . .  . . 2  . . . 
. . .  . 3 .  . 1 . 
4 . .  . . .  . . .


is interesting, because it has only 16 clues. It CANNOT be uniquely solvable, because it is missing both 8 and 9, and so they can always be exchanged in any final solution.

But, this turns out to be the only problem - the puzzle has exactly TWO solutions...



Pat (2005.Jul.22)

- Pat

.
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