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 A non-unique and yet unique solution 
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Posted on: Sun Nov 06, 2011 9:01 am




Posts: 48
Joined: Sun Oct 16, 2011 10:58 am
Post Re: A non-unique and yet unique solution
You can use the [ code ] tag for  
Code:
Although its not possible for 1 space.
  But it is for 2.
   Or 3,


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Posted on: Sun Nov 06, 2011 12:44 pm




Posts: 54
Joined: Thu Nov 03, 2011 8:52 am
Post Re: A non-unique and yet unique solution
Thanks. I thought that the Code tags would keep the indentations, but I felt it wasn't code. When one is a purist... ;)


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Posted on: Mon Nov 07, 2011 3:16 am




Posts: 54
Joined: Thu Nov 03, 2011 8:52 am
Post Re: A non-unique and yet unique solution
Patrick suggested that I post a snapshot of the 12x12 puzzle that started this thread once the solution was published. Here it is:

Image

I have highlighted the four cells that make the grid non-unique, although the CalcuDoku still has a unique solution.


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Posted on: Mon Nov 07, 2011 11:30 am




Posts: 2215
Joined: Thu May 12, 2011 11:58 pm
Post Re: A non-unique and yet unique solution
giulio wrote:
I have highlighted the four cells that make the grid non-unique, although the CalcuDoku still has a unique solution.

It depends on how you define a non-unique grid. In this case you could first define
an operation as you described:
- for 2 distinct cells in the grid, consider the 4 corner cells determined by those 2, and rotate once clockwise
Then define two grids to be equivalent if one such operation results in the same grid.
(what you call "non-unique").

That said, you can apply this operation more than once. Does this make the resulting grid "non-unique"?

And what about larger operations like this? For example, you could find this arrangement in a grid:
Code:
4 ... 2 ... 1
.     .     .
2 ... 1 ... 4
.     .     .
1 ... 4 ... 2

which you can rotate as well. Does this mean this grid is "non-unique"?

(in general, what set of permutations are you allowed to apply to the grid
such that you will still call it a non-unique grid).

Patrick


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Posted on: Mon Nov 07, 2011 1:01 pm




Posts: 54
Joined: Thu Nov 03, 2011 8:52 am
Post Re: A non-unique and yet unique solution
When I saw two pairs of digits that I could swap without violating the rules of unicity of rows and columns, I found it very surprising. Especially because I had stumbled on them by chance. But it is in fact a fairly common occurrence. Just by looking at the puzzle, I found two other pairs that behave in exactly the same way: b3/c3 and b4/c4. Then, I looked at one of my puzzles and, within less than a minute, I spotted the same configuration.

As you say, one can also imagine a whole series of increasingly complex swaps that would keep the unicity rules satisfied. A simple 'square' swap is just easier to spot. I agree that without a definition of unicity, such a statement is meaningless.

My apologies for saying that your puzzle had a non-unique grid. It sounded bad. At least, my post gave us the opportunity to talk about other things... :)


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Posted on: Mon Nov 07, 2011 1:05 pm




Posts: 2215
Joined: Thu May 12, 2011 11:58 pm
Post Re: A non-unique and yet unique solution
giulio wrote:
My apologies for saying that your puzzle had a non-unique grid. It sounded bad. At least, my post gave us the opportunity to talk about other things... :)

No, exactly, don't worry about it.
(catchy headlines to increase readership :-) )

I'd probably only define two grids to be identical if they can be transformed
into one another by rotation or mirroring (of the whole grid).
(I think using the word "unique" is confusing here)

Patrick


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