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From: phil@eurocontrol.fr (Philip Gibbs)
Subject: Re: How many eight puzzle states?
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Date: Tue, 14 Feb 1995 14:10:38 GMT
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In article <nagleD3yu1C.8zL@netcom.com>, nagle@netcom.com (John Nagle) writes:
> 
>       Half the possible states are legal, but I don't have a cite for that.
> 
>       Incidentally, the way to solve the 2^N-1 puzzle generally is as

you mean N^2-1

> follows:
> 
> 	1.  Solve the top row.  Never touch it again.
> 	2.  Solve the left column.  Never touch it again.
> 	3.  Recurse to solve the remaining subpuzzle.
>

If you know about signatures of permutations you can solve this problem. In
the case where N is even the signature of the permutation of the pieces is 
an invariant of the group of transformations. That is the permutation defined
by reading of the numbers left-to-right and top-to-bottom and ignoring the
space. Therefore only even permutations can be legal. To show that all
even permutations are legal you just have to prove that the solution John
described works for those cases.

Sorry that's a bit brief I'm in a rush.

  

