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From: nagle@netcom.com (John Nagle)
Subject: Re: How many eight puzzle states?
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Date: Tue, 14 Feb 1995 01:12:48 GMT
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cswellin@eos.ncsu.edu (Carol Smith Wellington) writes:
>I have discovered picking up and switching an odd number of pairs of tiles
>from a legal state gives an illegal state.  In addition, rotating any 4 tiles
>of a legal state gives an illegal state.  Unfortunately, not all operations
>which can cause illegal states always do.  For instance, rotating 5 tiles from
>a legal state may cause an illegal state, but might lead to another legal state.

>In general, for every legal state I can give at least one illegal state without
>worrying about duplicates from modifying other legal states.  For instance, for
>every legal state, the state generated by switching the first two tiles will be
>illegal and specific to this state.  Therefore, the number of legal states
>cannot be greater than 9!/2, but I cannot prove that is the exact number of
>legal states.

      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
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.

Once you know this, all the fun goes out of doing it by hand.  You
also realize it's really a rather lame puzzle.

						John Nagle
