From: Andrew Dudzik <adudzik@...>
Date: 2006-01-27T00:35:51+09:00
Subject: Re: [QUIZ.SUMMARY] Grid Folding (#63)

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>
> I think there is still some interesting math to pull out of this
> problem. For example, Andrew Dudzik ponders: "There are never two
> sequences that give the same perm. Does anybody know why this is?
> Seems like an interesting math problem." Myself, I wonder if he's
> right.
>

I think that this is true, and that it follows from Bill Dolinar's solution
for check_fold--the last fold is always uniquely determined by picking some
x and y coordinates and looking at the numbers on the top and bottom of the
stack--if, say, 4 and 7 are on opposite sides of the folded paper, they mus=
t
have been in the same sheet one fold ago--the direction of this fold is
determined by the relative orientation of 4 and 7 in the original, unfolded
paper.

Since there is only ever one possible direction to unfold, there can only b=
e
at most one sequence of unfolds that gives the desired 1--n^2 pattern.
Hence there is at most one sequence of folds that produces a given
permutation.

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