Newsgroups: sci.math
From: haoyuep@aol.com (Dan Hoey)
Date: Jan 31, 2000 11:48 PM
Subject: Re: geometric dissection problem with rectangles

[ Reposted to sci.math--the original went only to rec.puzzles. ]

David Bernier writes:
>...The general question relates to two rectangles with integral
>sides whose areas differ by one square unit;  suppose a,b,c and d
>are positive integers and that  axb = cxd -1; suppose R_1 is an
>axb rectangle and R_2 is a cxd rectangle;  when can we dissect
>R_1 into two pieces along grid lines which, when reassembled,
>give R_2 less a hole of size 1x1 ?  ...

>As a specific instance, what about the case:  a=13, b=8, c=21, d=5 ?

This case is not possible.  For suppose we have a rectangle A
consisting of 13 rows of 8 squares.  We wish to cut this to cover
a shape B consisting of a 21 x 5 rectangle minus one square.
Neither piece of A can be longer than 13 squares in any dimension.
B must have at least one square on its first row and one square on
its 21st row, and these must be covered by separate pieces of A
because of the length restriction. The restriction also implies that
no part of the top 8 rows of B can be covered by the piece that
covers the 21st row, and since there are only two pieces, this 8 x 5
rectangle (minus at most one square) must be covered by one piece
of A.  Similarly, the bottom 8 rows of B form an 8 x 5 rectangle, minus
at most one square, covered by the other piece of A.  There are
only two ways to cut the two 8 x 5 rectangles from A (even allowing a
one-square overcut), and in neither case is any of the unused part
of A contained in the oo x 5 strips formed by extending the 8x5
rectangles' long axis.  So there is no way to extend the pieces to
cover the central 5 rows of B, QED.

This is a fairly ad hoc way of deciding a two-piece dissection problem,
but it is good enough to show that no skip-2-Fibonacci rectangle (of
size F_(n+1) x F_(n-2)) can be cut in two sets of squares to cover the
corresponding adjacent-Fibonacci rectangle (F_n x F_(n-1)) minus one
square, for F_n > 5.

ObPuzzle: Find the part of the proof violated by F_n=5.

Dan Hoey <haoyuep@aol.com>