Newsgroups: geometry.puzzles
From: haoyuep@aol.com (Dan Hoey)
Date: Fri, 7 Jan 2005 17:34:59 +0000 (UTC)
Subject: Re: Rectangle-to-4-similar-triangles-puzzle

On Fri, 7 Jan 2005 13:37:56 +0000 (UTC), Bill Smythe wrote:
>It seems to me that, in the case of a square, the method you
>suggest can never leave you with no two triangles equal.

Certainly.  That does not deny any claim I intended to make.

The original problem was to divide a rectangle into similar
triangles so that not all the triangles were equal.  The method
you refer to, of dividing the rectangle into two equal triangles
and subdividing triangles, was suggested by John Berglund, not me.
I described to you how it would solve the original problem when
the rectangle is a square.

In the message you quoted, I wrote:

> The only open question left here is the smallest number of similar
> triangles into which you can split a square, such that _no two
> are equal_....

This is a new problem.  I proposed it only for a square, since John
Berglund's method can produce a pairwise unequal dissection of any
non-square rectangle.  It is true that his method does not solve this
new problem for a square, for exactly the reason you observe above.

I know of a seven-triangle solution for the new problem.  I think
I can prove that it can't be done with three triangles.  Any results
that narrow the bounds would be appreciated.

Dan