From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Jigsaw puzzle problem
Date: Fri, 7 Jan 2000 16:33:40 -0600
Newsgroups: sci.math
Summary: [missing]

On Sat, 8 Jan 2000, John R Ramsden wrote:

> Samu Aalto wrote:
> >
> > I once heard about a jigsaw puzzle, where using the same pieces,
> > you were able to form both perfectly square and perfectly circular
> > 2D planes. I remember, that minimum amount of pieces needed was
> > eventually something like 10^50. Can You point me to the
> > information over this topic?

M. Laczkovich, Equidecomposability and discrepancy; a solution of
Tarski's circle-squaring problem, J. Reine Angew. Math. 404 (1990),
77-117. (Mathematical Reviews 91b:51034)

> Hi Samu, I think you may be thinking of the Banach-Tarski paradox,
> although this is usually expressed in terms of reassembling one disk
> into another with a different area (which means you may not be thinking
> of this after all).

The Banach-Tarski paradox needs 3 dimensions. You can't reassemble a
plane figure into another with a different area; however, you can
change the shape, as Laczkovich showed in his solution of Tarkski's
old "circle-squaring problem".
==============================================================================

From: Pierre Bornsztein <pbornszt@club-internet.fr>
Subject: Re: Jigsaw puzzle problem
Date: Fri, 07 Jan 2000 23:47:34 +0100
Newsgroups: sci.math

John R Ramsden wrote:
> 
> Samu Aalto wrote:
> >
> > I once heard about a jigsaw puzzle, where using the same pieces, you
> > were able to form both perfectly square and perfectly circular  2D
> > planes. I remember, that minimum amount of pieces needed was eventually
> > something like 10^50. Can You point me to the  information over this
> > topic?
> 
> Hi Samu, I think you may be thinking of the Banach-Tarski paradox,
> although this is usually expressed in terms of reassembling one disk
> into another with a different area (which means you may not be thinking
> of this after all).

Hi, 

Here is what it is written in Stan Wagon's book "Unsolved problem in
mathematics" (sorry I don't have the name of the editor), and in
"Unsoved problems in Geometry" by Croft, Falconer and Guy (Springer) :

Miklos Laczkovich (Hungary) solved in 1990 affirmatively the following
problem :
"Can a circle be decomposed into finitely many sets that can be
rearranged to form a square (of equal area)."

Moreover, he is able to accomplish the decomposition using only
translations, which is quite surprising. 
Laczkovich estimates that his construction for squaring the circle
requires about 10^50 pieces. It is not difficult that at least three
pieces are required. Can closer bound be obtained?

Third, his work applies to polygons : Thus he improves Tarski's
set-theoretic version of the Bolyai-Gerwein theorem by showing that any
two polygons having the same area are equidecomposable using
translations alone.


Even more recently, Laczkovich showed that any pair of bounded
measurable subsets of |R^d of equal d-dimensional content are
translation equidecomposable, as are a ball and cube of equal volume in
|R^3. These construction depend heavily on the axiom of choice (can this
be dispensed with, perhaps with the pieces Lebesgue mesurable, or even
Borel sets?"

Obviously, Laczkovich's method are quite complex (using ideas from the
theory of uniform distribution of sequences)...

Reference : 
M.Laczkovich, "Equidecomposability and discrepancy : o solution to
Tarski's circle squaring problem".
J.Reine Angew.Math. 404 (1990) P.77-117.

Pierre.