# A Small Puzzle

I call the following visual proof that the sum of the first *n*
consecutive odd numbers is *n*^{2}, the "L"-decomposition.
Suppose we have two "L"-decompositions, can the pieces of these two
decompositions be reassembled into a single larger square?

Well, clearly it is necessary that the two "L"-decompositions we start
with are part of a Pythagorean triple.
(A triple of integers that satisfy the Pythagorean formula, for example
(3,4,5) or (5,12,13).)

In the case of the Pythagorean triple (3,4,5), a solution is given
below.

For *every* Pythagorean triple that I have tried, eventually
a solution was found. There is *no reason whatsoever* why
this should always be possible! But that is what seems to be the
case!

Here is the solution for the (5,12,13) Pythagorean triple.

### Update: Jan. 13, 1998

Erich Friedman has
conjectured that *every* sum of squares (i.e. things like
7^{2} + 7^{2} + 1^{2} + 1^{2} = 10^{2})
works this way. He has also described proofs that there are infinite
families of solutions that can be generated recursively.
The smallest triple that no one has found a solution for is (20,21,29).