Hi Kerry, Although you are asking about a 16x16 square dissection, there is a parity test that can be applied when attempting a tiling of an odd sized square with squares of given sizes; Firstly reduce the sizes of the squares by their greatest common divisor then count the number of odd squares. Joseph DeVincentis noticed that the numbers of odd squares that can appear are restricted to 1 mod 4 http://www.mathpuzzle.com/partridge.html " I noticed that many of my best solutions for odd-sided squares (which was all of them, because composite-sided squares could invariably be done best as a blown-up version of the solution for their smallest factor) had exactly 5 odd-sided squares, and this was the minimum number of such squares by a parity argument. The area of a square of odd integer side is congruent to 1 mod 4, and the area of an even square is 0 mod 4. Thus, to fill this square, you need a number of odd squares that is 1 mod 4, and the smallest value, 1, could only be done using the degenerate solution of filling the square with a square, because of the parity issue in every row and column. The next smallest possible value was 5, and many of my solutions exhibited this, with an arrangement of odd squares that consisted of one large odd square in each of two opposite corners, and 3 smaller squares that formed an L-shaped path between those two, arranged so that each column and each row contained either 1 or 3 odd squares." Stuart Anderson

Message: 3 Date: Wed, 29 Feb 2012 13:27:15 -0700 From: Kerry Mitchell <lkmitch@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Tiling a square

Hi all,

I know that 7 * 5^2 + 9 * 3^2 = 16^2, which suggests that 7 5x5 squares plus 9 3x3 squares can possibly tile a 16x16 square.  Is there a relatively simple way to figure out if either:  a) no, they won't, or b) yes, and here's how?  Ideally, I'd like to learn the process, not just the answer, as I will probably have other combinations to consider.

Thanks, Kerry Mitchell -- lkmitch@gmail.com www.kerrymitchellart.com