Each row (and each column) of any solution to the packing problem would give you a way to write 16 as a sum of 5s and 3s. Since there is only one such composition, 16=3+3+5+5, this could only work if the total area of the 3x3s were 6/16, and the total area of the 5x5s were 10/16, of the area of the 16x16. And it ain't. This kind of reasoning can only offer a proof of impossibility, of course. If more than one composition existed, you could get Diophantine constraints on how many rows of each type would need to exist, which might still show impossibility. Probably other people will have better ideas... --Michael On Wed, Feb 29, 2012 at 3:27 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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