Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that $T(kx,ky)
Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.
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$\begingroup$
combinatorics
number-theory
packing-problem
tiling
extremal-combinatorics
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0For some references and perhaps some computational ideas: http://oeis.org/A219158 – 2012-11-19
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0Of course not. (Waves hands and ducks) Maybe you can use some of the ideas in [Fourteen proofs of a result about tiling a rectangle-Wagon](http://mathdl.maa.org/images/upload_library/22/Ford/Wagon601-617.pdf) to prove this. – 2012-11-19
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1http://mathoverflow.net/questions/113899/minimum-number-of-integer-sided-squares-needed-to-tile-an-m-times-n-rectangle – 2012-11-21
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0Solutions up to 300×300 can be seen at [Minimally Squared Rectangles](http://demonstrations.wolfram.com/MinimallySquaredRectangles/) and [int-e.eu](http://int-e.eu/~bf3/squares/view.html#13,11). No counterexamples found yet. – 2013-04-03