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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)?

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    Solutions 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

3 Answers 3

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$T(m,n)$ is tabulated at the OEIS. Also, several references are given:

Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.

Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Bertram Felgenhauer, Tiling rectangles by minimal number of squares.

Maybe you could have a look at those, to see whether your question is considered (and, if it is, you could then report back).

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    No luck there..2012-11-20
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I've posted a 4944 possible counterexamples at Possible Counterexamples to the Minimal Squaring Conjecture. Odds are at least one of these is a valid counterexample.

Data and verified pictures up to 388 at Minimally Squared Rectangles

possible counterexample

Other information at tiling a rectangle with the smallest number of squares

According to my notes there, my minimal possible counterexample is currently

enter image description here

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    Yes, @EuYu. Rectangle $71 \times 81$ requires $10$ squares only. Edges (for example): $38$ (left-up), $43$(right-up), $33$(left-bottom), $28$(right-bottom), $20$(bottom), $8$, $8$, $5$, $4$ and $4$.2012-11-22
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Here's the Bouwkamp code of two simple perfect squared rectangles:

  • 13: 304x274 (141,78,85)(71,7)(92)(133,8)(51,28)(23,97)(74);
  • 15: 152x137 (83,69)(31,38)(54,29)(16,15)(8,30)(25,4)(1,22)(21).

There is no lower-order simple squared rectangle of either size, but there might be compound ones. If not then T(304,274) = 13, T(152,137) = 15, and T(304,274) < T(152,137).

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    The $152 \times 137$ rectangle can be subdivided into 12 squares with integer side lengths. http://pictat.com/show.php?i=/2012/11/29/43292squaredrec.png2012-11-29