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A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square shows, among other things, perfect squarings of the square with provably least number of smaller squares.

  • Regarding existence: I don't know any non-square rectangle that has a perfect squaring. If such rectangles do exist, how does one characterize those rectangles that admit perfect squaring? Or is it the case that any given rectangle has some perfect squaring, maybe with a very large number of squares?

  • If such rectangles do exist, are there rectangles for which a perfect squaring can be done using a lesser number of smaller squares than is needed when a square is squared? This seems unlikely.

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    $\begingroup$ Doesn't the wikipedia article already show a perfect squaring of a rectangle ? $\endgroup$ Apr 24 at 7:17
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    $\begingroup$ Not an answer, but might be useful: "Tiling with Incomparable Rectangles," The American Mathematical Monthly, vol. 81, no. 6, pp. 664-665. "Two rectangles are incomparable if neither can be placed inside the other when they are aligned so that the corresponding sides are parallel." "Any rectangle can be tiled with mutually incomparable rectangles. We prove this by constructing a covering of a rectangle by seven tiles each with the same area." $\endgroup$
    – JRN
    Apr 24 at 7:24
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    $\begingroup$ thanks HenrikRuping for pointing this out. indeed, there are subsets of the squared square layouts that are perfectly squared rectangles and i had overlooked them. $\endgroup$ Apr 24 at 8:03
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    $\begingroup$ As a partial answer to your more general characterization question: not every rectangle has a perfect squaring. A necessary condition is that the ratio between the lengths of the sides is rational, because otherwise there does not exist any tiling of the rectangle with squares. For a proof, see for instance the chapter "Tiling rectangles" in Proofs from THE BOOK (4th edition or later, chapter number varies between editions), or Chapter 12 of Matoušek's "Thirty-three Miniatures". $\endgroup$ Apr 24 at 23:52
  • $\begingroup$ Squaring.net claims that the aforementioned condition is also sufficient: "If a rectangle can be squared then there are infinitely many perfect squarings" (see here and here). The latter refers to Max Dehn's 1903 paper "Über Zerlegung von Rechtecken in Rechtecke" and the textbook "Mathematics, The Man-Made Universe" by Sherman K. Stein for proofs. However, I don't see the proof that the condition is sufficient in either source (though I may have overlooked it). Maybe it's obvious to experts? $\endgroup$ Apr 28 at 18:24

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Yes, there are non-square rectangles that admit a perfect squaring. The smallest number of squares in a perfect squaring of a rectangle is 9. On the other hand the smallest number of squares in a perfect squaring of a square is 21.

All perfect squarings of rectangles using between 9 and 17 squares have been found and are listed here. There appear to be many more rectangles that admit a perfect squaring than squares that admit a perfect squaring. See the quote by David Gale here.

The unofficial logo of the Department of Combinatorics and Optimization at the University of Waterloo is actually a perfect squaring of the 32 $\times$ 33 rectangle using 9 squares.

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  • $\begingroup$ Thanks for the answer. And the quote. So, characterization of rectangles that admit perfect squaring is the real challenge. Hope you could clarify if 9 is provably the least number of squares in any perfect squaring of any rectangle. And maybe rectangles with irrational ratio between length and width do not admit perfect squaring. $\endgroup$ Apr 24 at 8:07
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    $\begingroup$ Yes, 9 is provably the least number of squares in any perfect squaring of any rectangle. $\endgroup$
    – Tony Huynh
    Apr 24 at 8:23

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