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Ref 1: dividing a square into unique rectangles with the same perimeter

  1. https://arxiv.org/ftp/arxiv/papers/1307/1307.3472.pdf

Ref 1 asks if a square can be cut into some finite number of rectangles all of same perimeter but different areas. Ref 2 shows a rectangle that can be cut into 7 such rectangles and mentions those that can be cut into 8 or 9 such rectangles and makes the following guess:

Guess: The unit square cannot be cut into any finite number of rectangles all of same perimeter but different areas - at least when the dimensions of the pieces are rational.

Can a proof or counter be given for the above?

Further question: Are there rectangles that can be cut into 10 or more rectangles that are all of same perimeter but have different areas? More generally, how to characterize rectangles that allow such partition(s)?

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  • $\begingroup$ it seems quite hard even to cut a square into a finite "inclusion co-chain" of rectangles, that is in such a way that none of them is included in some other. I guess not less than 7 rectangles are needed. $\endgroup$
    – Pietro Majer
    Commented Aug 12, 2023 at 10:18
  • $\begingroup$ @Pietro, take a $3\times3$ square, cut it into a $1\times1$ square in the middle, and four $1\times2$ rectangles surrounding the small square. That's five rectangles, no one of which is included in some other. But perhaps I misunderstand your criteria. $\endgroup$
    – Gerry Myerson
    Commented Aug 13, 2023 at 4:09
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    $\begingroup$ Yes sorry, I mean no "inclusion up to rigid movement". Here the 1x1 square is included in any of the 1x2 rectangles, and each of them is included in the others. $\endgroup$
    – Pietro Majer
    Commented Aug 13, 2023 at 6:01

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