There's a solution for dividing a square into unique rectangles with the same area which is the blanche dissection. There's also a solution for dividing a square into unique rectangles with the same diagonal by Ed Pegg. In addition theres even a solution for dividing a square into similar rectangles where the solution even uses the plastic constant as a part of the solution. But what about a solution for dividing a square into unique rectangles with the same perimeter?
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$\begingroup$ The area problem is discussed at mathoverflow.net/questions/220567/… $\endgroup$– Gerry MyersonCommented May 15, 2018 at 12:52
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$\begingroup$ This question was touched upon in this 2013 document: arxiv.org/ftp/arxiv/papers/1307/1307.3472.pdf and a guess was made that the answer is negative (page 5). A "rectangle" that can be cut into 7 rectangles all of same perimeter but different areas is shown. A further question as to whether there are rectangles that can be cut into more than 9 rectangles all of same perimeter and different areas is also posed. $\endgroup$– Nandakumar RCommented Aug 12, 2023 at 9:20
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