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A perfect squared square is a squared square with all elements of different sizes with at least two elements. The "proportionateness" of a squared square is the ratio of the size of the size of the squared square to the side of the smallest component square. The first page of http://www.squaring.net/downloads/99-nice-pss.pdf contains the most "proportionate" squared square I know of, with a ratio of $1032/48 = 21.5.$ Is any better ratio possible? By "better" I mean lower.

Update: searched up to order $27,$ could not find any better one.

mathlander
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    Are you the maintainer of Squaring.net (the source of your link)? If not, have you tried contacting the maintainer directly (via the email address on the main page)? – Blue Aug 04 '22 at 20:38
  • I haven't done anything to contact the maintainer. – mathlander Aug 04 '22 at 20:48
  • A little more clarity of the definition, i.e. that a square is dissected into squares of different sizes, would improve the Question. However I'm voting to leave open in review. – hardmath Aug 11 '22 at 16:54

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This question shows that the number of square numbers that can add up to a square number is unlimited. The method can initiate with a square of side as small as $3$ (although the side of the initiating square may of course be larger than $3$), and accumulate unlimited numbers of larger squares which collectively sum to a (yet larger) square number. In other words, it is possible to find very large square numbers that are the sum of thousands, or even billions, of smaller squares.

Although that does not demonstrate that any such arbitrarily large set of square figures can be arranged so as to tile a square figure whose area equals their sum, I think it likely that among the infinite collection of such sets there will be examples where such tiling is achievable and the ratio of the side of the large square to the side of the smallest component square might become as large as one wishes.

Keith Backman
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