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I was inspired by this question Algorithm to find the largest number of identical squares that fit into a square with a specified area (ex: 972), where OP and we surmised that the solution had to use unit tiles (since the identical squares needed to have integer side lengths)

One question posed toward my solution of $\displaystyle \lfloor \sqrt{A}\rfloor^2$ for the max number of unit tiles that could fit inside a square of area $A$ was:

Is there some way to fit in more than that amount?

So I thought (for example purposes) there would be ample room to experiment with if $A=n^2-1$ for some integer $n$.

I don't know how to begin, and I personally don't think it's possible, since there is neither enough horizontal space for more than $\displaystyle \lfloor \sqrt{A}\rfloor$ tiles in any column (# of tiles touching some $x=x_1$) nor enough vertical space for more than $\displaystyle \lfloor \sqrt{A}\rfloor$ tiles in any row (# of tiles touching some $y=y_1$).

Wouldn't this necessarily bound the number of tiles to $ \lfloor \sqrt{A}\rfloor^2$?

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Packing problems are hard. There are often irregular configurations that allow denser packing. The Wikipedia page gives an example of packing five unit squares in a square of side $2+\frac 1{\sqrt 2} \approx 2.707$ The formula $\lfloor \sqrt A \rfloor^2$ would say you can only put four in. That formula assumes that all the packing squares are aligned with the outer square.

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  • $\begingroup$ very interesting, thank you! $\endgroup$ – Saketh Malyala Dec 3 '19 at 4:42

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