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I am having a hard time answering the following questions, despite them seeming elementary at first glance.

Is it true that the maximum number of (non-overlapping) squares with side lengths $x$ cm that fits inside a larger square with sides $y$ cm, is $\left(\left\lfloor\frac{y}{x}\right\rfloor\right)^2?$

Is it true that the maximum number of (non-overlapping) cubes with side lengths $x$ cm that fits inside a larger cube with sides $y$ cm, is $\left(\left\lfloor\frac{y}{x}\right\rfloor\right)^3?$

I think these are true, but I don't actually know how to prove them. On the other hand, I was thinking that if we make enough room then maybe we can fit one more square/cube in one of the corners in some cases?

Maybe this can be reduced to a linear programming problem, and/or the pigeonhole principle, and/or some inequalities like Cauchy-Schwarz might be useful.

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    $\begingroup$ It depends if you are stacking them nicely or not. Do the small squares or cubes have to be aligned with the large ones? If not you can pack five unit squares in a square with side $2 + \frac {\sqrt 2}2 \approx 2.707$ while you would claim the maximum is four. See en.wikipedia.org/wiki/Square_packing for examples $\endgroup$
    – Ross Millikan
    Commented Jun 8 at 23:33
  • $\begingroup$ @RossMillikan the squares do not have to be aligned, and so the article you linked to is exactly what I was looking for. Thanks. $\endgroup$
    – Adam Rubinson
    Commented Jun 9 at 0:26

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