I have a question on my homework that I cannot figure out. The question is: What is the largest number of identical squares whose areas are whole numbers that would fit in a square whose area is 972cm^2

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    $\begingroup$ So it seems clear that $1x1$ squares will get you the maximum, so now the question is how many $1x1$ squares fit into a square of side length $\sqrt{972}$. $\endgroup$ – Cheerful Parsnip Dec 3 '19 at 2:06

You will always maximize the number of squares you can fit by using those with side length $1$.

If the big square has area $n$, then the side length is $\sqrt{n}$. Then, you can fit $\lfloor \sqrt{n} \rfloor^2$ squares into it.

For example.. if the area is $972$, the side length is $31.17$ approx, which means you can fit a $31\times 31$ grid of squares inside it.

  • $\begingroup$ Ah, my bad. Read wrongly. $\endgroup$ – Calvin Lin Dec 3 '19 at 2:16
  • $\begingroup$ Does anything guarantee you can’t place them in such a way that more than $961$ fit? That would be incredibly sneaky, but you never know. $\endgroup$ – URL Dec 3 '19 at 4:14
  • $\begingroup$ @URL Clearly not. Considering the projections of the filling on each side, it's possible to improve any solution which isn't simply filling in $31\times 31$ squares greedily. $\endgroup$ – Don Thousand Dec 3 '19 at 4:34

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