Notes
stacked boxes in circles in squares solution

Solution to the Stacked Boxes in Circles in Squares Puzzle

Solution by Pythagoras' Theorem

As each pink square has area $1$, it has side length $1$, and so the diameter of each circle is $1$ and therefore the diagonal of each rectangle is $1$.

Consider the general case, and let $n$ be the number of little squares. Let $d$ be the side length of one of the little squares. Then Pythagoras' Theorem states that:

Therefore the total area of the $n$ little squares is $\frac{n}{n^2 + 1}$.

The sum of the areas is therefore:

Since $n \ge 1$, $n^2 + 1 \le n^2 + n$ so $\frac{n}{n^2+1} \ge \frac{n}{n^2+n} = \frac{1}{n+1}$. So this series is the harmonic series (except the first term) and so diverges.