# Solution to the [[Stacked Boxes in Circles in Squares]] Puzzle +-- {.image} [[StackedBoxesinCirclesinSquares.png:pic]] > If the area of each pink square is $1$, what’s the area of the blue squares in each box as the sequence continues? Is the total area of the blue squares finite? =-- ## Solution by [[Pythagoras' Theorem]] +-- {.image} [[StackedBoxCircleSquareLabel.png:pic]] =-- 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: $$ 1^2 = d^2 + (n d)^2 = (n^2 + 1) d^2 $$ Therefore the total area of the $n$ little squares is $\frac{n}{n^2 + 1}$. The sum of the areas is therefore: $$ \sum_{n=1}^\infty \frac{n}{n^2+1} $$ 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.