There are several ways to write the number $\dfrac{1}{2}$ as a sum of square reciprocals using distinct integers.
For instance, the numbers $\{2,3,4,5,7,12,15,20,28,35\}$ can be used:
$$\begin{align}\dfrac{1}{2} &= \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + \dfrac{1}{5^2} +\\ &\quad \dfrac{1}{7^2} + \dfrac{1}{12^2} + \dfrac{1}{15^2} + \dfrac{1}{20^2} +\\ &\quad \dfrac{1}{28^2} + \dfrac{1}{35^2}.\end{align}$$
In fact, only using integers between $2$ and $45$ inclusive, there are exactly three ways to do it, the remaining two being: $\{2,3,4,6,7,9,10,20,28,35,36,45\}$ and $\{2,3,4,6,7,9,12,15,28,30,35,36,45\}$.
How many ways are there to write $\dfrac{1}{2}$ as a sum of reciprocals of squares using distinct integers between $2$ and $80$ inclusive?
solutions/s152.md.