Find three integers $a,\, b,\,$ and $c$ such that $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{c^2+b^2}$, and $\sqrt{a^2+b^2+c^2}$ are all integers. How would you go about finding three nonzero integers $a,\, b,\,$ and $c$ such that $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{c^2+b^2}$, and $\sqrt{a^2+b^2+c^2}$ are all integers? Does anyone know if this is not solvable, and if so, is there an elementary proof of it? 

This is the "integer cuboid" or "Euler brick" problem. Currently wide open. 
