A Cube and a Square

Find two whole numbers such that the difference of their squares is a perfect cube, while the difference of their cubes is a perfect square.

Solution

The numbers $6$ and $10$ do both at once: $$\begin{gathered} 10^2 - 6^2 = 100 - 36 = 64 = 4^3,\\ 10^3 - 6^3 = 1000 - 216 = 784 = 28^2. \end{gathered}$$ The difference of the squares is the cube of $4$, and the difference of the cubes is the square of $28$.