Placing the base at the midpoints of the bottom base of the cube, the diagonal will be 2/2. If you extend the side and tilt it so that the other base touches the midpoints of the top of the cube (also 2/2), you'll have a side that is 5/2. Since this is not a square, you have to add a fixed number x to 2/2 and subtract the same fixed number from 5/2. So, (2/2 + x) = (5/2 - x), yielding x= (-2 + 5)/4. Substituting x into one of our expressions, we reveal that the side of our square is:
(2 + 5)/4
A square with this side will be the largest square you can fit inside a unit cube.
Problem Insight
