Originally posted by THUDandBLUNDER
That's correct, Acolyte.
While being more precise, I'm not sure if mentioning 'projections onto the plane' would actually makes things clearer to most people.
OK. In that case, what we want to do is maximize the projection area of the unit cube, and minimize the projection area of the bigger cube.
To maximize the projection area of the unit cube, we have to look for the longest diagonal, which is the one going from one corner of the cube to the opposite corner. This has length 3^(1/2). By symmetry, we know that a pair of these exist. So the biggest projection you could have would be a square having diagonal length 3^(1/2), resulting in a side length (3/2)^(1/2).
To minimize the projection area of the bigger cube, we have to look for the smallest diagonal. This is the diagonal going from one corner to the opposite corner on the same face. We set this equal to the diagonal length of the unit cube, or 3^(1/2), resulting in a side length of (3/2)^(1/2), which is approximately equal to 1.225.
So the biggest cube has side length (3/2)^(1/2).