1. The problem statement, all variables and given/known data A cube of sides a*a*a is in 3 dimensional space. All eight of its corners have integer coordinates. Prove that a is an integer. 2. Relevant equations - 3. The attempt at a solution First, I considered three corners of the cube p, q and r, with these, two vectors that run along the sides of the cube and have magnatude of a are q-p and r-p and these vectors have coordinates of (q1-p1, q2-p2 etc...). Taking the cross product of these two vectors, I arrive at a vector that has magnitude of a2. I had hoped that the expression for the length of (q-p)x(r-p) would turn out to be of the form (expression in p1, p2 etc...)2 and then this would show that a=that expression in the components of p, q and r therefore, since those are integers, a is an integer. Unfortunately, the expression for the length of that cross product is very, very ugly. I could, perhaps have put it into a computer programme and had that simplify it for me, but I'm sure there is some more efficient way of doing this. It is also possible to shift the cube such that one vertex is on the origin, this simplifies the calculation a lot, but it still isn't pretty and I still think that there is a more efficient method. There is a note beneath the question saying "This question is not assessed, so your only reward is aesthetical pleasure from solving it" which leads me to believe that the solution is clever somehow. Ideas? Thank you.