Think 2-D Is Cool? Wait Till You Try 3-D!

Geometry

A lattice cube is a cube in 3-dimensional space, whose vertices are lattice points. Describe the set of possible side lengths of a lattice cube.

Note: A lattice point is a point whose coordinates are all integers. $(0,0,1)$ is a lattice point, but $(\frac{1}{2}, \sqrt{2}, -0.2 )$ is not a lattice point.

For similar problems, you can read my note on Construction.

\{\sqrt{n}\mid n \text{ is the sum of 3 perfect squares.}\}

\{ n \mid n \text{ is a non-negative integer.} \}

\{ \sqrt{n} \mid n \text{ is a non-negative integer.} \}

\{ \sqrt[3]{n} \mid n \text{ is a non-negative integer.} \}