# Inspired by Roberto Nicolaides

**Geometry**Level 5

How many cubes with edge length 7 are there in \(\mathbb{R}^3\) such that all the vertices are in \(\mathbb{Z}^3\) and the origin is one of the vertices?

If you come to the conclusion that there are infinitely many such cubes, enter 666.

**Clarifications**:

\(\mathbb{R}^3\) is the set of all (ordered) triples \((x,y,z)\) of real numbers and \(\mathbb{Z}^3\) is the set of all (ordered) triples\((x,y,z)\) of integers.