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.


\(\mathbb{R}^3\) 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.



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