Putnam 2021, Problem A-3

Determine all positive integers $N$ for which the sphere
\[
x^2 + y^2 + z^2 = N
\]
has an inscribed regular tetrahedron whose vertices have integer coordinates.

Solution:
The integers $N$ with this property are those of the form $3m^2$ for some positive integer $m$.

In one direction, for $N = 3m^2$, the points
$$(m, m, m), (m, -m, -m), (-m, m, -m), (-m, -m, m)$$
form the vertices of a regular tetrahedron inscribed in the sphere $x^2 + y^2 + z^2 = N$.

Conversely, suppose that $P_i = (x_i, y_i, z_i)$ for $i = 1, \ldots, 4$ are the vertices of an inscribed regular tetrahedron. Then the center of this tetrahedron must equal the center of the sphere, namely $(0, 0, 0)$. Consequently, these four vertices together with $Q_i = (-x_i, -y_i, -z_i)$ for $i = 1, \ldots, 4$ form the vertices of an inscribed cube in the sphere. The side length of this cube is $(N/3)^{1/2}$, so its volume is $(N/3)^{3/2}$; on the other hand, this volume also equals the determinant of the matrix with row vectors $Q_2 - Q_1, Q_3 - Q_1, Q_4 - Q_1$, which is an integer. Hence $(N/3)^3$ is a perfect square, as then is $N/3$.