Magic square of squares

Definitions and mathematical statement

A $3\times3$ magic square is an array $(a_{ij})_{1\le i,j\le3}$ for which the three row sums, three column sums, and two principal diagonal sums all equal a common integer $M$. This problem requires each entry to be a positive integer square and all nine entries to be pairwise distinct. In explicit variables, ask whether there are $x_{ij}\in\mathbb Z_{>0}$ with pairwise distinct $x_{ij}^2$ and \[ \sum_jx_{ij}^2=M\ (i=1,2,3),\qquad \sum_ix_{ij}^2=M\ (j=1,2,3), \] \[ x_{11}^2+x_{22}^2+x_{33}^2 =x_{13}^2+x_{22}^2+x_{31}^2=M. \] The target is existence of one such array, or a nonexistence proof. The positive-square convention is the standard version intended here; allowing zero is a separate boundary variant. Distinctness rules out constant arrays and other trivial repeated-entry examples.

Rotations and reflections of an array preserve the conditions. Multiplying every entry by the same positive square also preserves them. Neither symmetry has to be quotiented out for the existence question. Checking only rows and columns gives a semimagic square and leaves two required diagonal equalities unverified. Both diagonals must be checked for a proposed example.

Short English statement

Is there a three-by-three magic square whose nine entries are distinct positive perfect squares?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3391 (OPG-16570), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Magic square of squares. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/magic_square_of_squares.