A proper Euler magic matrix of order $5$

Abstract

An Euler magic matrix is an integer matrix $M$ with $M{M}^t={\hat{I}}^{3}I$ whose squared entries sum to ${\hat{I}}^3$ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-$4$ proper example, and Müller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case. We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.

10 pages, plus verification source code