In the November 1996 Word Ways, I introduced the Euler square (almost known as a Graeco-Latin square), which is by its method of construction automatically magic. All rows and columns add up to a common sum since they merely consists of rearrangements of the same basic components.
"Self-Descriptive Euler Magic Squares,"
Word Ways: Vol. 30
, Article 15.
Available at: https://digitalcommons.butler.edu/wordways/vol30/iss3/15