Euler Square

A square Array made by combining $n$ objects of two types such that the first and second elements form Latin Squares. Euler squares are also known as Graeco-Latin Squares, Graeco-Roman Squares, or Latin-Graeco Squares. For many years, Euler squares were known to exist for $n=3$, 4, and for every Odd $n$ except $n=3k$. Euler's Graeco-Roman Squares Conjecture maintained that there do not exist Euler squares of order $n=4k+2$ for $k=1$, 2, .... However, such squares were found to exist in 1959, refuting the Conjecture.

See also Latin Rectangle, Latin Square, Room Square

References

Beezer, R. ``Graeco-Latin Squares.'' http://buzzard.ups.edu/squares.html.

Kraitchik, M. ``Euler (Graeco-Latin) Squares.'' §7.12 in Mathematical Recreations. New York: W. W. Norton, pp. 179-182, 1942.