Euler square

An Euler square is a square array made by combining n objects of two types such that the first and second elements form a Latin square. 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 are no Euler squares of order n = 4k + 2 for k = 1, 2, .... However, such squares were found to exist in 1959 by Bose and Shrikande, refuting the conjecture.