An Euler square of order 
is an 
array of ordered pairs formed
from two sets of 
symbols, where 
is a positive integer.
The first and second components each form a Latin square,
and each ordered pair occurs exactly once (Bose et
al. 1960, p. 190). Euler squares are also known as Graeco-Latin squares,
Graeco-Roman squares, or Latin-Graeco squares.
Euler squares exist for every positive integer 
except 
and 
(Bose et al. 1960, pp. 202-203). Euler's
Graeco-roman squares conjecture asserted their nonexistence for orders 
with 
, 2, .... Counterexamples found in 1959 refuted the conjecture
(Bose et al. 1960, pp. 189-190).
