Euler's Square

Leonhard Euler (1707-1783), one of the most famous mathematicians ever, spent the last years of his life dealing with the different possibilities of magic squares. He was faced with the special problem to combine two sets of n symbols each so that neither in a row nor in a line a pair of symbols occured twice. As Euler used Greek and Latin letters, those creations got also known as Graeco-Latin squares.

Here is an example of a Graeco-Latin square of order 4. Or should I call it a letter-number square?

a1 b2 c3 d4
b3 a4 d1 c2
c4 d3 a2 b1
d2 c1 b4 a3

In each column and in each row each letter only occurs once. The same goes for the numbers. And third, no combination of a letter and a number occurs twice.

Euler solved the problem for squares of order n odd and n a multiple of 4. He supposed that there won't be any solutions for the odd multiples of 2. But in 1959, E.T.Parker from Remington Rand Univac and R.C.Bose and S.S.Shrikhande from the University of North Carolina succeeded, with the help of their computers, in creating a square of order 10, which for 177 years had believed to be impossible.

Graeco-Latin square of order 10

In each column and in each row each colour of the outer square and each colour of the inner square only occurs once. No combination of two colours occurs twice.

April 4, 1995
Mark Dettinger