THEOREM OF THE DAY: The Existence Theorem for Bachelor Latin Squares

Theorem

Bachelor latin squares exist for all orders except for 1 and 3.

Click on the image.

The question of whether bachelor squares exist for all orders (except n = 1 and n = 3, which do not admit bachelors) goes back to Leonhard Euler who observed in 1779 that the addition table modulo n has no transversals when n is even. The case of odd-order latin squares is less tractable: a conjecture attributed to HJ Ryser says that these squares always have at least one transversal. 

However, Henry B. Mann showed in 1944 that bachelors exist for all n ≡ 1 (mod 4). The case n ≡ 3 (mod 4) was resolved more than 60 years later, in 2006, by Anthony B Evans and, independently, by Ian Wanless and Bridget Webb, who used the simple lemma, above right, to show that confirmed bachelors exist for all n , 1, 3.