Completing Latin squares

Abstract

Some of the oldest combinatorial objects, whose study apparently goes back to ancient times, are the Latin squares. To obtain a Latin square, one has to fill the n² cells of an n×n square array with the numbers 1, 2,..., n so that that every number appears exactly once in every row and in every column. In other words, the rows and columns each represent permutations of the set {1,..., n}. Let us call n the order of the Latin square.