The 6 × 6 Latin squares

The problem of the enumeration of the different arrangements of n letters in an n × n Latin square, that is, in a square in which each letter appears once in every row and once in every column, was first discussed by Euler(1). A complete algebraic solution has been given by MacMahon(3) in two forms, both of which involve the action of differential operators on an expanded operand. If MacMahon's algebraic apparatus be actually put into operation, it will be found that different terms are written down, corresponding to all the different ways in which each row of the square could conceivably be filled up, that those arrangements which conflict with the conditions of the Latin square are ultimately obliterated, and those which conform to these conditions survive the final operation and each contribute unity to the result. The manipulation of the algebraic expressions, therefore, is considerably more laborious than the direct enumeration of the possible squares by a systematic and exhaustive series of trials. It is probably this circumstance which has introduced inaccuracies into the numbers of 5 × 5 and 6 × 6 Latin squares published in the literature.