Number of ways that a football league can complete with all teams having the same number of points

Abstract

When $n$ teams play in a football league with home and away matches against every opponent there are $M = n \cdot (n-1)$ matches. There are 3 possible match results: a victory is awarded 3 points, a draw 1 point and 0 points for a defeat. Hence we have $3^M$ possible outcomes. In this paper the number of ways is determined that a football league can complete with all teams having the same number of points. An algorithm that works until $n=8$ is presented.