# Sunday Times Teaser 3004 – Going Up

*by John Owen*

#### Published Sunday April 19 2020 (link)

In our football league, the teams all play each other once, with three points for a win and one for a draw. At the end of the season, the two teams with most points are promoted, goal difference being used to separate teams with the same number of points.

Last season’s climax was exciting. With just two games left for each team, there were several teams tied at the top of the league with the same number of points. One of them, but only one, could be certain of promotion if they won their two games. If there had been any more teams on the same number of points, then none could have guaranteed promotion with two wins.

How many teams were tied at the top of the league, and how many of the remaining matches involved any of those teams?

My thanks to John Crabtree for offering and giving me a strategy for solving this teaser. Given that I dislike anything to do with football and football teasers, I have since spent an inordinate amount of time trying (and failing) to find a fast program to give the solution. The problem is that as the number of tied teams increases, the different number of ways these teams can play against each other (and other teams) increases rapidly and for each such arrangement we have to look at all win/loss combinations. This is time consuming but is fast enough for up to six tied teams, but at seven and beyond it becomes very slow. I have found some speed ups but all of them are essentially ‘cheating’ in that they simply bypass the final step, which is to find a number of tied teams for which no team that wins two matches can guarantee promotion. Its very unlikely that I will spend more time on this but Jim Randell has found the same issue in his solution where he indicates that he might do some more work on this.