## Teaser 3004: Going up

**From The Sunday Times, 19th April 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?

[teaser3004]

## Jim Randell 9:33 am

on18 April 2020 Permalink |I supposed there were several teams,

A, B, C, …, tied at the top. And we are looking for situations where teamAis looking at a guaranteed promotion, if they win both their games.Obviously any of the other teams tied at the top of the table would also be in with a chance of promotion if they win both their games (as they will have the same maximum number of points as

A).But if one of the teams were to win a match by an enormous number of goals, it would give them an unassailable goal difference. So

Acan only be guaranteed a win if there are fewer than three teams tied at the top after the games are played (so the goal difference rule doesn’t come into it).So we need to look at numbers of teams, such that there is an arrangement of remaining matches, where

A(and onlyA) is guaranteed a promotion if they win both their matches, but if there were one more team then there would be no such arrangement of matches.This Python program is a bit slow (it takes 8.9s), but it does find what seems to be a reasonable answer. I may see if I can come up with a more efficient program later.

Solution:There are 6 teams tied at the top. 7 of the remaining matches involve at least one of those teams.For six teams at the top:

If

AplaysBandCand wins both matches, then neitherBnorCcan achieve the maximum number of points, so they are out of contention.And there are three more teams at the top,

D, E, F, and they all play each other, then only one of them can achieve 2 wins, to tie withAat the top of the table.So

Ais guaranteed to finish in the top 2 if they win both their games, and get promotion.Introducing a seventh team,

G, would mean that a third team could get two wins, soA‘s promotion would not be guaranteed.LikeLike