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# Rugby World Cup: Several mathematical challenges in the design of tournaments

## That’s Maths: Knockout tournaments are dramatic in that each match results in immediate elimination of the losing team

On September 8th, I opened The Irish Times to find an A2 Poster with the programme for the Rugby World Cup. The plan showed the 20 teams who must do battle in which ultimate triumph requires survival through the preliminary rounds and victory in the quarter-finals, semi-finals and the final climax. We have reached the quarter-finals on seven occasions; this time, we are among the favourites to win the Webb Ellis Trophy.

Competing countries are divided into four pools of five teams. Teams in each pool play one another in a round-robin, with the top two teams advancing to the knockout stage. Our first big challenge is to get out of Pool B, which includes three of the top five teams, Ireland, South Africa and Scotland. We have already played and beaten Romania and Tonga. Only two of the five teams in each pool will advance.

#### Ensuring fair play

The design of tournaments presents several mathematical challenges. Ideally, the competition should determine the best team with a high probability, the final rankings should mirror the relative skills of the teams and spectators should enjoy exciting matches, culminating in a nail-biting final between the two best teams. There are several other constraints, and the structure of the tournament should reflect these.

The two basic patterns are round-robin tournaments and knockout tournaments, each having strengths and weaknesses. In a round-robin, each team plays each of the others once. In theory, this is the fairest way to determine the best team in the group.

The system is also better for ranking the teams, not just determining the winner. However, with many teams, these tournaments require large numbers of matches, increasing quadratically with size. For 20 teams, 190 matches are needed if each team plays each other once. The tournament could take months.

#### Brief and brutal

Knockout tournaments are more dramatic and can produce very exciting matches. Each match results in immediate elimination of the losing team, so the knockout is a brief and brutal scheme. With 20 teams, the match count is only 10 per cent of the round-robin count and increases linearly with the number of teams. With so many conflicting expectations, compromise is essential and, in their wisdom, the World Rugby Council has decided on a hybrid system of a round-robin followed by a knockout.

At the pool stage, a match won is worth four points, a draw is worth two and losses are worth zero. Bonus points are also awarded for scoring a minimum number of tries or for losing by a narrow margin. If two teams are level on points, tie-breaking criteria determine their final positions.

With eight teams advancing from the pools, the knockout stage has three rounds. There is also a play-off for third place making a total of 48 matches in the tournament. In each match tied at 80 minutes, extra time is played. If the scores are still tied, a sudden-death period will be played. If there is still no change, a kicking competition determines the winner.

A simple puzzle: in a pure knockout contest with 20 teams, calculate the minimum number of matches to select a winner. Easy: Ten first-round matches leave 10 teams; five in the second leave five; two third-round matches with a team getting a bye leave three; one lucky team getting a bye, the fourth round – the semi-final – eliminates one; the two remaining teams compete in the final. So, the count of matches comes to 19.

But there is a much simpler solution: every match produces a “loser” and, with 20 teams, there are 19 losers!

• Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com