The current Rugby World Cup format is four groups of five teams. The five teams in a group all play each other once, a total of ten games in each group. When all the matches in the groups have been played, the top two teams in each group (a total of eight teams) progress to the quarter finals. The other teams play no further matches.
In the quarter finals the top teams in each of the groups play a game against the second-placed team from another group.
The winners of the four quarter-final games progress to the semi finals, and the four losers play no further matches.
In the semi finals, the two winners progress to the final, with the winners of the final becoming World Champions; the two losers play a Third-Fourth play off.
The questions is, what is the largest number of teams that can finish the tournament having lost exactly one game?
First here is our solution, which we thought was good.
In the group stages, in each group one team beats all the other four teams, and the other four teams draw their games with each other. [An unlikely scenario, it is true, but it is possible]
This means the four group winners have a record of Played 4, Won 4, Lost 0. The other 16 teams have a record of Played 4, Won 0, Drawn 3, Lost 1.
There are various methods to determine which of the other four teams in the groups have come second, and these teams go forward to the quarter finals. The other twelve all leave the tournament having lost exactly one game.
In the quarter finals there are four team that have lost no games and four teams that have lost one game. The four group winners all lose to the four group runners up. Meaning the four group winners all leave the tournament having lost exactly one game, their quarter final.
At this point all 16 teams to leave the tournament have lost exactly one game
And there are four teams remaining that have lost one game.
The World Champion team will lose no further games, so they lose a total of one game. The losing finalist will lose one further game (the final), so they lose a total of two games. The third-placed team will lose one further game (the semi final), so they lose a total of Two games. And the fourth-placed team will lose two further games (the semi final and the play off), so they lose a total of three games.
This means that a total of 17 teams finish the tournament having lost exactly one game.
But there is a better solution in an hour by Thomas Cappleman, which you will not believe.