Another situation cropped up last night when a basic knowedge of quadratic equations proved very useful.
Richard Bacon posed a problem on Radio Five Live:
There is a round robin football tournament, where each team plays every other team once and once only. If there are 91 matches in total, how many teams take part?
If there are n teams, then they have to play the other n – 1 teams, making a total of n(n – 1), but each game is counted twice, so the total number of matches must be n(n – 1)/2.*
So: n(n – 1)/2 = 91
=> n(n – 1) = 182
=> n2 – n = 182
=> n2 – n – 182 = 0
=> (n – 14) (n + 13) = 0
So there must be 14 teams in the tournament, since a tournament with -13 teams does not make much sense.
*The number of matches can also be viewed as a simple arithmetic series:
Team 1 plays n-1 games
Team 2 plays n-2 games (not counting the game with team 1)
Team 3 plays n-3 games (not counting the games with teams 1 and 2)
.
.
Team n-1 plays 1 game with team n (not including the games counted before).
So the total number of games is:
1 + 2 + 3 + … + (n-2) + (n-1) = n(n-1)/2
