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Football Matches

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1. A set of football matches is to be organized in a "round-robin" fashion, i.e., every participating team plays a match against every other team once and only once.

If 66 matches are totally played, how many teams participated?
Answer: 12

Solution:

If 12 teams participated, then the first team plays matches against the other 11 teams. The second team has already played against the first team, and so has to play matches against only the other 10 teams. In this manner, the second-last team has to play against only one team, and the last team has already played against all the teams. Thus, the total number of matches is

11 + 10 + ........ + 2 + 1 = 66 .

If 66 matches are totally played, then 12 teams participated.

Food for thought:

Is there a formula to conveniently add
11 + 10 + ........ + 2 + 1 ?
Indeed, there is! It simply equals 11 × 12 / 2. Can you show why such a formula holds? Click here to find out more.

If n is the number of teams and m is the total number of matches, then the above formula provides the following relationship: n (n − 1) / 2 = m. Given m, the quadratic equation needs to be solved for n.

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