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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 36 matches are totally played, how many teams participated?
Answer: 9

Solution:

If 9 teams participated, then the first team plays matches against the other 8 teams. The second team has already played against the first team, and so has to play matches against only the other 7 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

8 + 7 + ........ + 2 + 1 = 36 .

If 36 matches are totally played, then 9 teams participated.

Food for thought:

Is there a formula to conveniently add
8 + 7 + ........ + 2 + 1 ?
Indeed, there is! It simply equals 8 × 9 / 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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