**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?

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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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