**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?

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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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