## Task 3 - 270823

A tournament in a chess group (AG "Schach") is reported: The tournament was played in several rounds. In each round, each group member played exactly one game against every other member. In this way, a total of 112 games were played in the tournament. More than two members participated.

Investigate whether a tournament is possible that meets these conditions, and whether the number of rounds and the number of participants can be uniquely determined from the given information! If this is the case, determine these numbers!
If such a tournament is possible, the same number of games must be played in each round; therefore, the number of games in a round must be a divisor of 112, i.e., one of the numbers

The following table shows the determination of values for the number of games in a round. For each number of participants, this number of games is found as follows:

If there are three participants A, B, C, then three games AB, AC, BC result. With each increase in the number of participants by one, as many games are added as the previous number of participants indicates (since the new participant must play against all previous ones).

| Participants | Games | Participants | Games |
| :---: | :--- | :---: | :--- |
| 3 | 3 | 10 | $36+9=45$ |
| 4 | $3+3=6$ | 11 | $45+10=55$ |
| 5 | $6+4=10$ | 12 | $55+11=66$ |
| 6 | $10+5=15$ | 13 | $66+12=78$ |
| 7 | $15+6=21$ | 14 | $78+13=91$ |
| 8 | $21+7=28$ | 15 | $91+14=105$ |
| 9 | $28+8=36$ | 16 | $105+15=120$ |

With further continuation of the table, only larger game numbers result.

The only game number that is a divisor of 112 is therefore 28. With this and because $112: 28=4$, it is proven:

A tournament of the mentioned type is possible; from the given information it follows uniquely: Exactly four rounds were played; exactly eight AG members participated.