In a chess tournament with three participants, a total of 3 games were played. Everyone played the same number

In a chess tournament with three participants, a total of 3 games were played. Everyone played the same number of games. How many games did each participant play?

Each chess game is played by two players. The two of them play the same game together. Consequently, the number of games played by all participants is half the sum of all games played by each of the participants in the tournament.

Since only 3 games were played, then for the sum of the games played by all the participants we get:

3 * 2 = 6.

Dividing this number by 3, we determine the number of games played by each of the participants:

6: 3 = 2.

Thus, each participant of the tournament played two games – one game each with the other two chess players.

Chess tournament with n participants
Consider a more general case when n chess players participate in a tournament, and each pair plays one game. How many games will be played in this case?

Each of the n chess players will play one game with all other participants, that is, n – 1 games. Multiplying this number by the number of participants, we get the sum of the games played by each of the participants:

M (n) = n * (n – 1). (1)

And the number of all parties will be 2 times less than the sum of all parties:

N (n) = M (n) / 2 = n * (n – 1) / 2. (2)

For example, for a tournament with 3, 4, 5 and 6 participants, we get, respectively:

N (3) = 3 * (3 – 1) / 2 = 3 (batches);
N (4) = 4 * (4 – 1) / 2 = 6 (batches);
N (5) = 5 * (5 – 1) / 2 = 10 (batches);
N (6) = 6 * (6 – 1) / 2 = 15 (batches).
Answer: each of the participants played 2 games.