4. Alina, Bogdan, Claudia, Denisa, and Eugen are playing doubles in table tennis. In each game, one of the children stands on the side while the other four play two against the other two. Each pair plays against each of the pairs formed from the remaining players exactly once.

a) How many games were played in total?

b) If Alina won 12 games, and Bogdan won 6 games, how many games did Claudia win? Find all possibilities and justify your answer.
## Solution:

a) Each player sits out 3 games on the sidelines: for example, when Alina sits out, Bogdan plays, in turn, in pairs with Claudia, Denisa, and Eugen, which is 3 games.

Therefore, with 5 children, each sitting out 3 games, in total,

![](https://cdn.mathpix.com/cropped/2024_06_07_41190f5fd3784decad28g-5.jpg?height=49&width=1471&top_left_y=455&top_left_x=298)
b) Alina played 12 games, so she won all of them. ....................... Bogdan played 3 games in pairs with Alina (Claudia, Denisa, and Eugen sat out, in turn). Since Alina won all her games, it follows that Bogdan won these 3 games. Additionally, when he played against Alina, he lost, so the other 3 games won by Bogdan were won when Alina was sitting out. Therefore, Bogdan won all 3 games in which Alina was sitting out.

Claudia won the 3 games in which she had Alina as her partner and lost all the games in which she played against Alina. In the three games in which Alina was sitting out, Claudia won when she played in pairs with Bogdan and lost when she played against him. Therefore, Claudia obtained 4 victories (the only possibility)