7.4. During the class chess championship, two participants, who played an equal number of games, fell ill and dropped out of the tournament, while the remaining participants finished the tournament. Did the dropped-out participants play against each other if a total of 23 games were played? (The tournament was played in a round-robin format: each player played one game against every other player).

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# Solution:

A tournament with 6 participants consists of 15 matches (6 participants $\cdot 5$ matches: 2), while a tournament with 7 participants consists of 21 matches, and with 8 participants, it consists of 28 matches. Therefore, either there were 6 participants in the tournament, in addition to those who dropped out, and the dropped-out participants played in 8 matches (23-15), or there were 7 participants, and the dropped-out participants played in 2 matches (23-21).

If they had played against each other, they would have played 7 matches against other participants in the first case, and 1 match in the second case, which means they could not have played an equal number of matches.

Answer: they did not play