The Problems of Ukrainian Authors

Question

Eight teams are playing single round-robin tournament, that is each team plays every other team once. And at that the schedule is arranged by rounds, in every round all teams play. What is the least possible number of rounds should finish to determine the best and the worst teams (that is, for any results of the games in uncompleted round, this teams will have strictly highest and strictly lowest numbers of points), if the tournament is:

a) volleyball (that is, winner receives 1 point, loser - 0 and there are no ties); b) handball (that is, winner receives 2 points, loser - 0, and for a tie each team receives 1 point)?

Accepted Answer

a) Lets show that at least 6 rounds are required. It is easy to construct an example for 6 rounds. We need to show that 5 rounds are not enough. Indeed, even if the best team won all games, to ensure first place, it should have at least 3 point advantage. Similarly, the worst team should have at least 3 point disadvantage. But such situation is impossible because the highest possible difference between the best and the worst team after 5 rounds is 5 points.

b) Lets show that at least 5 rounds required. It is easy to construct an example (one team gets 10 points, one - zero points and all other 6 teams get 5 points each. In the following 2 rounds each team can get at most 4 points). Absolutely similarly to the part a) it is possible to show that 4 rounds aren't enough.