Tournament Tables

May 2010
An even number of players compete in a round-robin tournament which is staged in rounds. Every player plays just once each round. Every pair of players meet exactly once in the tournament. In each match, the winner is awarded 1 point and the loser 0 points. No draws are possible.

a Show that a tournament with six players has five rounds.
b For a round-robin tournament of six players explain why all the players cannot have the same total score on completion of the third round.

The results of each round can be represented by a table. For example, a game consisting of the players A, B, C and D will have each of their names in a different box on in the top row, and the left column. A '1' means that the person in the row beat the person in the column. A 0 means that they have lost.

Ash, Bob, Cay, Jan, Ken and Lyn play a round robin tennis tournament. In the first round Ash defeats Jan, Bob defeats Ken and Cay defeats Lyn. In the second round Ash loses to Ken, Bob loses to Lyn and Cay loses to Jan.

c Draw the tables for rounds 1 and 2 using A, B, C, J, K, L to represent that players.
d Draw the tables of all the possible third and fourth rounds if the players have equal total scores on completeion of the fourth round.
Mar 2010
For part (a), consider the schedule of one particular player. He has to play each of the other players once and only once, which takes 5 rounds.

For part (b), notice that since there are six players, there are 3 games per round. Therefore a total of 9 points are awarded in three rounds, and these can not be evenly split among 6 players.

Part (c) is just a matter of filling in the table.

I don't see any obvious method for (d) except to start making tables: A can play either B, C, or L. If A plays B, then C has to play K, so J plays L. If A plays C, then B has to play J, so K plays L. But if A plays L, then B can play either C or J, with the third pairing being J-K or C-K respectively.

If the third round is A-B, C-K, and J-L, then A can play either C or L in the fourth round. If A plays C, then B plays J and K has to play L. Then we can have A, K, and J win in the third round and lose in the fourth or vice-versa.

But you run into problems if the third round is A-L B-J and C-K, In the fourth round, A can play B or C. If A plays B, then C can not play anyone (he has already played L, J, and K, and A,B are already playing).

- Hollywood
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