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.