Challenge 245: A Curious Competition
Triangle chess is a game for three players, so any game involves three people.
Five students (A, B, C, D, E) are taking part in a triangle chess tournament.
The organisers want each pair of students to play together in exactly one game.
They make this plan for the tournament.
game 1: A, B, C
game 2: A, D, E
game 3: B, D, C
game 4: E, A, B
However, this is not satisfactory for two reasons
- the pairs A-B, A-E and B-C play together in two of the games
- the pair C-E do not play together in any game.
a Explain to the organisers why they cannot hope to have each pair of students play together in exactly one game.
b Show that the organisers cannot hope to have each pair of students play together in exactly one game even if there are six players.
c Provide the organisers with a satisfactory plan for the tournament if there are seven players.
d Explain why tournaments cannot be organised in this way with an even number of players, nor with a number of players which is one less than a multiple of 3.
e Identify the next number of players above seven for which a tournament can be organised in this way, and provide a plan for the tournament in this case.
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