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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.

Submit your solution

Please do send in your solution to this problem to  weeklymaths@kcl.ac.uk  You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide you with feedback on your solution.

 

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