Problem 28 · 2019 Math Kangaroo
Stretch
Counting & Probability
careful-counting
Teams of three take part in a chess tournament. Each player plays against every player from every other team exactly once. For organisational reasons no more than 250 games may be played. What is the greatest number of three-player teams that can take part?
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Answer: E — 7
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Hint 1 of 2
Each game is between two players on different teams; count games as pairs of teams times players.
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Hint 2 of 2
Find the largest team count keeping games at most 250.
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Approach: count cross-team games and bound by 250
- With n teams of 3, each pair of teams plays 3 × 3 = 9 games, so total games = 9 × n(n−1)/2.
- For n = 7 that is 9 × 21 = 189 (allowed); for n = 8 it is 9 × 28 = 252 (too many).
- So at most 7 teams can take part.
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