A local tennis club has 15 members. The organizer wants every member to play the same number of matches at an upcoming event, with no one playing the same opponent more than once.
The organizer is considering two possible schedules:
Every member plays five matches against five different opponents.
Every member plays four matches against four different opponents.
The club president says that one of those schedules is mathematically impossible. Which schedule is impossible? Bonus question: Why is it impossible?
Here is one way to arrange a tournament of 15 players where everybody plays four matches: arrange the players in a circle. Everybody will play the two players to their left and the two players to their right.
It is impossible for each of the 15 players to play five matches against five different opponents. If you ask all 15 players how many matches they each played and add up their answers, the total will be 15 × 5 = 75 when every player tells you they played five matches. If player A and player B played a match, then that match will be reported twice: once when you ask player A how many matches they played and then again when you ask player B. This double counting occurs for every match; that 75 number is actually double the total number of matches played. Dividing it by 2 yields 37.5 total matches, but fractional matches are impossible, so this schedule is mathematically impossible. In general, in any one-on-one tournament, if you ask all of the players how many matches they played and sum up their answers, the result must be an even number because it represents double the total number of matches played.
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