20 teams and 10 competition stations

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Question from Joel, a teacher:

I have 20 teams. There are 10 competition stations.
Each station will accommodate exactly 2 teams to compete with each other at a time.

Each of the 20 teams needs to compete at each of the 10 stations exactly one time.

Here's the catch I cannot solve: how to schedule it so that each team plays EVERY OTHER TEAM once and only once, while never repeating a station.

Joel,

You’re having trouble because what you want isn’t possible. Each team has 19 opponents, and so needs to play 19 games in order to play every other team exactly once. After 10 games (at most) a station must be repeated. I don’t know if it possible to schedule the competition you want so that no station is used more than twice. It might be.

—Victoria

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