Alyssa, Bill, and Clayton are playing ping pong. The rules are as follows:
The first game is between Alyssa and Bill.
Whenever a game ends, the next game is played with the winner against the person who was sitting out.
No ties are possible.
For example, one possible sequence of results is as follows:
Alyssa defeats Bill.
Clayton defeats Alyssa.
Clayton defeats Bill.
Alyssa defeats Clayton.
After some number of games have been played, they count how many times each player won; Alyssa won 10 times and Bill won 21 times.
How many times did Alyssa and Bill play against each other?