This Labor Day weekend I’m going to the wedding of one of my oldest friends from my hometown. Combine the fact that the number of my friends who aren’t married is dwindling dangerously close to zero and the fact that I wasn’t able to attend this year’s fantasy football draft so I had to autodraft and the fact that this weekend marks the end of the summer and OMG WHAT IS HAPPENING TO MY LIFE.

This weekend’s prize: you guessed it — free access to the Math Guide Beta. Make sure you don’t comment anonymously if you want to win.

In a certain youth soccer league, each team plays each other team exactly one time per season. If, during a certain season, there are 231 games played, how many teams were in the league that season?

First correct answer in the comments wins Beta access, and the undying adoration of my throngs of loyal readers.

UPDATE: Nice work, “AP FRQ Solutions” (whoever you are). I’ve shared the Beta document with you. Use it in good health.

Solution below the cut.

If you look back at my post about matching questions, you’ll notice a neat little tidbit hidden in there: if you’re looking for the total number of matches that can be made between a certain number of teams, you can take the number of teams, subtract 1, and take the sum of that number and each number lower than it to find the number of matches. In other words, if you have 7 teams, they will need 6+5+4+3+2+1 matches to each play each other once. Neat, huh?

So you can use that to solve this one quick and dirty:

1+2 = 3

1+2+3 = 6

1+2+3+4 = 10…

…

1+2+3+4+…+20+21 = 231.

So there must be 22 teams in the league.

Note: even if you didn’t know this little factoid, you might have been able to solve this way by finding the number of matches needed for smaller leagues, and looking for patterns in the number of matches required as the leagues get bigger.

If you’re looking for a mathier explanation, I think AP FRQ Solutions did a decent job in the comments.

## Comments (4)

22 teams

Nice work. Welcome to the Beta!

if n is number of teams, games played is n(n-1)/2 because each team will play all the other teams except itself (why is why there is n-1) and then you divide by 2 because team 1 playing team 2 is the same as team 2 playing team 1.

21 teams