You know that athletic events can be a lot of fun. But they can also help solve math problems! Let’s see how it works.

Some tennis tournaments are played using a “round robin” system: everybody plays everybody.
Winning more games than the others makes you the tournament winner.

1. How many games are played by each person if the number of participants is

   1. 3?
   2. 4?
   3. 11?
   4. n?
2. How many games are played altogether if the number of participants is
   1. 3?
   2. 4?
   3. 11?
   4. n?
3. The total number of games must be a whole number. Prove that the formula obtained in problem 2, part (d), always yields whole numbers.
4. Consider n points. Each point is connected by a segment to all others. How many segments are there?
5. How many diagonals does a regular n-gon have?
   Use the result of problem 4.
   

There are no hints for this problem sequence.

1. 1. 2
   2. 3
   3. 10
   4. n - 1
2. 1. 3
   2. 6
   3. 55
   4. 
3. See solution.
4. 
5. 

1. If there are 3 participants, say Abby, Ben, and Carlos, then each will play 2 games. For example, Abby plays Ben and Abby plays Carlos. Since each participant plays everyone else (except himself or herself), each must play n-1 games.
2. Here are two possible ways to approach this problem. One is to notice that each of the n players plays (n - 1) games, which gives us n(n - 1) games. But when Abby plays Ben, Ben plays Abby--and this is just one game, not two. Therefore, the answer is not n(n - 1), but .
   Alternatively, in a two-player tournament, there is only 1 game. If one more player joins the tournament (for the total of 3 players), that player must play all others, adding 2 more games to the total. If one more player joins in, bringing the total number of players up to 4, it adds 3 more games, and so on. The nth player must play (and adds to the tournament) (n - 1) more games. Therefore, in a tournament with n players, there would be 1 + 2 + + (n - 1) = games.
3. Either n or (n - 1) must be an even number, so n(n - 1) is even; that is, divisible by 2 without a remainder--which means the answer is a whole number.
4. Imagine that the n participants in the round-robin tournament are standing on the points, one person for each point. The segments connecting two points could then represent a game played between the two people at those points. The total number of segments is equal to the total number of games played: .
5. The diagonals of an n-gon are the segments connecting each point to each non-adjacent point. A segment connecting two adjacent points is a side of the n-gon. Since there are segments connecting each point to every other point, and n of those are not diagonals, there are - n diagonals. This can be simplified to diagonals.