Choosing Players for a Card Game If there are 5 players that want to play a card game that requires two teams of  2 players, how many games must be played so that each person will be partnered with each other person?
I assumed there would be $$ \binom{5}{2}$$ possible teams, and since each game requires two teams of two players, the final result would be 
$$ \frac{\binom{5}{2}}{2} $$ games until each person would be partnered with every other person. Is this correct? If not, how can this be calculated?
edit: Each person would be partnered with every other person within 4 games, so is my result the number of games until each pair of players plays all other pairs of players?
 A: Yes, you're reasoning is correct! Your answer, $5$, is the minimum number of games in which each player can be with every other person. However, the minimum number of games in which one player can be with every other person is $4$. If you are still unsure about what you've done, try listing all of the teams out. Let's represent the players as a set $\{ a, b, c, d, e \}$. Here's all of the teams:


*

*$\{ a, b \}$

*$\{ c, d \}$

*$\{ a, c \}$

*$\{ d, e \}$

*$\{ a, d \}$

*$\{ b, e \}$

*$\{ a, e \}$

*$\{ b, c \}$

*$\{ c, e \}$

*$\{ b, d \}$


If you make each game the first and second pair as listed, and then the next game the third and fourth pair as listed, and so on until you've gone through $5$ games, you will have gone through all of the pairs.
A: It's me, the OP for this question! I was reminded of this problem recently and five years, a CS degree, and a math degree later, I've given it enough thought to come up with a general solution. I wrote a blog post about it, but the gist is this: while my original answer happened to work for the specific case of selecting two-player teams from a group of five players, it does not work for any given number of players and team size. For that, we need to take a trip through some advanced-ish (algebraic!) graph theory to prove a key fact; at the end we find that, for $n$ players and teams of size $k$, the minimum number of games required for every distinct team of players to play a game is $$ \left\lceil \frac{\binom nk}{2} \right\rceil, $$ which is the edge covering number of the Kneser graph $K_{n:k}$.
Looking back now, this question truly fanned my nascent mathematical flame – asking questions like this let me know that math was beyond the AP calc test. Glad I asked it, and glad I came back to it.
