Among $n$ students where every pair of students was matched head-to-head once, every student had won exactly the same number of games. What is the possibilities of $n$ that this could have happened? Show the set of $n$.
So there are $\dfrac{n(n-1)}{2}$ matches and everyone has to win $\dfrac{n-1}{2}$ matches. How would i gaurantee that this could happen? i started with $n=2, 3$, and these could not happen with $n=2, 3$; is there some theorem i can use to help me with this question?
Hint
The idea is $\frac{n-1}{2}$ should be an integer. For example, if $n=2$, $\frac{n-1}{2}=1/2$ which is not an integer. However, when $n=3$, $\frac{n-1}{2}=1$ which is valid. So what all values are possible for $n$?
In other words, solve for $n$, assuming $\frac{n-1}{2}=k$, for some integer $k$.