TITLE: When can a mathematician have $3$ friends?
QUESTION [5 upvotes]: At a mathematics convention there are $n$ mathematicians. Each one has $3$ friends (friendship is symmetric). For what values of $n$ is this possible?

REPLY [2 votes]: Base case $1$ if you have $4$ people make them all know each other.
Base case $2$ if you have $6$ imagine you have two columns of $3$ people
and make each person know every person in the opposite column.
Now for the general case if you have $4n$ people seperate them in $n$ groups 
of $4$ people and apply case $1$
If you have $4n+2$ people apply case $2$ for $6$ people so now you are left with $4n -4= 4(n-1)$ people. And the previous can be applied.