Mathematicians are good at counting.  We can count the number of ways to roll  a 7 with two dice.  (Answer = 6.) We can count the number of ways to shuffle a deck of cards so that every card is not in its original position.  (Answer = $!52$ = 29672484407795138298279444403649511427278111361911893663894333196201.)  We can count the number of lines on a cubic surface in $\mathbf{CP}^3$. (Answer = 27.)   Sometimes we can count, but we don’t really know what the actual number is, such as when we count the minimum number of guests that must be invited so that at least $m$ will know each other or at least $n$ will not know each other.  (Answer = the Ramsey number $R(m, n)$.)  Many times, it is better to use an asymptotic estimate, such as Stirling’s formula.  When we can’t estimate, we can bound, such as Conrey’s result that more than two-fifths of the zeros of the Riemann zeta function are on the critical line.  Lately, we (along with our colleagues in other disciplines) have started counting citations.   Continue reading →