Putnam 1997 B1

Let $\{x\}$ denote the distance between the real number $x$ and the nearest integer. For each positive integer $n$, evaluate $$F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).$$ (Here $\min(a,b)$ denotes the minimum of $a$ and $b$.)