A stick 5 cm long, a stick 9 cm long, and a third stick $n$ cm long form a triangle. What is the sum of all possible whole number values of $n$?
Using the Triangle Inequality, we see that $n > 4$ and $n < 14,$ so $n$ can be any integer from $5$ to $13,$ inclusive. The sum can be calculated in several ways, but regardless, $5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = \boxed{81}.$