How many ordered pairs of positive integers $(x, y)$ satisfy $\frac{1}{x} + \frac{1}{y} = \frac{1}{6}$?

Clearing denominators gives $(x - 6)(y - 6) = 36$. Each positive divisor $d$ of $36$ gives one pair, with $x = 6 + d$, and $36 = 2^{2} \times 3^{2}$ has $(2+1)(2+1) = 9$ divisors. Negative factors would make $x$ or $y$ non-positive.