Here's a cute little computation. Suppose you pick two uniform random numbers. What's the probability that their ratio rounds to an even number? It's straightforward enough to get "this is a sum that looks like it's probably famous." But it turns out that different definitions of rounding pull in different transcendental constants. The spoilery answer that, if you round to the *nearest* integer, you get probability $\frac{5-\pi}{4} \approx 0.46$. But if you *floor* to an integer, you get $1-\frac{\ln 2}{2} \approx 0.65$. It's kind of wild that, of all the uncountable transcendental numbers, the sorts of problems we build tend to generate just the same handful of them.