everything looks like a π

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.