Source

I got this problem from Rustan Leino, who got it from Sophia Drossopoulou and Alex Summers.

I solved it and wrote up my solution.

Problem

There's a certain kind of egg about which you wonder: What's the highest floor of a 36-story building from which you can drop an egg without it breaking? All eggs of this kind are identical, so you can conduct experiments. Unfortunately, you only have 2 eggs. Fortunately, if an egg survives a drop without breaking, it's as good as new—that is, you can then conduct another dropping experiment with it. What's the smallest number of drops that's sure to determine the answer to your wondering?

Solution
Reveal

Eight is the smallest number of drops required. Here's how to do it.

Drop eggs from floors 8, 15, 21, 26, 30, 33, 35, and 36, stopping whenever the egg breaks. When the egg breaks, you'll have $9-n$ possibilites for your answer, where $n$ is the number of drops you've performed. (For instance, if the egg breaks on the 1st drop, from floor 8, then the $9-1=8$ possibilities are 0–7. If the egg breaks on the fourth drop, from floor 26, then the $9-4=5$ possibilities are 21–25.) The lowest possibility doesn't need to be checked because you've already determined it to be safe for egg dropping. So, you only have to try the other $8-n$ possibilities. Make sure to try those possibilities in increasing floor order, so that you don't break the egg before getting your answer.