Mathematics
The Brachistochrone Problem and Calculus of Variations
Quick fact
In 1696, Johann Bernoulli challenged the world's mathematicians with the brachistochrone problem; Newton solved it in a single evening and published the answer anonymously, but Bernoulli recognized the author immediately by the brilliance of the solution.
Why this is interesting
You've probably learned that the shortest path between two points is a straight line. But if you're a ball rolling downhill, the fastest path isn't straight at all—and it's not even a circular arc. What curve makes a bead fall from here to there in the least time?