Isaac Newton was made aware of Jean Bernoulli's "brachistochrone" problem #OTD in 1697. He solved it later that evening, showing that the curve of fastest descent between two points was given by a portion of a cycloid.

Newton published his solution anonymously in the Philosophical Transactions of the Royal Society of London. Bernoulli would later say that the identity of the author was obvious; he recognized "the lion by his claw."

The problem was also solved by Jacob Bernoulli, Leibniz, and L'Hôpital. Jacob Bernoulli's solution was especially elegant, an important early step in using variational calculus to solve physical problems.

Fwiw, I’ve seen a few dates attached to this result. Some sources say Newton received Bernoulli’s “programma” announcing the problem on 1/29; others say it was late 1696 and then the solution appeared in Philosophical Transactions in January 1697.

Newton’s niece Catherine Conduitt recalled him receiving the problem in January 1697. She said he returned home from the Tower of London at 4pm tired from “the hurry of the great recoinage,” then staying up until 4am to solve the problem.

Anyway, here's another brachistochrone.

Someone asked me what it meant that a brachistochrone is "part of a cycloid." A cycloid is the path traced out by a point on the rim of a rolling circle. Here's how a cycloid gives the brachistochrone from the first tweet.