Two gladiators are about to do battle in the coliseum. Before the battle, they enter separate rooms and pray to Ares, the god of war. Ares blesses them each with random amount of power, which you can think of as a real number between $0$ and $1$. If the gladiators are unsatisfied with their blessing, then they may pray at most one more time, receiving a new amount of power which replaces the old one. They then fight, and the gladiator with the greatest blessing wins.
How can a gladiator guarantee that they win with probability at least $1/2$?
The reason I like this puzzle is because there is one strategy which is "obviously correct:" namely, pray again if your first blessing is less than $0.5$. Surprisingly, there are strategies which beat this more than half the time. As a warning, this is a bit more computation-heavy than your average clever math puzzle.