Here is an interesting problem I came across.
Let us play a game of cards, shall we? I will be the dealer, and you will do the betting. The rules are simple. I take a standard deck of 52 playing cards and shuffle it well. I begin to flip over the cards in the deck, in sequence, one at a time. Before each flip, you have the opportunity to bet any amount of money that you have, from $0 to everything you have, on the color of the next card. That is, you can bet any amount of money up to what you have that the next card will be red, or bet that it will be black. A correct bet of $x wins you $x; an incorrect bet you lose your $x.
You begin the game with $100. At any point in the game, you can recall perfectly the sequence of cards that has been flipped. When you choose to bet, you can bet any positive amount of money, and are not restricted to betting just multiples of cents.
Now the question is: What is the maximum amount of money you can be guaranteed to have once the deck is through, and what betting strategy should you use to achieve this outcome?
As a place to start, notice that there is a simple way to guarantee $200. Just wait until I am about to flip the 52nd card, and at that point, bet your entire $100 on the color you know that card to be.
How much better can you do?