Let us play a game of cards, shall we? I will be the dealer, 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. Immediately 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 card that I am about to flip. So, for instance, if you have $5 and I am about to flip a card, you may either do nothing, bet any amount of money up to $5 that the card will be red, or bet any amount of money up to $5 that the card will be black. A correct bet of x wins you an incorrect bet costs you 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. To simplify things, assume that dollars are continuously divisible. That is, whenever you chose to bet, you can bet any positive number of dollars; you need not stick to multiples of a cent.

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?

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