Card Betting

Continuing on my plan of posting the puzzles I found in the paper "Games People Don't Play", I have a card flipping puzzle this time. Back when I started this blog I posted a card flipping problem that Bart had told me, and the puzzle this time is something of a continuation of that one:

There is a deck of 52 cards, 26 red and 26 black. You will be playing a game where you bet on what is the next card in the deck. You begin with $1 and can make a bet of any real number between zero and the amount of money you have on whether the next card in the deck is red or black. The top card of the deck is then revealed and if you were correct you gain the amount you bet, if you were incorrect you lose the amount you bet. Then the game continues with betting on the next card in the deck. The game ends when there are no cards remaining in the deck to bet on. What is the optimal betting strategy assuming that the deck is being arranged by an adversary?

I really like the sort of puzzle that seems flat out impossible to do anything at first, putting the player up against an adversary seems like a pretty insurmountable obstacle, but you still have some wiggle room.