# Spiked Math: MPF - Betting on the Next Card

by Greg Bernhardt
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 Admin P: 8,447 It's Math Puzzle Friday! Yay! We gave this problem to the students at Math Night on Wednesday. It appears in Peter Winkler's book Mathematical Puzzles: A connoisseur's Collection (on page 67) and is stated as follows: See if you can solve it. If you don't know where to start, try to solve it for a deck of 4 cards (2 black/2 red). What about 6 cards (3 black/3 red)? Problem 2: What's the maximum amount he can assure himself of winning for a deck of 2n cards (with n black and n red)? More...
 P: 2,163 Spoiler I haven't worked out the answer even to the 4 card version. However, I know a way to guarantee that you end up withat least $2.66 as follows: Don't bet the first card, say it turns up red. Bet$0.33 that the second card will be black. If the second card is black you will have $1.33 and there will be a red and a black card remaining in the deck. Don't bet on the third card. Bet$1.33 on the fourth certain card for a final total of $2.66 If the second card is red you will have$0.67 and there will be 2 black cards remaining in the deck. Bet everything on black twice for \$2.68 Obviously, if the first card turns up black, a symmetric strategy will have the same result.

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