Spiked Math: MPF - Betting on the Next Card

by Greg Bernhardt
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Greg Bernhardt
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Mar2-12, 04:30 AM
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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)?

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Jimmy Snyder
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Mar2-12, 06:52 AM
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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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