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Coins in a Row

Puzzle

30 coins of arbitrary denominations are laid out in a row. Simran and Tavleen alternately pick one of the two coins at the ends of the row so as to pick up as much money as possible. If Simran makes the first move, could Tavleen ever collect more money than Simran, if Simran makes the optimal choices?

Source

From Mathematical Puzzles: A Connoisseur's Collection (163 pages, 2003) by Peter Winkler. Solving this puzzle for even 4 coins requires the right insight.

Solution

When the total number of coins is even, the first player could pick all coins in odd-numbered positions or all coins in even-numbered positions, whichever set is larger in value.

At a restaurant, how can Veronica choose one out of three desserts with equal probability with the help of a coin? What if the coin is biased and the bias is unknown?

Next Puzzle: Ants in a Circle

Suppose n ants are placed on a circle with a diameter of one meter. Each ant location is chosen independently, uniformly at random. An ant chooses between clockwise or anti-clockwise direction, uniformly at random, and starts scampering along the circle. All ants move at the same speed: one meter per second. When two ants bump into each other, they reverse their direction of travel. One of the ants is named Alice. What is the probability that Alice returns to the same point as she started, one minute after the ants start their scampering?

9 Dec 2008
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