Blet: A Mathematical Puzzle

Blet is a puzzle. One starts with an even number of coins, laid out in a circle. At first, the coins are laid out with heads and tails alternating (HTHTHT. . .HT). Each turn, you are allowed to take three consecutive coins that show tails-heads-tails and flip them over, getting heads-tails-heads. This increases the total number of heads by one. You may do the opposite, flipping a heads-tails-heads pattern to get tails-heads-tails. The object of the game is to get as many heads as possible. A secondary goal is to reach this maximum in as few moves as possible. Playing with 4, 6, or 8 coins, it’s easy to reach the maximum by being greedy, always converting THT to HTH and never converting HTH to THT. With 10 coins, you can only get 7 heads by being greedy, but there is a way to get 8 heads. Can you find it? In this article we’re going to spoil your fun by figuring out what the maximum number of heads is for any starting size, and devising a strategy for reaching that number. Before reading on, you might want to try solving the 10-coin puzzle (Blet-10) on your own. (You may prefer using a 2-color counter, such as are used in Othello, instead of coins. Or you can use pencil and paper. An electronic version, with 28 “coins” labeled 0 or 1, is available at http://www.ma.utexas.edu/users/voloch/blet.html).