## i. checkmate

There is an old story about the original creator of the game of chess, a wily mathematician who submits his invention to the ruler of the country. Asked by the delighted queen what he would require by way of reward, the mathematician requests to be paid in gold. He proposes the queen place one single coin on the first square of the chessboard, two on the second square, four on the next, eight on the next, doubling the number of coins on each successive square up to the sixty fourth. The queen, perplexed that the mathematician would ask such