Second Half of the Chessboard

exponential

The rice and chessboard problem is a mathematical problem: If a chessboard were to have rice placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of rice would be on the chessboard at the finish? The answer is 18,446,744,073,709,551,615, which would be a heap of rice larger than Mount Everest.

This problem (or a variation of it) demonstrates the quick growth of exponential sequences. In technology strategy, ‘the second half of the chessboard’ is a phrase, coined by Ray Kurzweil, in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization’s overall business strategy. While the number of grains on the first half of the chessboard is large, the amount on the second half is vastly larger. The first square of the second half alone contains more grains than the entire first half.

While the story behind the problem changes from person to person, the fable usually follows the same idea: When the creator of the game of chess (in some tellings an ancient Indian mathematician, in others a legendary noble Tamil named Sessa showed a new invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The man, who was very wise, asked the king this: ‘that for the first square of the chessboard, he would receive one grain of rice, two for the second one, four on the third one, and so forth, doubling the amount each time.

The ruler, arithmetically unaware, quickly accepted the inventor’s offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the rice to the inventor. However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would be impossible to give the inventor the reward. The ruler then, to get back at the inventor who tried to outsmart him, cut off the inventor’s head to discourage such trickery.

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