The wheat and chessboard problem (the problem is sometimes expressed in terms of rice instead of wheat) is a mathematical problem in the form of a word problem:
If a chessboard were to have wheat 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 wheat would be on the chessboard at the finish?
[…]The total number of grains equals 18,446,744,073,709,551,615, which is a much higher number than most people intuitively expect.
[…]There are different stories about the invention of chess. One of them includes the geometric progression problem. Its earliest written record is contained in the Shahnameh, an epic poem written by the Persian poet Ferdowsi between c. 977 and 1010 CE.
When the creator of the game of chess (in some tellings an ancient Indian Brahmin mathematician named Sessa or Sissa) showed his 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 clever, asked the king this: that for the first square of the chess board, he would receive one grain of wheat (in some tellings, 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 wheat 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 take more than all the assets of the kingdom to give the inventor the reward. The story ends with the inventor becoming the new king. (In other variations of the story the king punishes the inventor.)
[…]In technology strategy, the second half of the chessboard is a […] 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 (232 > 4 billion times) larger.
The number of grains of rice on the first half of the chessboard is 1 + 2 + 4 + 8… + 2,147,483,648, for a total of 4,294,967,295 (232 − 1) grains of rice, or about 100,000 kg of rice (assuming 25 mg as the mass of one grain of rice). India’s annual rice output is about 1,200,000 times that amount.
The number of grains of rice on the second half of the chessboard is 232 + 233 + 234… + 263, for a total of 264 − 232 grains of rice (the square of the number of grains on the first half of the board plus itself). Indeed, as each square contains one grain more than the total of all the squares before it, the first square of the second half alone contains more grains than the entire first half.
As a moral story the problem is presented to warn of the dangers of treating the finite as infinite. As Carl Sagan said when referencing the fable, “Exponentials can’t go on forever, because they will gobble up everything.”