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The concept of how rapidly the total increases when each successive deposit is double the one that precedes it is over 1000 years old.
(largely from Wikipedia):
The wheat and chessboard problem (the problem is sometimes expressed in terms of grains of rice instead of wheat) is a mathematical problem in the form of a word problem:
If a chessboard were to have wheat grains 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 problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615 (that is, 18.45 x 10^18, approx.) which is a much higher number than most people intuitively expect.
The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 2^0 + 2^1 + 2^2 + 2^3... and so forth up to 2^63. The base of each exponentiation, "2", expresses the doubling at each chessboard square, while the exponents represent the position of each square (0 for the first square, 1 for the second, etc.).
Origin and storyThere 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. A.D. 977 and 1010.
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.)
Pedagogical applicationsThis exercise can be used to demonstrate how quickly exponential sequences grow, as well as to introduce exponents, zero power, capital-sigma notation and geometric series.
Derivatives of the problem can be used to explain more advanced mathematical topics, such as hexagonal close packing of equal spheres. (How big of a chessboard would be required to be able to contain the rice in the last square, assuming perfect spheres of short-grained rice?)
On the entire chessboard there would be 2^64 − 1 = 18,446,744,073,709,551,615 grains of rice, weighing 461,168,602,000 metric tons, which would be a heap of rice larger than Mount Everest. This is around
1,000 times the global production of rice in 2010 (464,000,000 metric tons).
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