Wheat and chessboard problem

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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 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, 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: 20 + 21 + 22  + 23... and so forth up to 263. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, etc.).

Solutions[edit]

The simple, brute-force solution is to just manually double and add each step of the series:

T_{64} = 1 + 2 + 4 + \cdots + 9,223,372,036,854,775,808 = 18,446,744,073,709,551,615
where T_{64} is the total number of grains.

The series may be expressed using exponents:

T_{64} = 2^{0} + 2^{1} + 2^{2} + \cdots + 2^{63}

and, represented with capital-sigma notation as:

\sum_{i=0}^{63} 2^i.\,

It can also be solved (much more easily) using:

T_{64} = 2^{64}- 1. \,

A proof of which is:

s = 2^{0} + 2^{1} + 2^{2} + \cdots + 2^{63}.

Multiply each side by 2:

2s = 2^{1} + 2^{2} + 2^{3} + \cdots + 2^{63} + 2^{64}.

Subtract original series from each side:

2s - s = - 2^{0} + 2^{64}
\therefore  s = 2^{64}- 1. \,

Origin and story[edit]

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[1][2] 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 being beheaded. (In other variations of the story, the inventor becomes the new king.)

Macdonnell,[3] also investigates the earlier development of the theme.

[According to al-Masudi's early history of India], shatranj, or chess was invented under an Indian king, who expressed his preference for this game over backgammon. [...] The Indians, he adds, also calculated an arithmetical progression with the squares of the chessboard. [...] The early fondness of the Indians for enormous calculations is well known to students of their mathematics, and is exemplified in the writings of the great astronomer Āryabaṭha (born 476 A.D.). [...] An additional argument for the Indian origin of this calculation is supplied by the Arabic name for the square of the chessboard, (بيت, "beit"), 'house'. [...] For this has doubtless a historical connection with its Indian designation koṣṭhāgāra, 'store-house', 'granary' [...].

Pedagogical applications[edit]

This 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 a chessboard would be required to be able to contain the rice in the last square, assuming perfect spheres of short-grained rice?)

Second half of the chessboard[edit]

An illustration of the principle.

In technology strategy, the second half of the chessboard is a phrase, coined by Ray Kurzweil,[4] 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 (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).[5] India's annual rice output is about 1,200,000 times that amount.[6]

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.

On the 64th square of the chessboard alone there would be 263 = 9,223,372,036,854,775,808 grains of rice, or more than two billion times as much as on the first half of the chessboard.

On the entire chessboard there would be 264 − 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).[7]

Moral story[edit]

As a moral story the problem is presented to warn of the dangers of treating large but finite resources as infinite, i.e., of ignoring constraints that are distant but absolute and inevitable. As Carl Sagan said when referencing the fable, "Exponentials can't go on forever, because they will gobble up everything."[8]

The usage as a moral fable was re-ignited with the release in 1972 of The Limits to Growth where the story is referenced to present the unintended consequences of exponential growth. "Exponential growth never can go on very long in a finite space with finite resources."[9]

See also[edit]

References[edit]

  1. ^ "Shaolin - A beautiful story of chess". Chess.com. Retrieved 2013-12-08. 
  2. ^ "info | Lahur Sessa". Xklsv.org. Retrieved 2013-12-08. 
  3. ^ Art. XIII.—The Origin and Early History of Chess, A. A. Macdonell, Journal of the Royal Asiatic Society, Volume 30, Issue 01, January 1898, pp. 117-141, DOI: 10.1017/S0035869X00146246, |url=http://journals.cambridge.org/abstract_S0035869X00146246
  4. ^ Raymond Kurzweil (1999). The Age of Spiritual Machines. Viking Adult. ISBN 0-670-88217-8. 
  5. ^ "Rice CRC - Size and Weight". 2006-08-23. Archived from the original on 2006-08-23. Retrieved 2011-09-16. 
  6. ^ "Rice production may touch 100 mn tonnes in 2010-11". Retrieved 2013-02-13. 
  7. ^ PTI Jun 24, 2011, 02.34pm IST (2011-06-24). "World rice output in 2011 estimated at 476 mn tonnes: FAO". Articles.economictimes.indiatimes.com. Retrieved 2013-12-08. 
  8. ^ "(Thoughts On Life And Death At the Brink Of The Millennium by Carl Sagan" (PDF). Retrieved 2013-12-08. 
  9. ^ [Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) The Limits to Growth. New York: University Books. ISBN 0-87663-165-0]

External links[edit]