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:


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.




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:


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.


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


Pedagogical applications[edit]


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.


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]


  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. 

External links[edit]