wheat and chessboard problem
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The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as: 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: and so forth for the 64 squares. The total number of grains can be shown to be 264−1 or 18,446,744,073,709,551,615 (eighteen
quintillion Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-E ...
, four hundred forty-six quadrillion, seven hundred forty-four trillion, seventy-three billion, seven hundred nine million, five hundred fifty-one thousand, six hundred and fifteen, over 1.4 trillion metric tons), which is over 2,000 times the annual world production of wheat. This exercise can be used to demonstrate how quickly exponential sequences grow, as well as to introduce exponents, zero power,
capital-sigma notation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, ma ...
and
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
. Updated for modern times using pennies and a hypothetical question such as "Would you rather have a million dollars or a penny on day one, doubled every day until day 30?", the formula has been used to explain compound interest. (Doubling would yield over one billion seventy three million pennies, or over 10 million dollars: 230−1=1,073,741,823).


Origins

The problem appears in different stories about the invention of
chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
. One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by
Ibn Khallikan Aḥmad bin Muḥammad bin Ibrāhīm bin Abū Bakr ibn Khallikān) ( ar, أحمد بن محمد بن إبراهيم بن أبي بكر ابن خلكان; 1211 – 1282), better known as Ibn Khallikān, was a 13th century Shafi'i Islamic scholar w ...
. Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed. Macdonnell also investigates the earlier development of the theme.


Solutions

The simple, brute-force solution is just to manually double and add each step of the series: : T_ = 1 + 2 + 4 + ..... + 9,223,372,036,854,775,808 = 18,446,744,073,709,551,615 ::where T_ is the total number of grains. The series may be expressed using exponents: : T_ = 2^0 + 2^1 + 2^2 + \cdots + 2^ and, represented with capital-sigma notation as: :\sum_^ 2^i.\, It can also be solved much more easily using: : T_ = 2^- 1. \, A proof of which is: : s = 2^0 + 2^1 + 2^2 + \cdots + 2^. Multiply each side by 2: : 2s = 2^1 + 2^2 + 2^3 + \cdots + 2^ + 2^. Subtract original series from each side: : \begin 2s - s & = \qquad\quad \cancel + \cancel + \cdots + \cancel + 2^ \\ & \quad - 2^0 - \cancel - \cancel - \cdots - \cancel \\ & = 2^ - 2^0 \\ \therefore s & = 2^- 1. \end The solution above is a particular case of the sum of a geometric series, given by :a + ar + a r^2 + a r^3 + \cdots + a r^ = \sum_^ ar^k= a \, \frac, where a is the first term of the series, r is the common ratio and n is the number of terms. In this problem a = 1, r = 2 and n = 64. The exercise of working through this problem may be used to explain and demonstrate
exponents Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
and the quick growth of
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
and geometric sequences. It can also be used to illustrate
sigma notation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
. When expressed as exponents, the
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
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, and so on.). The number of grains is the 64th
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
.


Second half of the chessboard

In
technology strategy Technology strategy (information technology strategy or IT strategy) is the overall plan which consists of objectives, principles and tactics relating to use of technologies within a particular organization. Such strategies primarily focus on the ...
, the "second half of the chessboard" is a phrase, coined by
Ray Kurzweil Raymond Kurzweil ( ; born February 12, 1948) is an American computer scientist, author, inventor, and futurist. He is involved in fields such as optical character recognition (OCR), text-to-speech synthesis, speech recognition technology, and e ...
, 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 wheat on the first half of the chessboard is , for a total of 4,294,967,295 (232 − 1) grains, or about 279 tonnes of wheat (assuming 65 mg as the mass of one grain of wheat). The number of grains of wheat on the ''second'' half of the chessboard is , for a total of 264 − 232 grains. This is equal to the square of the number of grains on the first half of the board, plus itself. The first square of the second half alone contains one more grain 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, more than two billion times as many 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 wheat, weighing about 1,199,000,000,000 metric tons. This is over 1,600 times the global production of wheat (729,000,000 metric tons in 2014 and 780.8 million tonnes in 2019).


Use

Carl Sagan titled the second chapter of his final book ''The Persian Chessboard'' and wrote that when referring to bacteria, "Exponentials can't go on forever, because they will gobble up everything." Similarly, ''
The Limits to Growth ''The Limits to Growth'' (''LTG'') is a 1972 report that discussed the possibility of exponential economic and population growth with finite supply of resources, studied by computer simulation. The study used the World3 computer model to simula ...
'' uses the story to present suggested consequences of
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
: "Exponential growth never can go on very long in a finite space with finite resources."Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III (1972). . New York: University Books. . Retrieved 2015-04-05.


See also

* Legend of the Ambalappuzha Paal Payasam *
Malthusian growth model A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert ...
* Moore's law *
Orders of magnitude (data) An order of magnitude is usually a factor of ten. Thus, four orders of magnitude is a factor of 10,000 or 104. This article presents a list of multiples, sorted by orders of magnitude, for units of information measured in bits and bytes. The byt ...
*
Technology strategy Technology strategy (information technology strategy or IT strategy) is the overall plan which consists of objectives, principles and tactics relating to use of technologies within a particular organization. Such strategies primarily focus on the ...


References


External links

*
Salt and chessboard problem
- A variation on the wheat and chessboard problem with measurements of each square. * {{Wikiversity-inline, Math Adventures/Wheat and the Chessboard Mathematical chess problems Exponentials
Chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...