Wheat and chessboard problem

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.

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.

[5] [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 [sic] (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' [...].The simple, brute-force solution is just to manually double and add each step of the series: The series may be expressed using exponents: and, represented with capital-sigma notation as: It can also be solved much more easily using: A proof of which is: Multiply each side by 2: Subtract original series from each side: The solution above is a particular case of the sum of a geometric series, given by where

The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences.

In technology strategy, the "second half of the chessboard" is a phrase, coined by Ray Kurzweil,[6] in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.

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.

[8] Carl Sagan titled the second chapter of his final book "The Persian Chessboard" and wrote, referring to bacteria, that "Exponentials can't go on forever, because they will gobble up everything.

By the time that the fifth square is reached on the chessboard, the board contains a total of 31, or , grains of wheat.
The sum of powers of two from zero up to a given positive integer power is 1 less than the next power of two (i.e. the next Mersenne number )
A chessboard with each square labeled with the number of wheat grains according to the problem. A red line divides the chessboard in half.
An illustration of Ray Kurzweil 's second half of the chessboard principle. The letters are abbreviations for the SI metric prefixes .