﻿ Wheat and chessboard problem

# Wheat and chessboard problem

Sissa ibn Dahir (also: Sessa ) allegedly lived in the third or fourth century AD in India and is, according to legend as the inventor of chess or his Indian archetype Tschaturanga.

His name is also connected with the WeizenkornLegende. The story is a " parable for the variety of chess " (Martin Beheim - Schwarzbach ), which symbolizes the inexhaustible possibilities of the game and gradients in chess. The legend can be understood both as a tribute to the game of chess, as an illustrative example of mathematical teaching as well as socially critical work. The anecdote is commonly associated with exponential functions mention and is the oldest example of how a mathematical question on the chess is based - and thus an anticipation to the field of chess mathematics.

## Legend

The Indian rulers Shihram tyrannized his subjects and plunged his country into poverty and misery. To draw the attention of the king on his mistakes without kindle his wrath, created Dahirs son, the wise Brahmin Sissa, a game in which the king can do as the most important figure without the help of other pieces and pawns nothing. The lessons in chess, made ​​a strong impression on Shihram. He was mild and had spread the game of chess, so that all of them would take note. To thank for the ideological doctrine of wisdom and at the same time Entertainment, he granted the Brahmin a free request. This wished wheat grains: In the first field of a chessboard he wanted a grain, on the second field twice, two, on the third turn, double the amount, four and so on. The king laughed and was also angry about the supposed modesty of the Brahmins.

When Shihram few days later inquired whether Sissa took his reward in receiving, he had to hear that the Reckoner had not yet calculated the amount of wheat grains. The head of the granary reported after several days of continuous work, that he could not afford this amount of cereal grains throughout the empire. In all fields of a chessboard together it would be 264-1 or 18,446,744,073,709,551,615 grains of wheat. Now he faced the question of how the promise could be redeemed. The Reckoner helped the rulers from the embarrassment by recommending him to simply let Sissa ibn Dahir include the cereal grain by grain.

There are alternative narratives, which it had instead acted to wheat grains rice grains.

## Mathematical Calculation

Calculating the number of the wheat grains is carried out as follows:

Or

It follows

To illustrate this gigantic amount, a little thought experiment:

Suppose a grain of wheat weighs approximately 0.05 g ( thousand grain mass from 40 to 65 g) and the total amount of wheat is to be loaded into trucks with 8840 kg payload ( for example, grain trucks the company DAF, ATI Grain Tipper, 95.360 ), the length of a exhibit of 7.6 meters, then fit into a van 176.8 million grains. Overall, it therefore requires about 104 billion transporters to load everything. Placed bumper to bumper, give these transporters a distance of about 793 million kilometers; with an assumed earth's circumference of 40,000 km yields the total of nearly 20,000 times around the earth ( or 5.3 astronomical units, ie 5.3 times the Earth-Sun distance ).

The total quantity of wheat which befände on the chessboard (about 922 billion tonnes ) is roughly equivalent to 1500 times the global wheat harvest in 2004 ( 624 million tonnes).

## Origin of the anecdote

For this story, there is no evidence in Indian sources. It is attributed to the Arab culture. There are both the biographer Ibn Khallikan (1211-1282) as well as the writer and poet Al- Sabhadi, who lived in the Middle Ages in Baghdad, designation of origin. Al- Sabhadi to have already given the answer.

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