The Wheat Problem

WheatfieldChess is inextricably linked with Mathematics. One of the first  to use the chess board in a math problem was the scholar and mathematician, Ibn Khallikān. Born in Iraq around the year 1211 AD, he is best known for The Obituaries of Eminent Men. A work that is often referred to as “The Biographical Dictionary” of the ancient Muslim world.  It is so expansive that an English translation of the book covers more than 2700 pages. Ibn Khallikān studied in Mosul (northern Iraq) and then emmigrated westward to Egypt. In Cairo he took up a professorship and served as a deputy to the Chief Judge. He also held the post of Chief Judge twice in the the city of Damascus (Syria).

In the year 1256 he proposed the following problem based on a myth regarding the creation of chess:

The inventor of Chess, Grand Vizier Sissa Ben Dahir, presented his work to King Shirham of India as a gift. The ruler was so pleased that he gave the inventor the right to name his prize for the achievement. Sissa, being a very wise man asked the King:  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, who was not strong in mathematics, quickly accepted the inventor’s offer, even getting offended by his perceived notion that Sissa was asking too low a 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 to account for his tardiness. The treasurer gave him the result of the calculation, and explained that it would be impossible to give the inventor his reward. The ruler then, to get back at Sissa who tried to outsmart him, told the man that in order for him to receive his reward, he was to count every single grain that was given to him, in order to make sure that the King was not stealing from him.

The Wheat & The Chessboard

The solution to the problem is that Sissa would end up with a sum total of 18,446,744,073,709,551,615 seeds or grains of rice. To say this number out loud start with 18-quintillion—446-quadrillion—744-trillion, etc.  If Sissa were to store his reward he would require a building with the following dimensions: 25 miles (40 km) long,  25 miles wide, and 984 feet (300 meters) high. The math that gives us this result:

The point of the exercise shows us the power of exponential numbers. A modern day example you may be familiar with, asks:

If your employer offered you a million dollars for a month of
work, or a penny a day doubled—which would you take?

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