Wheat and chessboard problem

Wheat and chessboard problem

The wheat and chessboard problem is a classic example of exponential growth and is often used to illustrate the concept of geometric progression. The problem is also known as the "rice and chessboard problem" and is a popular mathematical puzzle.

Problem Statement

The problem is set on a standard chessboard with 64 squares. The challenge is to place wheat grains on the chessboard such that the number of grains doubles on each subsequent square. Starting with one grain on the first square, the number of grains on each square would be as follows:

• 1st square: 1 grain
• 2nd square: 2 grains
• 3rd square: 4 grains
• 4th square: 8 grains
• ...
• 64th square: 2^63 grains

Mathematical Explanation

The total number of grains on the chessboard can be calculated using the formula for the sum of a geometric series. The sum \( S \) of the first \( n \) terms of a geometric series with the first term \( a \) and common ratio \( r \) is given by:

\[ S = a \frac{r^n - 1}{r - 1} \]

In this problem, \( a = 1 \), \( r = 2 \), and \( n = 64 \). Therefore, the total number of grains \( S \) is:

\[ S = 1 \frac{2^{64} - 1}{2 - 1} = 2^{64} - 1 \]

This results in a total of 18,446,744,073,709,551,615 grains of wheat.

Historical Context

The wheat and chessboard problem is often attributed to the legend of the invention of chess in India. According to the legend, the inventor of chess presented the game to the king, who was so impressed that he offered to grant the inventor any reward he desired. The inventor asked for grains of wheat to be placed on the chessboard in the manner described above. The king, initially thinking the request modest, soon realized the enormity of the number of grains required.

Applications

The wheat and chessboard problem is frequently used in computer science and information theory to explain the concept of exponential growth. It also serves as a cautionary tale in economics and finance about the potential pitfalls of exponential growth in contexts such as compound interest and population growth.

See Also

• Exponential growth
• Geometric progression
• Mathematical puzzle
• Chessboard
• Legend of the invention of chess