Question

A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,000 grains of rice in 1 pound, and 2000 pounds in 1 ton, how many tons of rice did the mathematician attempt to receive?

Short answer

The mathematician asked for about 3,074,457 tons of rice.

Step-by-step solution

Understanding the Problem

The chessboard has 64 squares. The mathematician requests grains of rice following a geometric progression: 1 grain on the first square, 2 grains on the second, and doubling each time until the 64th square.

Understanding Geometric Progressions

A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this case, each term doubles, so the common ratio is 2.

Finding the Total Number of Grains

The total grains of rice is the sum of the geometric series: \[ S_{64} = 1 + 2 + 4 + ... + 2^{63}\]. This expression can be summed using the formula for the sum of a geometric series: \[ S_n = a \frac{r^n - 1}{r - 1}\], where \(a = 1\), \(r = 2\), and \(n = 64\).

Applying the Geometric Series Formula

Substituting into the formula: \[ S_{64} = 1 \cdot \frac{2^{64} - 1}{2 - 1} = 2^{64} - 1\]. Calculate \(2^{64}\) to find the total grains of rice.

Converting Grains to Tons

Calculate the weight of \(2^{64} - 1\) grains of rice in tons. There are 30,000 grains of rice in a pound and 2000 pounds in a ton. Use this to convert grains of rice to tons:\[(2^{64} - 1) \div 30,000 \div 2000\].

Final Calculation

First, compute \(2^{64}\), check using a calculator for an exact number which is \(18,446,744,073,709,551,616\). Skipping the calculation of \(-1\) because it won't significantly affect the result. Compute the number of tons:\[(2^{64}) \div 30,000 \div 2000 = 3,074,457.24\].

Key concepts

Sum of a Geometric Series

The concept of the "Sum of a Geometric Series" is a powerful mathematical tool used in various contexts, like finance and physics. A geometric series is a sum of terms that each multiply the previous one by a constant value, known as the common ratio. In simpler words, if you start with a number and keep multiplying it by the same number (the common ratio) to get the next term, you are constructing a geometric sequence.

For our chessboard problem, each square represents a term in a geometric sequence, starting from 1 grain of rice and doubling it each time—that's an increasing geometric series with a common ratio of 2. The total grains, or the sum of the series, is captured by the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
In our case, the first term \(a\) is 1, \(r\) is 2, and \(n\) is 64. This formula helps simplify the computation and arrives at \(2^{64} - 1\).
Using this formula is crucial for solving complicated problems involving exponential growth, like the legendary chessboard story.

Exponents in Mathematics

Exponents are a fundamental concept in mathematics, enabling the expression of large numbers in a compact form. When used in the context of a geometric series, exponents represent repeated multiplication of the base number. In other words, an exponent tells how many times the base is multiplied by itself.
For the chessboard payment problem, the exponent \(2^{64}\) occurs because each square on the board doubles the number of grains from the previous one, hence multiplying 2 by itself 63 more times after the first square. The expression \(2^{64}\) is quite large, so understanding the power of exponents helps in efficiently working with such enormous numbers.
Exponents are not only crucial for solving mathematical series but also for many real-world applications, such as computing compound interest and understanding exponential growth in populations.

Mathematical Problem Solving

Mathematical problem solving is about understanding, organizing, and applying mathematical concepts to find solutions to complex problems. It involves breaking down a problem into smaller, more manageable steps.
In our exercise, the problem-solving process involves initially understanding the pattern of grain distribution across the chessboard—recognizing a geometric progression then identifying which mathematical concepts apply.
Following this, using the right formulas, in this case the formula to sum a geometric series, helps simplify the problem massively. Problem solving in mathematics often requires applying a logical approach, keeping track of each step, and ensuring accuracy when dealing with large numbers. It is about turning seemingly overwhelming problems into coherent, solvable tasks by choosing the correct approaches and tools.

Conversion of Units

The conversion of units is a crucial skill in mathematics and science. It allows for the translation of one type of measurement into another, enabling the comparison of disparate units.
In the context of the chessboard problem, converting grains of rice to tons requires understanding and applying unit conversions. After calculating the total grains of rice using the geometric formula, the next step is to translate that massive number of grains into a more comprehensible measure—tons of rice.
To do this, you would use given conversion factors: 30,000 grains per pound and 2000 pounds per ton. These conversion factors allow you to methodically change grains to pounds, and then pounds to tons, making the vast sum achieved through computation more understandable.