Exponential Functions
The Legend of Chess and Rice Grains
The Legend of Chess and Rice Grains
The Story
Long ago in ancient India, the game of chess was invented - a strategic game played on a board with 64 squares. The Indian Emperor was so delighted with this ingenious game that he wanted to reward its inventor generously.
The emperor told: “Ask for any reward you desire, but do not be too modest!”
The humble request surprised everyone: “My lord, I ask only for rice grains. Place 1 grain of rice on the first square of the chessboard, 2 grains on the second square, 4 grains on the third square, and for each following square, double the number of grains from the previous square.”
The emperor felt almost insulted by such a modest request - he had no idea what he was agreeing to!
Let’s Investigate This “Simple” Request
Pattern Recognition
Let’s see how the rice grains accumulate on the first few squares:
| Square Number \((n)\) | Rice Grains \((a_n)\) | Calculation | Power Notation |
|---|---|---|---|
| 1 | 1 | 1 | \(2^0\) |
| 2 | 2 | \(1 \times 2\) | \(2^1\) |
| 3 | 4 | \(2 \times 2\) | \(2^2\) |
| 4 | 8 | \(4 \times 2\) | \(2^3\) |
| 5 | 16 | \(8 \times 2\) | \(2^4\) |
| 6 | 32 | \(16 \times 2\) | \(2^5\) |
| 7 | ? | ? | ? |
| 8 | ? | ? | ? |
Your Task: Complete the table for squares 7 and 8. Can you spot the pattern?
Finding the General Formula
From the pattern above, we can see that for square \(n\), the number of rice grains is: \[a_n = 2^{n-1}\]
This is called an exponential function with:
- First term: \(a_1 = 1\)
- Common ratio: \(q = 2\) (we double each time)
The Reality
Calculate the 64th Square
Using our formula \(a_n = 2^{n-1}\), let’s find how many rice grains would be needed for the 64th (final) square:
\[a_{64} = 2^{64-1} = 2^{63}\]
Using a calculator: \(2^{63} = 9{,}223{,}372{,}036{,}854{,}775{,}808\)
Total Rice Grains
The total number of rice grains for all 64 squares would be:
\[S_{64} = a_1 + a_2 + a_3 + \ldots + a_{64} = 2^{64} - 1\]
\[S_{64} = 18{,}446{,}744{,}073{,}709{,}551{,}615\]
Putting This in Perspective
Weight Calculation
Let’s estimate the actual weight:
- Approximately 100 rice grains weigh 3 grams
- So 1 rice grain weighs about 0.03 grams
For the 64th square alone: \[\text{Weight} = 9{,}223{,}372{,}036{,}854{,}775{,}808 \times 0.03\text{ g} \approx 277 \text{ billion tons}\]
For all 64 squares: \[\text{Total Weight} \approx 553 \text{ billion tons}\]
Reality Check: In 2006, the entire world produced only 634 million tons of rice. The “modest” request would require about 873 years of the world’s entire rice production!
Understanding Exponential Growth
Key Insight: This story demonstrates the incredible power of exponential growth. Even though each step seems small (just doubling), the cumulative effect becomes large very quickly.
This is why exponential functions are so important in mathematics - they appear in:
- Population growth
- Compound interest
- Radioactive decay
- Viral spread
- Computer processing power