You probably know this story everyone studied it at some point, either in physics or mathematics.
One of the most amusing stories that illustrates the difference between exponential growth and linear growth is the story about the origin of the game of chess.
When the game was first invented by Sissa ibn Dahir, he presented it to one of the ancient Persian kings. The king was overjoyed and decided to reward the inventor generously.
As we know, a chessboard has 64 squares 8 across and 8 down.
Sissa asked for a reward that seemed extremely modest and simple: a quantity of rice to be determined by the chessboard as follows:
On square number 1 place one grain of rice.
On square number 2 double that, two grains.
On square number 3 four grains.
On square number 4 eight grains.
And so on, with each square containing double the number of grains as the previous one, until reaching square number 64.
Before we continue, let’s take a quick look at what linear growth would look like.
In that case, square 1 would have 1 grain, square 2 would have 2 grains, square 3 would have 3, square 4 would have 4, and so on up to 64 grains on the 64th square.
If you add all those numbers (64 + 63 + 62 … + 3 + 2 + 1), the total would be around 1,000 grains of rice.
And since one kilogram of rice contains about 10,000 grains, that’s just a small bag of rice.
But in Sissa’s case, with one grain on the first square, two on the second, four on the third, and so on by the 8th square, the total would already reach 128 grains.
By the 11th square, it would exceed 1,000 grains.
By the end of the second row, there would be a square that alone would need an entire bag of rice and the next square would need two!
By the third row, there would be a square requiring an amount of rice the size of a small house.
By the fifth row, one square alone would need enough rice to fill a football stadium.
And finally, the 64th square would require billions of billions of grains to be exact, 2⁶⁴, which equals 18,446,744,073,709,551,615 grains of rice!
That’s more rice than exists on the entire planet Sissa effectively asked for all the rice produced on Earth for over two thousand years.
This is what’s known as exponential growth.
And if you compare the number of grains in the linear case to the exponential case, you’ll see that exponential growth increases at an absolutely mind boggling rate.
