Once upon a time..
Once upon a time, there was a King who ruled over a region in India.
He was a brilliant (and undefeated) chess player and challenged anyone to a game.
One day a travelling Sage came by.
To everyone’s surprise, he beat the King.
After accepting his defeat, the King offered the Sage any reward he wanted.
Being a simple man, the Sage made a humble request:
“Today, give me one grain of rice for the first square of this chessboard
Then tomorrow, two grains for the second square
The day after, four grains for the third square
Thereafter, eight grains for the fourth square and so on…”
The King being a wealthy man, laughed and happily agreed.
On day five he laid out and paid 16 grains of rice.
On day six he laid out 32, and so on.
However, after filling out the last square on the second row (32,768), he began to realise something was amiss.
The King consulted his Treasurer.
To get to the last square, the King would need a total of more than eighteen quintillion grains to settle his debts (18,446,744,073,709,551,615).
This was more than all the rice in the world!
With current annual rice production, it would take the King over 2000 years to settle this debt.
The wise Sage had taught not only the King, but also the Kingdom a powerful maths lesson in exponential growth.
In real life
In technology strategy, the “second half of the chessboard” is a phrase, coined by Ray Kurzweil.
It refers to the point where an exponentially growing factor begins to have a significant economic impact on an organization’s overall business strategy.
While the number of grains on the first half of the chessboard is large, the amount on the second half is vastly (2^32 > 4 billion times) larger.
In finance, this concept provides the answer to the question:
“Would you rather have a million pounds or a penny on day one that is doubled every day until day 30?”
And thus helps explain the concept of compound interest.
(Doubling would yield over ten million pounds).
For my older students, it’s a reminder to start saving as early as possible.
The maths
There are 64 squares on a traditional chessboard.
This story is just an example of a simple geometric series. We can either do a brute force calculation for the number of grains needed:
Or express it using exponents:
The base of “2” just represents the doubling feature.
I.e. on day one the King pays 2^0 (2 to the power of 0), which equals 1 grain of rice. On day two this is 2^1 = 2 , then 2^2 = 4, and so on.
Alternatively, we can represent it using the following Sigma notation:
On the 64th square of the chessboard (the final piece), there would need to be 263 = 9,223,372,036,854,775,808 grains, more than two billion times as many as on the first half of the chessboard.
So the next time somebody in a meeting says things are growing exponentially, you might want to challenge exactly what they have in mind :)
Either that, or invest in it straight away!
I’m currently working as a Maths Teacher at a challenging secondary school. I comment on themes from Education & Learning, and how they might benefit you, and the next generation.
Subscribe to follow my journey from banker to teacher: www.theedletter.com
