2023-01-16 The Astonishing Exponential growth

You probably heard it a million times: “this is growing exponentially!”. That’s often used to mean, “this is growing really fast”. But most people struggle with getting the feel of how actually fast exponential is.

There is a very old story you probably know in some form or another. Here’s my version.

Once upon a time, in a far away kingdom, lived a king.

This was an intelligent and enlightened king, who one day had to go fight a war. He came home victorious, but alas his first born son, who was leading the cavalry, died on the battlefield.

The king was heartbroken, and could not be consoled. He retired in his castle and mourned, and felt guilty for having made some mistake in leading the war that had cost him his son’s life.

One day, a wise old man walked from his little hut in the woods to the castle, and asked to talk to the king. When he was in front of him, he pulled out a chessboard and chess pieces, and asked the king to play a game. After some resistance, the king accepted.

The wise man, playing black, repeated the strategy of the enemy in the real war where the prince had been killed. At one point, the king was sure he could win the game, but only if he sacrificed one of his knights. He immediately realised what the old man was doing.

“Let’s play again”. The king played a different strategy, the old man used the same he did before, which was the same the actual enemy used. The king found himself in the same situation as before. They played and played, but no matter how hard the king tried, he’d either lose the game, or get to a point where sacrificing that knight was unavoidable.

The king understood: to win the war, his son had to die. But if he lost the war, his whole people would die or be enslaved. He felt relief for the first time in weeks. The king wanted to reward the old man and asked him to name anything, and it would be his.

The wise old man replied, “I don’t want much. Just a little rice. Put one grain of rice on the first square of the chessboard, two grains on the second, four on the third, eight on the fourth, and so on. The king found it amusing that this old man only wanted a little rice, when he could ask anything.

Ok, let’s stop here and run some numbers for a minute, and see how much rice exactly the king now owes to the old man.

What the old man asked for is a quantity of rice that grows exponentially, doubling on every square.

Just so you know, there are about 66,000 grains of rice in a kilogram (kg) of rice (that’s about 2 pounds, if you are weirdly non-metric), as [this guy proves](Roughly 66000 grains in 1 kg (here’s a video: https://www.youtube.com/watch?v=V1C0cFSYyqk).

Ok, let’s get counting rice grains.

• On the first square of the chessboard, obviously, there’s only one single grain of rice.
• On the last square of the first row, the king would need to put 256 grains. That’s about half a gram.
• On the last square of the second row, 65,536 grains, which is pretty much exactly 1kg.
• By the end of the fourth row, on a single square, the king would have to pour 32 metric tons of rice. That’s half a space shuttle, in weight.
• On the last square of row 6 we get at 2132,386 tons. More that 20 blue whales worth of rice.
• On the very last square, he’d need 139,748,061,212 metric tons of rice. That’s about 140 Mount Everests of rice.

So summing it all together, that’s 2,794,961,223,289 metric. tons. of. frigging. rice.

According to FAO, the total world rice production in 2020 was 756,743,722 metric tons. It would take the king and his descendants 3694 years. That’s how fast exponential goes.

The story ends, in my version, with the king breaking his word, and the wise man telling him next time not to make promises before thinking about the cost of things and returning to his hut in the woods.

But there’s more I want to say about exponential growth, and how it applies to technology. More tomorrow!