The rice and chessboard problem: why exponential growth fools your brain
Exponential vs linear growth: why your brain guesses so low
Guess how much rice a chessboard needs when each square gets double the one before. A bag? A truck? The real answer is bigger than centuries of the world's harvest. Meet exponential growth and linear growth, and the habit of the brain that mixes them up.
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1
Rono thought one bag would cover the board, and you thought so too. You were both wrong by billions of billions of grains. How can a guess be so far off?
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2
Your brain makes things grow by adding. Add one grain on each square, and you get 1, 2, 3, 4. Doubling starts almost the same way: 1, 2, 4, 8. For a few squares, the two look alike.
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3
Then doubling runs away. Adding gives a straight line, which is called linear growth. Doubling gives a curve that bends up faster and faster, which is called exponential growth.
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4
Here is the secret of doubling. Each new square holds one grain more than all the squares before it put together. Squares 1 to 4 hold 15 grains. Square 5 alone holds 16.
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5
So the last square holds more rice than the 63 other squares together. Half of the rice of the whole board sits on that one square.
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6
Now your guess makes sense. If each square got just one grain more than the one before (1, 2, 3, up to 64), the whole board would need only 2,080 grains, about a handful. Your brain was counting a straight line. The board was doubling. Just then, someone knocks on the door.
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Quick check
A sheet of paper, folded again and again.
Show the answer
The 10th fold. It doubles the thickness, so it adds as much as the whole stack had before it: more than all the other folds added together.
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Go further
Where the word comes from, and a fold too far.