The rice and chessboard problem: why exponential growth fools your brain
The rice and chessboard problem: one grain of rice, doubled on every square
One grain of rice on the first square of a chessboard, two on the next, then four, then eight. It sounds like a tiny gift. Try it with a real 1 kg bag of rice, and the bag is empty before the second row is done. So how much rice would all 64 squares need?
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1
An old legend tells of the man who invented chess. The king loved the game so much that he let the inventor choose his own reward. The inventor pointed at the board.
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He asked for rice. One grain on the first square, two on the second, four on the third. Each square gets double the square before, up to the 64th square. The king laughed at such a small wish.
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On a rainy Saturday, your friend Rono tells you this legend while he teaches you chess. He has a 1 kg bag of rice. Surely that is enough for the whole board, he says.
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You start with the first row. It takes only 255 grains, less than a spoonful. Rono smiles: the bag is still almost full.
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Then the heaps grow fast. Squares 1 to 15 already take about 33,000 grains. Square 16 alone needs 32,768 more. A kilo of rice holds only about 40,000 to 50,000 grains, so the bag runs out partway through square 16.
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And the last square? It alone would need about 9 billion billion grains. Written out, the number has 19 digits.
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7
The whole board would need about 18 billion billion grains. That is more rice than the whole world grows in 500 years. In the legend, the king could never pay. He had laughed too soon, and so had Rono.
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Go further
Two other ways to say it.