On a board of 64 squares, grains of rice are placed like this: - 1 in the first box,
- 2 in the second,
- 4 in the third,
- and so on, always doubling the amount in the previous box. How many grains of rice would there be in total at the end of the board?
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Chess and the grain of rice
The legend of the board and the rice is one of the best ways to teach exponential growth: what seems modest at the beginning becomes enormous in the end.
Hints
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- Think in powers of 2: each square doubles the previous one.
- The total after 64 squares is a geometric sum.
- Use the formula $1+2+4+\cdots+2^n = 2^{n+1}-1$.
Solution
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The quantities form a geometric progression:
$$ 1,2,4,8,\dots,2^{63}. $$ The total sum is $$
1+2+4+\cdots+2^{63}=2^{64}-1.
$$ Therefore, the total grains is
$$ 2^{64}-1 = 18\,446\,744\,073\,709\,551\,615. $$ The lesson of the problem is that doubling an amount seems innocent at first, but it grows incredibly quickly. $$
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