Rice on a chessboard
1) Explain how the courtier was able to fool the Persian King.
2) Calculate the doubling time of something (100?) growing 10% per year.
3) For the math wizards: are you able to calculate / reason the amount of
rice on the final square?
4) Why is it important to understand exponential growth, according to the
article? Can you think of reasons why?
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www.Docentgeschiedenis.nl - 2011
Rice on a chessboard
According to legend, a courtier presented
the Persian king with a beautiful, handmade chessboard. The king asked what he
would like in return for his gift and the
courtier surprised the king by asking for
one grain of rice on the first square, two
grains on the second, four grains on the
third etc. The king readily agreed and
asked for the rice to be brought. All went
well at first, but the requirement for
2 n − 1 grains on the nth square demanded
over a million grains on the 21st square,
more than a million million (aka trillion)
on the 41st and there simply was not
enough rice in the whole world for the
final squares.
When a quantity such as the rate of consumption of a resource (measured in tons per
year or in barrels per year) is growing at a fixed percent per year, the growth is said to
be exponential. The important property of the growth is that the time required for the
growing quantity to increase its size by a fixed fraction is constant. For example, a
growth of 5% (a fixed fraction) per year (a constant time interval) is exponential. It
follows that a constant time will be required for the growing quantity to double its
size (increase by 100 %). This time is called the doubling time T2 , and it is related to
P, the percent growth per unit time by a very simple relation that should be a central
part of the educational repertoire of every human being.
T2 = 70 / P
As an example, a growth rate of 5% / yr will result in the doubling of the size of the
growing quantity in a time T2 = 70 / 5 = 14 yr. In two doubling times (28 yr) the
growing quantity will double twice (quadruple) in size. In three doubling times its
size will increase eightfold (23 = 8); in four doubling times it will increase sixteenfold
(24 = 16); etc. It is natural then to talk of growth in terms of powers of 2.
We can now see that this astounding observation is a simple consequence of a growth
rate whose doubling time is T2 = 10 yr (one decade). The growth rate which has this
doubling time is P = 70 / 10 = 7% / yr.
The reason all this is important has to do with our society; energy usage, population
growth and technological developements. Undertanding the exponential function and
the doubling time of something growing exponentially, helps us understand major
problems in our society.
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www.Docentgeschiedenis.nl - 2011