Attempt the chessboard problem in real life and you will find yourself rapidly running out of rice, let alone space on the chessboard!

The chessboard problem is a problem from antiquity depicting a function that grows unexpectedly fast. The basic formulation of the problem is as follows:

If I receive one grain of wheat for the first square of a chessboard, two grains for the second square, and so on, doubling the number of grains for each square up to all 64, how many grains of wheat will I receive total?

This problem is usually dressed up in a story of the inventor of chess meeting the emperor of India or Persia and presenting him the game, then asking for one grain of wheat for the first square, two grains for the second square, and so on as above. The answer turns out to be 264 - 1 = 18,446,744,073,709,551,615 grains, which was an inconceivably large amount of grain and would bankrupt the emperor. Even today that much grain would take up to 2,000 years for the world to produce.

While slow by googological standards, examples of exponential growth were essential to the exploration of larger numbers, and are still useful as a learning tool today.

Alternative scenarios Edit

Another scenario commonly presented to express this doubling phenomenon is that you win a lottery and are given the choice of receiving $100,000 per day for 30 days, or $1 the first day, $2 the second day, and in general $2n-1 the nth day. In the first scenario, you will receive "only" $3,000,000, but in the second scenario you will win a total of $1,073,741,823 and probably bankrupt who does the lottery.