The Lesson of Exponential Growth: Rice and the Chess Board

The inventor of chess, Sessa, pleased the current king so much, that he was asked to name his own prize. His request seemed modest. One grain of rice doubled for each square on the board.

"If a chessboard were to have rice placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?"

On the first half of the board, a total of 4,294,967,295 (2³² − 1) grains of rice, or about 100,000 kg of rice (assuming 25 mg as the mass of one grain of rice) were counted. India's annual rice output is about 1,200,000 times that amount. Not that bad.


But on the 33 square the number of grains of rice was now more that the first 32 in total.

On the 64th square alone, 9,223,372,036,854,775,808 grains of rice were due, or more than two billion times as much as on the first half of the chessboard.

So, all of the rice due for the total 64 squares was 18,446,744,073,709,551,615 grains of rice, weighing 461,168,602,000 metric tons, which would be a heap of rice larger than Mount Everest. This is around 1,000 times the global production of rice in 2010 (464,000,000 metric tons).

In the world of technology, the doubling of processor speed is occurring approximately every 20 months. That means every 5 years or so, the amount of processing speed increases by a factor of 8. So, if every square represented a 5 year period, we would have 1 unit of processing power, followed by 8, then 64, 512, 4096, 32768. This would be for 2013, 2018, 2023, 2028, 2033 and 2038. And in 2043, the processing power would be over 256,000 time as great as it is today.

Certainly its not clear what will happen so far out, however trends for exponential growth in processor speed have been fairly constant since the early 1970s. There is no reason to believe that we will not be able to achieve these numbers. The bigger question is what does it mean to society when nearly every assumption about life will change? The study of such growth and its implications is Exponology.