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## Exponential Thinking

by | Feb 27, 2017

As humans, we have a much easier time understanding linear progression versus exponential progression.  Here’s a famous fable illustrating our troubles grasping the exponential:

In the fable, a wise man invents the game of chess and presents it to his king. Pleased, the king allows the man to name his reward. The wise man responds that he wishes only modest compensation, following a simple rule. He would have one grain of rice on the first square of the chessboard, two on the second, four on the third, and so on, doubling each time for each of the sixty-four squares. The king chuckles at the apparent measliness of these amounts and says yes. It soon becomes clear that he has made quite a big mistake. After two rows the king owes nearly 33,000 grains of rice and is not chuckling quite so much. By the last square of the first half of the chessboard the amount involved is enormous, totaling more than 2 billion grains, or nearly 100,000 kg, of rice –but it is not yet absurd. Yet on the first square of the second half the king must pay that entire sum again, and then twice that, until he owes a Mount-Everest-sized pile of rice.  On the sixty fourth square the king would have had to put more than 18,000,000,000,000,000,000 grains of rice which is equal to about 210 billion tons and is allegedly sufficient to cover the whole territory of India with a meter thick layer of rice and is about 1,000 times the current global production of rice.

The second half of the chessboard is a phrase,  in reference to the point where an exponentially growing factor begins to have a significant impact.

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