How to become rich with simple math!

There's a legend telling that, long time ago, a king of Persia played chess with the ambassador. After defeat, the king, for gratitude, invited the ambassador to express any desire, which would have been undoubtedly fulfilled. The ambassador said he wanted wheat: "one grain on the first square of the chessboard, two grains on the second, four on the third, and continuing so on up to the last square, doubling the number of grains on each subsequent square". "What an easy task!" the king replied, astonished that the request was so miserable, then ordered the court treasurer to provide.

The story doesn't tell us how it ends, but we can certantly say that the king never paid the ambassador; here is why:

Doubling the number of grains up to the sixty-fourth square, we have:

1st square: 1 grain = 2⁰
2nd square: 2 grain = 2¹
3rd square: 4 grain = 2²
4th square: 8 grain = 2³
5th square: 16 grain = 2⁴
[...]
14th square: 8.128 grain = 2¹³
15th square: 16.384 grain = 2¹⁴
[...]
63th square: 4.611.686.018.427.387.904 grain = 2⁶²
64th square: 9.223.372.036.854.775.808 grain = 2⁶³

It's a geometric series; the sum of all the grains is 2⁶⁴-1 = 18.446.744.073.709.551.615.

According to the Food and Agricolture Organization, the international wheat production in 2014 was 729 millions of metric tons¹. So, if we assume, on the low side, 10 grains weighing 1 gram, we can estimate that the total amount of grains to put on the chessboard equals the annual world production for 2500 years!

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Sources: image1, image2