prize13-2, Exercise 2: The wheat legend

Model answer (check the details):

The first square has 1 grain (2^0), the second 2 (2^1), and so on, so the nth square has 2^(n-1) grains. The 64th square has 2^63 grains, which is 9,223,372,036,854,775,808, about 9.2 x 10^18.

The total on all 64 squares is 2^64 - 1 = 18,446,744,073,709,551,615, about 1.8 x 10^19 grains.

A check in Python:

    print(2**63)
    print(sum(2**i for i in range(64)))   # equals 2**64 - 1

The figure is far larger than any harvest in history, so no ruler could have paid. That is the point of the story: it teaches how fast doubling grows. It is a legend attached to the game in later sources, not a record of an event.

Care: the arithmetic is exact, but the earliest source of the story is uncertain.