Wheat and Chessboard Problem/Historical Note

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Historical Note on Wheat and Chessboard Problem

Note on $18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 615$

The story goes that the inventor of chess, Sissa ben Dahir, was offered a reward by King Shirham of India.

King Shirham asked Sissa to name his reward.

Sissa replied:

"Please give me one grain (of rice, or wheat, etc.) placed on the first square of the chessboard. Then two grains placed on the second square. Then four placed on the third square and eight placed on the fourth square, and so doubling the number of grains on each subsequent square until all $64$ squares are so covered."

King Shirham acceded to what he viewed as a foolishly paltry request, until such time came as to attempt to actually carry out such a task.

By Sum of Geometric Sequence, The total number of grains evaluates to $2^{64} - 1$, which is more grains of wheat than existed.

It is a popular classroom exercise to require students to calculate the volume or weight of grain that Sissa asked for.

Note that the number of grains is the same number as that of the number of moves needed to complete the task of the Tower of Hanoi.

Edward Kasner and James Newman proceed to point out, in their Mathematics and the Imagination of $1940$, that if it is assumed there have been $64$ generations since $0 \text {CE}$, this is also the number of ancestors of each person alive since that time.

As David Wells puts it in his Curious and Interesting Puzzles of $1992$:

The ratio of $2^{64}$ to the actual population of the earth at that time is therefore a measure of the amount of unintentional interbreeding that has taken place.

$19$th Century Version

The question was modernized in Cassell's Book of In-Door Amusements, Card Games and Fireside Fun, as The Pin in the Hold of the "Great Eastern" Steamship.

It is calculated that $200$ pins go to the ounce.

Suppose that for the $52$ weeks in the year:

one pin were dropped into the hold during the first week
two in the second
four in the third

and so on.

By the end of the year the weight of the whole would be no less than $628 \, 292\, 358$ tons of pins.

As the Great Eastern steamship was built to carry $22 \, 500$ tons only, it follows that to carry all the pins there would be required $27 \, 924$ ships of the size of the Great Eastern.