# 18,446,744,073,709,551,615

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## Number

$18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 615$ is:

$3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$

$2^{64} - 1$

The number of moves it would take the monks to move all the disks onto a different peg in the Tower of Hanoi.

## Historical Note

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.