Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc.

How many grains total will be placed on an chessboard?The answer is 18446744073709551615.

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First, we show that $\sum\limits_{i=0}^n 2^n = 2^{n+1}-1$ . This can be shown easily via induction, but more intuitively by considering the following: $\begin{aligned} \sum_{i=0}^n 2^n &= 1+2+4+8+\cdots+2^n \\ &= 2+2+4+8+\cdots + 2^n - 1 \\ &= 4 + 4 + 8 + \cdots + 2^n - 1 \\ &= 8 + 8 + \cdots + 2^n - 1 \\ &\;\vdots \\ &= 2^n + 2^n - 1 \\ &= 2^{n+1}-1 \end{aligned}$

Then, the sum of the rice on our chessboard is $\sum\limits_{i=0}^{63}2^i = 2^{64}-1 = \boxed{18{,}446{,}744{,}073{,}709{,}551{,}615}$