9th Math Riddle 2019

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 $\sum\limits_{i=0}^{63}2^i = 2^{64}-1 = \boxed{18{,}446{,}744{,}073{,}709{,}551{,}615}$

On each square, $n$ , you will have $2^{n-1}$ grains of wheat. Since there are 64 squares on a chess board, the total amount of grains will be $\displaystyle \sum_{i=1}^{64} 2^{n-1} = 18,446,744,073,709,551,615$ .