Porisma

Worked solution

Total the doubling reward on a chessboard

In the source legend, a chess inventor asks for one grain of rice on the first square, two on the second, four on the third, eight on the fourth, and twice as many on each following square. How many grains are required in total for all 6464 squares?

CJC · Sequences and Series · Unnumbered rice-grain and chessboard question

Answer

264−1=18 446 744 073 709 551 6152^{64}-1=18\,446\,744\,073\,709\,551\,615 grains.

Compare your reasoning

Identify the finite geometric sum

S64=1+2+4+8+⋯+263=∑j=0632j\begin{aligned}&S_{64}\\ & =1+2+4+8+\cdots+2^{63}\\ & =\sum_{j=0}^{63}2^j\end{aligned}

The reward has a=1,r=2 and 64 terms, from exponent 0 through 63.

Use the finite-sum formula

S64=1(264−1)2−1=264−1=18446744073709551615\begin{aligned}&S_{64}\\ & =\frac{1(2^{64}-1)}{2-1}\\ & =2^{64}-1\\ & =18446744073709551615\end{aligned}

The exact integer is an authored calculation applying the formula introduced immediately after the legend.

How did this question go?

Solution notes

The source poses the question and introduces the general GP-sum derivation; it does not print this exact integer. The legend’s historical and rice-production claims are retained as story context, not verified measurements.