Question

A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chess board. On the second square the King would place two grains of wheat, on the third square, four grains of wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining squares, how many grains of wheat should be placed on square ? Also find the total number of grains of wheat on the board at this time and their total weight in pounds. (Assume that each grain of wheat weighs 1/7000 pound.)How many grains of wheat should be placed on square 18?  How many total grains of wheat should be on the board after the the grains of wheat have been placed on square 18 ?What is the total weight of all the grains of wheat on the board after the grains of wheat have been placed on square 18 ?  enter your response here pounds(Round to the nearest tenth as needed.)

Answer 1

First we have to find a formula for the number of grains on a given wheat. If we writte the first numbers: 1 2 4 8.... we notice that this is a exponential function in the way:

`2^k`

where "k" is the wheat number from zero. In the first square, we replace k=0 and we got 1 grain, for the second square, k=1 and we got 2 grains, and so on.

The total grains in the square 18 is:

`2^(17)=131072`

Then the answer for first question is 131072

Hence the total number of grans in the board can be estimated as:

`2^n-1`

where n is the number of quares filled. For the first question, we replace n=18 as follow:

`2^(18)-1=262143`

Hence the second answer is 262143

Next, the total weight is calculated multiplying the latter value for the weight of a grain:

`262143*(1)/(7000)=37.449\text{ pounds}`

Then the third answer is 37.4 pounds.