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Doubling Grains of Wheat

Date: 10/7/96 at 23:46:47
From: Ernie & Nichole Drumm
Subject: Double Grains of Wheat on Chessboard...

A man asked for 1 grain of wheat for the 1st square on a chess board, 
2 grains for the 2nd square, 4 for the 3rd, 8 for the 4th, 16 for the 
5th ...and so on for each of the 64 squares on the board.

How do I find the total number of grains of wheat? How many 6-foot-
deep swimming pools would that many grains of wheat fill up?  How 
would you go about figuring something like that out?

Thank you,  
   Nichole Drumm

Date: 10/8/96 at 6:3:23
From: Doctor Pete
Subject: Re: Double Grains of Wheat on Chessboard...

The number of grains of wheat on the n(th) square is 2^(n-1), or 2 to 
the power of n-1.  This is because the first square has 2^0 = 1 grain, 
the second has 2^1 = 2, and the n(th) square has twice as many as the 
previous. Thus the total number of grains of wheat is

     S = 1 + 2 + 4 + 8 + ... + 2^63.

Since this is a geometric sequence with common ratio 2, the sum is

         2^64 - 1
     S = -------- = 2^64 - 1 = 18446744073709551615.
           2 - 1

You can use a calculator to evaluate this approximately, or multiply 
it by hand for a few minutes, or if you have a symbolic math program 
like Mathematica or Maple, you can find it exactly as I have here.  
It's approximately 2(10^19) grains of wheat.

How many 6-ft-deep swimming pools will this fill?

Well, let's see.... how large is a grain of wheat?  If we assume 
an average grain of wheat has dimensions of 2mm x 2mm x 5mm, and an 
average swimming pool is a rectangular prism of dimensions 
15ft x 30ft x 6ft = 4.572m x  9.144m x 1.8288m, then we have

       20 (mm)^3                     20 * 1.845 * 10^19
     ------------ * 1.845 * 10^19 = --------------------
     76.455 (m)^3                      76.455 * 10^9

     = 4.825 * 10^9 swimming pools.

So it would take about 5 billion swimming pools of the above 
dimensions to contain all those grains of wheat.  Of course, depending 
on how you figure it, that value can be off by as much as an order of 
magnitude.  But the whole point is that 10^19 grains of wheat is a LOT 
of bread to bake....

-Doctor Pete,  The Math Forum
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Associated Topics:
High School Sequences, Series

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