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 Check out our web site! http://mathforum.org/dr.math/
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