Squaring Pennies

Date: 9/20/95 at 15:59:01
From: Mike Bortz 
Subject: Fwd: Re: math problem

I teach 7th and 8th grade math. These questions are from my students. 
Is it better to have $1,000,000 or 1 penny doubled every day for a 
month?

Date: 9/22/95 at 15:28:15
From: Doctor Andrew
Subject: Re: Fwd: Re: math problem

I remember hearing an ancient parable (Babylonian, says Dr. Ethan) about 
a king and one of his wise subjects who agreed to do a favor for him.  The
king offered him a fair reward in return for this favor, and his subject
asked him for one piece of grain to be placed on one square of a chessboard,
two on the second, four on the third, and so on until each square on the
chessboard was full.  The king laughed at him for his foolishness, thinking
that this reward would cost him nothing.  But, when the time came to reward
his subject for his service, the king found that he did not have enough grain
in all his kingdom to fulfill his subject's request.  After all, there are 64
squares on a chessboard (8 x 8 = 64), so the king needed to put 2^63 pieces
of grain on the final square (1 = 2^0 on the first, 1 * 2 = 2^1 on the second, 
1*2*2 = 2^2 on the third, etc).  Considering that 2^20 is about a million 
(1048576 to be exact), 2^63 is more than a million cubed (2^63 = 2^20 * 2^20
* 2^20 * 2^3 = (2^20)^3 * 2^3), this is a lot of grain.  There is some 
historical debate over whether the king honourably sold his kingdom to pay 
his debt or whether he beheaded his subject (and whether Babylonians played 
chess).

A little more math for you here:  The king actually had to put 2^64-1 total
pieces of grain on the chessboard, when you add up the grain in all the
squares.  In fact, 1+2+4+..+2^n = 2^(n+1) - 1 for all n > 0.  It's a tough
but interesting problem to show that this is true.

With regard to your specific question, hopefully you can now figure out 
how many pennies you would receive in a month.  Here's a useful fact:  
2^10 is about 1000 (1024 to be exact).  I think that there are nearly that 
many pennies in circulation, but finding enough rolls to hold all of them 
could be a chore.

Thanks for your question.  Keep in touch.

-Doctor Andrew,  The Geometry Forum