A Mule and a Donkey

Date: 04/08/99 at 00:45:38
From: Justin Rittgasser
Subject: Euclid, "A mule and a donkey"

I am in Grade 5 and I am doing research on Euclid. I came across a 
puzzle posted on another site.  It is supposed to be a famous puzzle 
attributed to Euclid. It goes: A mule and a donkey were talking. The 
mule said, "I'm carrying more than you. In fact, if you gave me one of 
your sacks, then I would have twice as many as you. If I gave you a 
sack, our loads would be equal." How many sacks was each animal 
carrying?  

I am struggling a bit in math, but I like to do these brainteasers.  
Can you guide me? Is there a place to find out more about this 
puzzle? Is it solveable?

Thanks for your time and help.

Justin
(typed by my sister, Nicole)

Date: 04/08/99 at 12:11:15
From: Doctor Peterson
Subject: Re: Euclid, "A mule and a donkey"

Hi, Justin.

This problem is pretty easy to do using algebra; but for you and 
Euclid, neither of whom probably know algebra, it's a brain teaser!

Here's how I'd do it with algebra, if you can follow it:

Let's call the number of sacks on the mule and the donkey M and D 
respectively. Then the two statements mean this:

   If you gave me one of your sacks, then I would have twice as many 
   as you. (That is, if the mule had one more and the donkey one less, 
   the mule would have twice as many.)
   M + 1 = 2 * (D - 1)

   If I gave you a sack, our loads would be equal.
   (That is, if the mule had one less and the donkey one more, they 
   would have the same amount.)
   M - 1 = D + 1

From the second statement, I know that the mule has 2 more than the 
donkey:

   M = D + 2

I can put this fact into the first statement and rewrite it using 
rules of algebra:

   (D + 2) + 1 = 2 * (D - 1)

    D + 3 = 2 * D - 2
    D + 3 - D = 2 * D - 2 - D
    3 = D - 2
    5 = D

So the donkey has 5 sacks. The mule has 2 more, or 7.

Check this out: if the donkey gave one to the mule, they would have 4 
and 8; if the mule gave one to the donkey, they would both have 6.

The ancient Greeks and Egyptians spent a lot of time on problems like 
this, because they didn't have any easy way to work with them. In a 
few years you'll be able to do things Euclid had trouble with!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/