4- and 9-liter Pails: How to Measure 7 Liters

Date: 05/07/2002 at 01:27:30
From: Cynthia
Subject: Volume and Liters

I have a problem that needs to be solved. I tried using cups to do 
this problem but without the ability to measure I just can't seem to 
think it out properly. Can you help?  

Here it is:

You are trapped in a cave that has an endless supply of water. You 
have to create an explosive device to get out. You need to add exactly 
7 liters of water to a pre-made mixture. You only have a 4-liter pail 
and a 9-liter pail. Neither pail has markings on it and they are not 
symmetrical. You do not have the use of any kind of marking device. 
How do you get out?

Date: 05/07/2002 at 09:09:59
From: Doctor Peterson
Subject: Re: Volume and Liters

Hi, Cynthia.

Here are a couple of similar problems from our archives that may give 
you some ideas:

   Using 3C and 5C Pails to get 4 Cups
   http://mathforum.org/library/drmath/view/58594.html 

   Six Quarts of Water, 2 Containers
   http://mathforum.org/library/drmath/view/58685.html 

What you want (and these pages don't give) is a strategy for solving 
this sort of puzzle. Let me suggest one.

One thing we need is a way to keep track of what we are doing. I like 
to use a chart that shows how much is in each container:

    (4)   (9)
    ---   ---
     0     0

Now at each step we can either fill a pail, empty a pail, or pour from 
one to the other until we either empty the former or fill the latter. 
We can indicate on the chart what we did and what the result was:

     0     9  fill (4)
     4     5  pour into (4)
     0     5  empty (4)

You might just draw arrows to show what happened; that's harder to do 
in print.

Now we have to figure out how to choose the steps to take. There are 
only a few things to do at each step, so you might just make several 
charts starting with different first steps, and see where each one 
takes you; whenever you have a choice to make, start a new chart at 
that point if your first choice doesn't work out.

It will also be helpful to know the goal. You should try working 
backward from the end of the chart, to see what the next to last step 
should look like. You want to end with 7 liters in the 9-liter pail. 
How could you get to that state?

     0     7

You might either have poured 4 into (9):

     4     3
     0     7  pour into (9)

or poured 2 from (9) into (4):

     2     9
     4     7  pour into (4)

So you want to see if you can get amounts of 3 or 2, knowing that if 
you can get either of those, you can get 7. (That is, 7 = 9-2 and 
7 = 3+4.)

This should make the work a lot easier. You might find it interesting 
to keep a list of what amounts you are able to get in either bucket, 
and how many steps it takes.

This page may help in solving the problem; it lets you simulate the 
problem and try out ideas, and also gives some theory behind it:

   2 Pails Puzzle - Cut-the-Knot, Alexander Bogomolny
   http://www.cut-the-knot.org/ctk/CartWater.shtml 

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