Sugar Cubes and Coffee Cups

Date: 09/26/2002 at 02:11:42
From: Robert Kim
Subject: Coffee cup question

Hello Dr. Math,

In Introduction to Calculus, my teacher asked us an interesting 
question and asked us to find a solution.  He said that it is vital 
that we should know the reason behind the solution.

If there are 20 blocks of sugar and only 3 cups of coffee, how 
many blocks of sugar must be put into each cup of coffee if there 
have to be a total of an odd number of blocks in each cup?

(Note that the blocks of sugar cannot be broken up into smaller 
ones.)

If you can give me a solution to this question I will be very 
grateful.

Thanks a lot,
Robert

Date: 09/26/2002 at 09:46:22
From: Doctor Ian
Subject: Re: Coffee cup question

Hi Robert,

In problems like this, it's crucial to make sure that you have the 
wording correct, so that you can identify all your assumptions. 

For example, let's state a couple of assumptions that seem 
reasonable:

  1) Every cube must go into exactly one cup.

  2) Every cup is distinct from every other cup (e.g., 
     you can't have one cup inside another). 

Now, suppose we put A cubes into the first cup, B cubes into the 
second cup, and C cubes into the third cup. If each of these numbers 
is odd, then the following condition must hold:

  (2A + 1) + (2B + 1) + (2C + 1) = 20

                2(A + B + C) + 3 = 20

                    2(A + B + C) = 17

Which is to say, UNDER THESE ASSUMPTIONS, the problem has no solution.  

Which ISN'T to say that these assumptions must hold! For example, if 
we relax assumption (1), we can solve the problem by putting 1 cube 
into each cup, and leaving 17 of them on the table. 

Or if we relax assumption (2), then we can do something like this:

  |  |      |  |     | c c c   |
  |  |      |  |     | c c c c |
  |  |   c  |  |     | c c c c |
  |  +------+  |     | c c c c |
  |            |     | c c c c |
  +------------+     +---------+

Now each of the cups on the left has an odd number (1) of cubes, and 
the cup on the right has an odd number (19) of cubes. 

As Ayn Rand used to say, if you think you've found a contradiction, 
examine your asssumptions. One of them will be wrong.  

Does this help? 

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

Date: 04/08/2003 at 12:34:29
From: Peter Calvert
Subject: Re: Coffee cup question

Dividing 20 cubes into three groups each with an odd number of cubes 
in the group has a simple solution

Place 8 cubes in the first cup in a 2x2x2 arrangement - you can now 
see it as 1 cube or 9. The remaining twelve can be split 1/11, 3/9, 
or 5/7. This problem can work with as few as 10 cubes.

Date: 04/08/2003 at 15:20:25
From: Doctor Douglas
Subject: Re: Coffee cup question

Hi, Peter,

thanks for writing!

Your solution is very neat, because it doesn't invoke any other
information that hasn't been given - such as the "normal" number of
cubes to put in coffee.

Well done!

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