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/
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