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### Multiple Solutions

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Date: 02/08/2002 at 09:27:43
From: Ryan Morris
Subject: Why?

Why is there more than one way of doing one type of math problem?
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Date: 02/08/2002 at 10:53:40
From: Doctor Ian
Subject: Re: Why?

Hi Ryan,

That's an interesting question!

In one sense, math is the search for easier ways to solve certain
kinds of problems. For example, suppose you're in a library, and you
want to know how many books it contains.

There is a simple solution: Count them, one by one. But that takes
time, and if you lose track in the middle, you have to start all over
again. Is there a better way?

There is. You can count the number of books on a few shelves, and find
the average number of books per shelf. Then you can multiply that by
the number of shelves. You lose a little accuracy (your answer won't
be _exactly_ right, but the more shelves you use in finding an
average, the closer your answer will be to the truth); but you gain
quite a lot of speed. But if there are lots of shelves, this can still
be tedious, especially if you need only an approximate answer. Is
there a _better_ way?

There is. You can estimate the volume of a single book, compute the
volume of the space taken up by the books, and divide the latter by
the former. And you can improve that solution by correcting for the
fact that some of the volume of any bookcase is left unfilled; and so
forth, and so on.

Now, when you come up with a _new_ way to solve the problem, what
happens to all the _old_ ways? Do they just disappear?

more, or if you have any other questions.

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