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### Guess and Check

```Date: 06/17/2002 at 18:12:40
From: Johnathan Minix
Subject: Problem Solving Straegy: Guess and Check

This is an example of a question

1. Sum of two numbers = 15
Difference of the numbers = 3
Find the numbers.
What is the product?

Thank you
```

```Date: 06/17/2002 at 21:08:28
From: Doctor Peterson
Subject: Re: Problem Solving Straegy: Guess and Check

Hi, Johnathan.

I'll demonstrate some ideas using a slightly different problem:

sum = 31
difference = 7

One "guess-and-check" strategy is just to try lots of numbers
randomly, and see WHETHER each pair works. That's extremely
inefficient.

A better strategy is to try a pair, and see HOW WELL it works,
getting an idea for a better guess. For example, if you try 16+15 (a
natural first choice, since it's in the middle), the difference is
only 1. So you know you have to try numbers that are farther apart,
so you might try 8+23 or something. The difference now is 15, which
is too big, so you have to try a pair that are closer. And so on...

Often you can do better than that, and actually use the error in the
HOW the problem works. In this case, you need to increase the
differe nce from 1 to 7, an increase of 6. What happens if you change
the numbers you use by 1? By adding 1 to the larger number and
subtracting 1 from the smaller number, you keep the sum the same, but
increase the difference by 2. Since we have to increase the
difference by 6, we need to do this 3 times. So we add 3 to 16 and
subtract 3 from 15, giving 19+12=31 as our sum, and 19-12 = 7 as our
difference. We've got it!

Maybe we can call these three strategies "guess and check", "guess
and improve", and "guess and solve". Which you use depends on how
hard the problem is, how much you feel like thinking, and how much
time you want to take.

If you need more help, please write back and show me how far you got.

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