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 
first guess to find the correct answer directly, by thinking about 
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/