Find the Pattern, Complete the Combinations

Date: 01/13/2002 at 20:52:14
From: Tracy Payne
Subject: Patterns

I got this question from a fellow teacher. We cannot solve it. Can you 
help?

Find the pattern: 

   3*4->5, 8*4->0, 3*7->2, 1*2->9.  

Use the pattern to complete the combinations.  

   5*5->?, 4*4->?, 5*7->?

This is the way it is presented. The -> are arrows and I am assuming
that the * is just some operation of some sort. I have tried using
numbers in circles, different combinations of operations, but cannot
figure it out. Thanks.

Date: 01/14/2002 at 10:04:28
From: Doctor Peterson
Subject: Re: Patterns

Hi, Tracy.

Let's write the data as a table:

    3*4 = 5
    8*4 = 0
    3*7 = 2
    1*2 = 9

Now, without any idea what sort of operation "*" might mean, it really 
could be anything at all. We have to assume it's something very 
simple, and look for something that fits. One kind of simple operation 
would be a linear one, that is, one for which a constant change in one 
argument produces a constant change in the result. Let's look for 
clues to such a rule.

Compare 3*4 and 8*4; when you increase the first argument by 5, the 
result decreases by 5. That suggests that we are subtracting the first 
argument from something.

Now compare 3*4 and 3*7; when you increase the second argument by 3, 
you decrease the result by 3. So we seem to be subtracting the second 
argument from something.

See if you can use those clues to find a definition of "*" that fits 
all four given facts; then use your definition to find the answers.

If we didn't have such nice sample facts, we would have had to define 
"*" as the linear operation

    x*y = ax + by + c

and use the data to write a system of equations to solve to find a, b, 
and c. If there was no solution, we would have to try another kind of 
operation, since we would have shown it is not linear. As it is, it 
doesn't take any algebra to find an answer.

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