Finding the Operation

Date: 06/21/99 at 22:53:08
From: don yehling
Subject: Series problem

This is from Max Sobel's _Teaching Mathematics_ (page 28.) (I have 
substituted ? for * to avoid confusion with multiplication.) The 
object is to find the operation '?'.

Given:

     3 ? 4 ... 5
     4 ? 7 ... 1
     8 ? 4 ... 0
     1 ? 2 ... 9

Then he asks:

     5 ? 5 ... 
     4 ? 1 ... 
     6 ? 2 ... 

I find that the first three are 32 minus the product of the two 
numbers divided by four, but my solution breaks down with the fourth 
pair of numbers. Sobel adds that this problem came from a third grade 
math text, so you have to use operations a third grade student would 
be familiar with. I'm stumped.

Date: 06/22/99 at 02:34:30
From: Doctor Pete
Subject: Re: Series problem

Hi,

This is an interesting problem. No doubt you noticed that the largest 
pair, 8?4, gives the least result, 0, and the smallest pair, 1?2, 
gives the greatest result, 9. Clearly we are dealing with a function 
where subtraction is involved. The next hint you provided was that 
this is a "third grade" level problem, so the key here is not to look 
too hard. So let us not consider division. A bit of computation shows

     3 + 4 = 7           3 * 4 = 12
     4 + 7 = 11          4 * 7 = 28
     8 + 4 = 12          8 * 4 = 32
     1 + 2 = 3           1 * 2 = 2

Clearly these are good places to start because the operations of 
addition and multiplication preserve the property that the larger the 
given pair, the larger the result. But we note that in the latter 
case, were we to subtract these values from a given number, the 
differences between the results would be too large. On the other hand, 
a bit of thought shows that

     12 - 7  = 5
     12 - 11 = 1
     12 - 12 = 0
     12 - 3  = 9

And so we have found the rule: Take the two given numbers, add them, 
and subtract the sum from 12. Algebraically speaking, the binary 
operation x ? y is given by the function

     x ? y = f(x,y) = 12 - (x+y).

It is now straightforward to find

     5?5 = 12-(5+5) = 2,
     4?1 = 12-(4+1) = 7,
     6?2 = 12-(6+2) = 4.

Now, I say this problem is interesting, because it demonstrates an 
important method of thinking: that of understanding how certain 
familiar operations (addition, subtraction, etc.) can be thought of as 
actions on "inputs," and furthermore, how the result depends on the 
sizes of the inputs for a given operation. It also introduces on an 
intuitive level algebraic concepts, where there is some sort of 
"reverse" thinking involved.

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