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### Finding the Operation

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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

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.
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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/
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