Finding Input-Output Rules

Date: 10/08/2003 at 09:49:20
From: Tammy and Jamie
Subject: Number Patterns

Hello,

I am the parent of a 3rd grader who is having trouble in math.  This
week he's working on Patterns in Numbers.  For example, 

  In  12 20 35
  Out 15 23 38 41 57 62

How can he explain the "changing" from in numbers to out numbers?  He
said he added 15 each time.  Of course all were marked wrong.  Please
help us.

Thanks,
Tammy and Jamie

Date: 10/08/2003 at 09:57:32
From: Doctor Ian
Subject: Re: Number Patterns

Hi,

The idea in a problem like this is to make up a rule that you can
apply to a particular input number to get the corresponding output number.
  
If you have the rule, making up the lists is pretty straightforward. 
For example, suppose my rule is 'multiply by 2', and my input list is

  1, 3, 7, 12

To make the output list, I apply the rule to each input:

  Input  ->  Apply rule  =  Output

     1          2 *  1         2
     3          2 *  3         6
     7          2 *  7        14
    12          2 * 12        24

So my input and output lists look like this:

    In   1,  3,  7, 12
    Out  2,  6, 14, 24

Does this make sense so far?  What's less straightforward is to go in
the other direction, i.e., to start with the two lists, and figure out
what the rule is.  

Let's think about how we might do that.  Suppose we're given the lists

    In    2,  4,  6,  9
    Out   4, 16, 36, 81

The simplest kind of rule will look like

   add ___

Here's an example of that:

    In   4, 11, 15, 19
    Out  6, 13, 17, 21

How can we find the rule?  Try subtracting every input from its output:

    In        4, 11, 15, 19
    Out       7, 14, 18, 22
    Out - In  3,  3,  3,  3

If the difference is always the same, we know we have a rule of this
type.  Do you see why?  Another simple rule would be the one I
mentioned earlier:

   multiply by ___

In a case like that, we can find the rule by dividing the outputs by
the inputs:

    In        1,  3,  7, 12
    Out       2,  6, 14, 24
    Out / In  2,  2,  2,  2

Let's try these with our mystery list:

    In         2,  4,  6,  9
    Out        4, 16, 36, 81
    Out - In   2,  8, 30, 72          <- Not a constant difference
    Out / In   2,  4,  6,  9          <- Not a constant ratio

So neither of these rules will work. 

However, sometimes trying out simple rules can help you find a more
complicated one!  In this case, let's look only at the ratios:

    In         2,  4,  6,  9   <-----
    Out        4, 16, 36, 81         |  These are the same!
    Out / In   2,  4,  6,  9   <-----

What this is telling us is that if we divide the output by the input,
we get the input again.  That is, 

   Out / In = In

which means that 

        Out = In * In

In other words, the rule we're looking for is 'multiply the input by
itself'.  

Finding these rules is more of an art than a science, and (as with
most things) the more experience you get with them, the easier they
become.  There are particular techniques for dealing with certain
kinds of patterns, but when you don't know what else to do, it's good
to start with simple rules, like the ones I've discussed here.  Often
they'll work, but even when they don't, the results you get from
trying them can give you ideas about other rules to try. 

I hope this helps!  Write back if you'd like to talk more about this,
or anything else. 

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