Multi-Step Patterns

Date: 09/04/2003 at 01:17:51
From: Mrs. C
Subject: About patterns 

My daughter has been working using tables to find patterns.  Here is
an example problem. 

Find the rule that will give the second number if you know the first. 
Then fill in the rest of the table according to the rule.
  
   2/7
   3/10
   5/16
   8/?
  22/?
   ?/22

I thought: 2+5=7, and 3+7=10, but I don't know where to go from there. 

Date: 09/04/2003 at 12:37:15
From: Doctor Peterson
Subject: Re: About patterns for my ll year old.

Hi, Mrs. C.

This kind of problem is really a tricky puzzle that can only be 
solved by guesswork and intuition. You don't have to feel bad about 
not being able to do it; and I hope it is not assigned as if everyone 
should be able to figure it out!

You are probably looking for one possible type of pattern, namely 
that the same number is always added. The actual pattern is a little 
different. One way to discover it is to look at the differences 
between numbers:

  x      y
  -     --
  2 -->  7
  3 --> 10    increasing by 1 increased result by 3
  5 --> 16    increasing by 2 increased result by 6

The result seems to increase by 3 times as much as the number you put 
in; so there must be multiplication by 3 going on somewhere. Suppose 
we assume that the first step is multiplying by 3, and then compare 3 
times the input with the output:

  x     3x      y
  -     --     --
  2 -->  6 -->  7
  3 -->  9 --> 10
  5 --> 15 --> 16

Can you see how 3 times x and y are related? This means that the 
pattern takes two steps, first multiplying and then adding. That's 
not easy to see if you haven't seen a lot of these, as I have. (If 
your daughter has seen enough of them, and knows what kind of pattern 
to expect, then it makes sense to assign this problem; but the 
teacher needs to make the connection clear, and not just drop this 
puzzle on them out of thin air.)

There's some more work left to do, but this solves the hardest part 
of the problem.

If you have any further questions, feel free to write back.

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