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### Multi-Step Patterns

```Date: 09/04/2003 at 01:17:51
From: Mrs. C

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/
```
Associated Topics:
Elementary Puzzles
Middle School Puzzles

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