Finding the Next Number in a Series

Date: 07/22/2002 at 20:31:52
From: Kim Reger
Subject: Finding the Next number in a series

Hi there,

I haven't been able to find much information about an approach or 
method in determining the "next" number is a given series of numbers,
e.g., 

  9, 5, 45, 8, 6, 48, 6, 7...   What is the next number?

I can usually figure it out but if there is a formal way that makes 
it easier I would love to know about it!

Thanks,

Kim Reger

Date: 07/22/2002 at 23:22:09
From: Doctor Peterson
Subject: Re: Finding the Next number in a series

Hi, Kim.

Mathematically speaking, problems like this are impossible. Literally!

That's because there is no restriction on what might come next in a 
sequence; ANY list of numbers, chosen for no reason at all, forms a 
sequence. So the next number can be anything.

A question like this is really not a math question, but a psychology
question with a bit of math involved. You are not looking for THE
sequence that starts this way, but for the one the asker is MOST
LIKELY to have chosen - the most likely one that has a particularly
simple RULE. And there is no mathematical definition for that.

If you just wanted _a_ sequence that starts this way, but can be 
defined by _some_ mathematical rule, there is a technique that lets 
you find an answer without guessing. This is called "the method of 
finite differences," and you can find it by searching our site (using 
the search form at the bottom of most pages) for the phrase. It 
assumes (as is always possible) that the sequence you want is defined 
by a polynomial, and finds it. Sometimes this is what the problem is 
really asking for.

But often, especially when many terms are given, there is a much 
simpler rule that is not of polynomial form. Then you are being asked 
to use your creativity to find a nice rule. Sometimes starting with 
finite differences gives you a good clue, even if you don't end up 
with a polynomial; just seeing a pattern in the differences can 
reveal something about the sequence. Other times it is helpful to 
factor the numbers, or to look at successive ratios. Here you are 
doing a more or less orderly search, in order to find something that 
may not turn out to be orderly.

Some puzzles like this are really just tricks. The "rule" may be that 
the numbers are in alphabetical order, or that each number somehow 
"describes" the one before, or even that they are successive digits 
of pi. In such cases, you have to ignore all thoughts of rules and 
orderly solutions, and just let your mind wander. This is sometimes 
called "lateral thinking," and it's entirely incompatible with 
"formal methods"!

I first assumed the specific sequence you gave was just a random list 
of numbers, rather than a real problem, so I shouldn't bother looking 
for a pattern. But glancing at it, I see that it is not random:

    9, 5, 45, 8, 6, 48, 6, 7, ... 

I see some multiplications here:

    9 * 5 = 45, 8 * 6 = 48, 6 * 7 = __

I can't recall what chain of reasoning my mind went through to see 
that, but it may have helped that my kids asked me to go through a 
set of multiplication flash cards an hour or two ago. And focusing on 
the few larger numbers, thinking about how large numbers might pop up 
(multiplication makes bigger changes than addition), probably led me 
in the right direction. I don't recall seeing anything quite like 
this presented as a sequence problem before, but seeing factors, one 
of my usual techniques, was the key.

In this case, you were apparently just asked to find the NEXT number, 
so we're done as soon as you fill in my blank. It may well be that 
there is no pattern beyond that; the choice of 9, 5, 8, 6, 6, 7 may 
be random. That's a good reminder that we have to read the problem 
carefully and not try to solve more than we were asked. We weren't 
told that there was any pattern beyond the next number!

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