Thinking about Number Pattern Problems

Date: 08/09/2004 at 16:12:35
From: Teresa 
Subject: Number Pattern

How do I figure out the next 2 numbers in the pattern 1, 8, 27, 
64, ____, ____?

Using addition, the pattern doesn't add up.  If I use multiplication,
I'm stumped.  I get 1x1; 4x2; 9x3; 16x4;, but I don't understand what
the next number would be to use for ___x5 and then ____ x6.

Thanks!

Date: 08/10/2004 at 14:50:11
From: Doctor Wilko
Subject: Re: Number Pattern

Hi Teresa,

Thanks for writing to Dr. Math!

Sequence or pattern problems can be anywhere from very challenging to 
fairly easy.  This is a matter of perspective and depends on how many 
problems like this you've been exposed to.  The more you do and see, 
the less challenging these types of problems become.

I'll tell you how I approach a problem like this. 

The first thing is that I know there is a pattern, so that is my goal,
to find the pattern.

I might see if there is a pattern using addition.  

          1, 8, 27, 64, __, __, 
           \/ \/  \/  \/  \/
           +7 +19 +37  ?   ?

Hmmm, I don't see any pattern using addition.  

Next, I might try to use multiplication.  

The question is:  Is there a number that I can multiply each term by 
to get the next term?  To get from 1 to 8, there is only one number
that I can multiply 1 by to get 8 - it's 8.  But I also notice that I
can't multiply 8 by any other number to get exactly 27 (no remainder).

            1, 8, 27, 64, __, __, 
             \/ \/  \/  \/  \/
             *8  *?

So far, I don't see a pattern with using addition from term to term 
and I don't see a number that I can multiply each term by to get the 
next.

I would quickly do a similar check for subtraction and division, but I
don't find anything that works there, either. 

Now, I'm going to look at the position of each number, as sometimes 
this will reveal a pattern.  1 is in the 1st position, 8 is in the 
2nd position, ...(see below)

            1,   8,   27,   64,   __,   __, 
            |    |     |     |     |     |
            1    2     3     4     5     6  

I might ask,
 
  Does 1 have anything to do with 1?
  Does 2 have anything to do with 8?
  Does 3 have anything to do with 27?
  Does 4 have anything to do with 64?

This might go somewhere!

  1 is a divisor of 1.
  2 is a divisor of 8.
  3 is a divisor of 27.
  4 is a divisor of 64.

So maybe multiplication will work, but not in the way I tried above!  
I'm going to factor each term in the sequence to look at the divisors.

   1:    1     =      1
   8: (4 * 2)  = (2 * 2 * 2)
  27: (9 * 3)  = (3 * 3 * 3)
  64: (16 * 4) = (4 * 4 * 4)  

The 4s could be factored further, but I'm starting to see a pattern 
emerge, so I'll keep it as (4 * 4 * 4).

If this pattern continues, can you find the 5th and 6th term of the 
sequence?  (I rewrote the 1's to match the pattern.)

   1  =  (1 * 1 * 1)  
   8  =  (2 * 2 * 2)
  27  =  (3 * 3 * 3)
  64  =  (4 * 4 * 4)
   ?  =  (? * ? * ?)
   ?  =  (? * ? * ?)

Does this help?  The basic idea in number pattern problems is to keep
trying possible patterns.  As I said at the top, the more of these you
do the easier it gets to see the patterns.  As with most things,
experience and practice pays off!

Please write back if I can help any more.

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