Finding a Sequence of Numbers

Date: 08/13/2002 at 18:28:28
From: Kirsti
Subject: Algebra

There are six digits in a sequence: two 4's, two 5's, and two 6's.  
There is one digit between the two fours; there are two digits between
the two fives; and there are three digits between the two sixes.  
Write this sequence of numbers.  

My problem is I can't figure the answer to this question; that's why 
I came here.

I have tried many things to figure it out and I have looked in books 
from the library, I have asked my teacher, I have asked my parent's, 
and I have asked my friends. Please help me. 

Thank You!

Date: 08/13/2002 at 21:22:09
From: Doctor Ian
Subject: Re: Algebra

Hi Kirsti,

It's important to attack a problem like this systematically, so that 
you don't end up trying the same answers over and over again, like 
someone wandering lost in the woods. 

The sequence will look like this:

  _ _ _ _ _ _

There is one digit between the two 4's, so that means the following 
are the only possibilities:

  4 _ 4 _ _ _

  _ 4 _ 4 _ _

  _ _ 4 _ 4 _

  _ _ _ 4 _ 4

Now, there are two digits between the two 5's. Let's look at each of 
our possibilities, and see which ones can be extended to take this 
second condition into account. In each case, there is only one 
possibility:

  4 _ 4 _ _ _   =>  4 5 4 _ 5 _

  _ 4 _ 4 _ _   =>  _ 4 5 4 _ 5

  _ _ 4 _ 4 _   =>  5 _ 4 5 4 _

  _ _ _ 4 _ 4   =>  _ 5 _ 4 5 4

In which of these sequences can you insert a pair of 6's that will be 
separated by three digits? 

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