Laying a Brick Walkway

Date: 04/22/2002 at 09:25:38
From: Dawn
Subject: Math Word Problem

I have to build a walkway, 2ft. by 20ft., out of bricks that are 1ft. 
by 2ft.  The bricks can lie vertically and horizontally but in no 
other direction.

How many different ways can I build this walkway?

I know there has to be a pattern, but I do not see it.  Any help would 
be great.

Dawn

Date: 04/22/2002 at 12:30:51
From: Doctor Peterson
Subject: Re: Math Word Problem

Hi, Dawn.

When I don't see how to solve a problem like this, I start by playing 
with the ideas involved in the problem. That usually means 
simplifying the problem by working with smaller, easier numbers - 
actually, that's more or less what children's play is, isn't it? By 
trying things out in a smaller case, I get a feel for how it works. 
And in this case, that leads directly to a solution.

How many ways can you make a 2-by-1 walkway (silly as it sounds!)? 
One, of course:

    |
    |

How about a 2-by-2 walkway? Two ways:

    | |    ---
    | |    ---

How about 2-by-3? You can build on what you have already done: if the 
first brick you place is vertical (across the walk), you have 2 more 
feet to fill in, using one of the solutions for the 2-by-2. What 
happens if the first brick is horizontal (lengthwise)? See if you can 
use this to identify a pattern.

You will recognize a relationship to the Fibonacci sequence, which 
you can read about in the Dr. Math FAQ:

   Golden Ratio, Fibonacci Sequence
   http://mathforum.org/dr.math/faq/faq.golden.ratio.html 

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