Building a Brick Wall

For more information refer to: Fibonacci Numbers and Brick Wall Patterns by Ron Knott

As you consider the different ways to make a wall with a certain number of bricks, look for a pattern or a system to predict what will happen with any number of bricks.

Brick wall made with one brick

Since the wall must be 2 units high, there is only one possibility.

Brick wall made with two bricks

There are two possibilities if the wall can be made with 2 bricks. A height of 2 units can be made either using two horizontal bricks or two vertical bricks.

Brick wall made with three bricks

There are three possibilities if the wall can be made with 3 bricks. Now the pattern begins. If you take the two possibilities that could be made using 2 bricks and add a vertical brick in front of both of them you will have the first two figures to the left below. Then look at the brick wall that could be made with 1 brick (2 before the one you are currently working on). Put two horizontal bricks in front of it and that will give you the third possibility.

Brick wall made with four bricks

There are five possibilities if the wall can be made with 4 bricks. If you take the three possibilities that could be made using 3 bricks and add a vertical brick in front of each of them you will have the first three figures. Then look at the brick walls that could be made with 2 bricks (2 before the one you are currently working on). Put two horizontal bricks in front of them and that will give you the fourth and fifth possibilities.

Brick wall made with five bricks

There are eight possibilities if the wall can be made with 5 bricks. If you take the five possibilities that could be made using 4 bricks and add a vertical brick in front of each of them you will have the first five figures. Then look at the brick walls that could be made with 3 bricks (2 before the one you are currently working on). Put two horizontal bricks in front of them and that will give you the sixth, seventh and eighth possibilities.

Brick wall made with n number of bricks

If you look at the pattern that emerges if you are finding the number of configurations for any number, n, bricks:

you add 1 vertical brick to all of the possibilities listed under (n-1)
you add 2 horizontal bricks to all of the posibilities listed under (n-2).
This can be stated as a function: F(n) = F(n-1) + F(n-2). This is the same function that generates the Fibonacci numbers.