extra credit problem

[taytortot725]

Ok, this is for my trig class.

Al and Betty arehard at work laying paving stones for a path along their house. The path is to be exactly two feet wide, and each paving stone is rectangular with dimensions of 2 feet by one foot.

You might think this is easy: simply lay one stone after another across the path. But there is more than one way to lay the stones.

So of course, al and betty want to know how many different ways there are to lay out the stones. The path is 20 feet long altogether. Al and Betty start to analyze the problem using 1 by 2 inch tile set within a 2 by 20 inch rectangle, but are overwhelmed with possibilities.

Can you help? You might want to start with shorter paths and look for patterns in the number of cases.

You do not need to show the patterns except to explain your thinking.

 

[mmm4444bot]

Perhaps Betty and Al are overwhelmed because they don't know how to organize the information they gathered from whatever analysis they completed before quitting.

There are only two ways in which to lay a stone; either its two-foot side spans the width of the path or it's rotated 90 degrees.

Let W represent a stone laid with its two-foot dimension spanning the width of the path. Each W contributes one foot to the 20-foot length of the path.

Let L represent two adjacent stones laid to form a two-foot by two-foot square. Each L contributes two feet to the 20-foot length of the path.

Now we can type different arrangements of W and L from left to right that form a 20-foot path.

WWWWWWWWWWWWWWWWWWWW -- There is only one way to lay down this arrangement.

LLLLLLLLL -- There is only one way to lay down this arrangement.

LWLWLWLWLW -- This arrangement is 18 feet long. We need to add another L somewhere along the line. There are 13 different locations in which to place this remaining L.

This is how I'm interpreting the exercise. Soroban's interpretation (below) is different.

Next, we realize that the different arrangements can be categorized in terms of how many Ls and Ws they each contain.

10 Ls requires 0 Ws along the way in order to add up to 20 feet.
9Ls require 2 Ws along the way in order to add up to 20 feet.
8Ls require 4 Ws along the way.
7Ls require 6 Ws along the way.
6Ls require 8Ws.
5Ls require 10 Ws.
And so on, all the way to 0 Ls require 20Ws.

In each of these catagories, how many different ways are there to arrange the Ls and Ws?

For example, in the category with 1 L and 18 Ws, there are 19 different arrangements because the L can go in one of 19 different locations.

In the category with 2 Ls and 16 Ws, there are 17 locations for the first L, and for each of those placements, there are 17 locations for the second L that do not result in non-distinct arrangements (i.e., no duplicates). That's 17 times 17 equals 289 different arrangements in that category.

Complete a count for each category, and sum the results.

Another question which has the same answer is: "How many different ways can you arrange the digits 1 and 2 from left to right such that all digits sum to 20?"

Soroban's interpretation yields 1,024 arrangements; mine yields closer to 2,000,000.

 

[soroban]

Hello, taytortot725!

Two pavers will form a 2-by-2 square
. . and there are two possible orientations for these squares.

       Horizontal            Vertical

      * - - - - *         * - - * - - *
      |         |         |     |     |
      |         |         |     |     |
      * - - - - *         |     |     |
      |         |         |     |     |
      |         |         |     |     |
      * - - - - *         * - - * - - *

Each pair of pavers will be placed "horizontally" or "vertically."

There are ten pairs of pavers and each has two choices of orientiation.

\(\displaystyle \text{Therefore, there are: }\:2^{10} \,=\,1024\text{ possible arrangements.}\)