# Investigate Borders - a fencing problem.

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Introduction

Borders

Borders

Aim

My aim is to investigate Borders. I will be drawing borders to different squares and finding a formula for each one and finally I will find a Universal Formula.

Introduction

This is what the task tells me:

Here are 2 squares with squares added on each side to make a border, which surrounds the starting squares.

You can then add another border as shown:

Investigate Borders.

Method

First I will find out how many squares needed for the border to a 1x1 square, then 2x1 and so on up to 5x1. Then I will find a formula for the border to each square and also test the formula out to prove that it works. I will predict how many squares needed for the 6th border and find out if my prediction was correct.

Next, I will find out how many squares needed for the border of a 1x2 square, then 2x2, and so on up to 5x2, then 1x3, 2x3, and so on up to 5x3. Again I will find formulas to them and prove that all the formulas work.

Finally, I will put all the formulas together in three groups and find one overall formula for each of the groups. Then, I will get those three formulas and get one Universal formula in the end.

Diagram of Borders of square: 1x1

Table of results for Borders of square: 1x1

Formula

You can always find ‘the nth term’ using the Formula:

‘a’ is simply the value of

Middle

nth term = 4 x 6 + 8 = 32

Common Difference nth Term

Results

My prediction was 32 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order.

Formula to find the number of squares needed for each border (for square 1x2):

Formula = Simplification =

I have already found out the Formula for 1x2 so there’s no need for the Diagrams & Tables, and I have already proved that the Formula works.

Diagram of Borders of square: 2x2

Table of results for Borders of square: 2x2

Formula to find the number of squares needed for each border (for square 2x2):

Common difference = 4

First term = 8

Formula = Simplification =

Experiment

I will try to find the number of squares needed for border number 6 using the formula, I found out, above:

nth term = 4 x 6 + 4 = 28

Common Difference nth Term

Results

My prediction was 28 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order.

Diagram of Borders of square: 3x2

Table of results for Borders of square: 3x2

Formula to find the number of squares needed for each border (for square 3x2):

Common difference = 4

First term = 10

Formula = Simplification =

Experiment

I will try to find the number of squares needed for border number 6 using the formula, I found out, above:

nth term = 4 x 6 + 6 = 30

Common Difference nth Term

Results

Conclusion

5x3 = 4n + 12

Formula for nx3th term:

4 y + ( 2 n + 2 )

e.g. for border number 6:

First replace ‘n’ with 6.

4 y + ( 2 x 6 + 2) = 4 y + 14

Now you can replace the ‘y’ with an ‘n’ to have the formula, to find the number of squares, for border number 6.

4 n + 14

The Universal formula:

The formula for Length x Width = 4 n + 2 L + 2 W – 4 = B (Border)

e.g. for the 6th border of a 5x3 rectangle:

First replace ‘n’ with the border number = 6, ‘L’ with the Length = 5, and ‘W’ with the Width = 3. Then add brackets where necessary.

( 4 x 6 ) + ( 2 x 5 ) + ( 2 x 3) – 4 = B

Then multiply out the brackets:

24 + 10 + 6 – 4 = 36

36 is the correct answer.

Formulas for nx1

1x1 = 4n

2x1 = 4n + 2

3x1 = 4n + 4

4x1 = 4n + 6

5x1 = 4n + 8

Formulas for nx2

1x1 = 4n + 2

2x1 = 4n + 4

3x1 = 4n + 6

4x1 = 4n + 8

5x1 = 4n + 10

Formulas for nx3

1x1 = 4n + 4

2x1 = 4n + 6

3x1 = 4n + 8

4x1 = 4n + 10

5x1 = 4n + 12

Conclusion

In the time available to me, I believe I have researched Borders to the full extent of my ability. I found formulas to squares nx1. I then extended this to squares nx2 and nx3, and I then was able to construct my Universal Formula, which will tell you the number of squares in any border of square nxn, which could be anything from 2x2 to 10x15. I also found that many of my predictions I made along the way turned out to be correct.

I would say that this investigation has been a success. I began this investigation with the aim to find formulas to nx1, nx2 & nx3 and then a Universal Formula and they were achieved.

Created by Syed Islam, 11Q

Syed Islam, 11Q, 3819 – Mr. Abadji

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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