# A dark cross-shape has been surrounded by white squares to create a bigger cross-shape. The next cross-shape is always made by surrounding the previous cross-shape with small squares.

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Introduction

## Assignment

## A dark cross-shape has been surrounded by white squares to create a bigger cross-shape. The next cross-shape is always made by surrounding the previous cross-shape with small squares. Investigate to see how many squares would be needed to make any cross-shape built up in this way.

## Introduction

## For my mathematics coursework I have been asked to do a two part investigation. Part one requires me to investigate a formula to see how many squares would be needed to make any two-dimensional cross-shape. The second part requires me to extend the investigation into three dimensions. Each time the cross shape increases in size, border squares are increased. These border squares cover the outside of the original cross shape thus creating a new, larger one. An example is shown below.

## Equipment

In order to investigate the assignment, I will make use of Lego bricks to use them to build my shapes so that I can gather all of the information for finding the formula, which is a required part of this coursework. This will allow me to model the problem and gather preliminary data.

## Two Dimensional Investigation

## Investigating the 2D formula

I am going to start by building a 1x1 cross-shape and adding on the borders. The complete shape will be known as the cross-shape (this includes the border squares). The diagram for this cross-shape and the others that I investigated are shown below. The original shape is shown shaded, with the border squares clear.

A 1x1 cross-shape

(Shape 1)

A 3x3 cross-shape

(Shape 2)

A 5x5 Cross-shape

(Shape 3)

The order of squares going from left to right are:

Shape 2: 1 + 3 + 1 = 5 (see diagram below for example)

Shape 3: 1 + 3 + 5 + 3 + 1 = 13

Middle

5x5 Cross-shape

a n 2 + b n + c

2 x 4 2 - 2 x 4 + 1

Answer: 25

##### Status: Correct

1x1 Cross-shape

a n 2 + b n + c

2 x 2 2 - 2 x 2 + 1

Answer: 5

Status: Correct

7x7 Cross-shape

a n 2 + b n + c

2 x 5 2 - 2 x 5 + 1

Answer: 41

Status: Correct

3x3 Cross-shape

a n 2 + b n + c

2 x 3 2 - 2 x 3 + 1

Answer: 13

Status: Correct

9x9 Cross-shape

a n 2 + b n + c

2 x 6 2 - 2 x 6 + 1

Answer: 61

Status: Correct

So far, all of the Cross-shapes that I have tested, have been correct. This shows that the formula works for these shapes. I can’t, however, prove that it will work for any shape, though by testing some different sized cross-shapes, I can discover how many total squares they consist of and then draw them out, to see if the formula was correct. This will still not prove that it works for any number, however, by extending my two dimensional investigation, I will whether is more reliable. This will go further to proving that the investigation works with any sized shape, though will not make it certain.

Below are two other cross shapes. They are supposed to be 85 and 113 squares. I will now draw them out to test that this is correct.

#### An 11x11 Cross-shape

a n 2 + b n + c

2 x 7 2 - 2 x 7 + 1

#### Answer: 85

Counted Answer: 85

Status: Both Correct

A 13x13 Cross-shape

a n 2 + b n + c

2 x 8 2 - 2 x 8 + 1

Formula Answer: 113

Counted Answer: 113

Status: Both Correct

## Summary

There is no way to make sure that the formula will work for every sized shape but I tried to make it more reliable by showing that it works with all the ones tested and two other cross-shapes that I had not investigated before, which were all correct.

The only way to be certain that the formula works with every sized cross shape is by testing it on every possible sized one, which would be very difficult. This is why I feel that I have generated enough information to show that the formula that I have found out, 2 n 2 - 2 n + 1, gives the expected results.

## ( 3D Investigation on next page )Three Dimensional Investigation

## Introduction

I will now investigate to find the three dimensional formula. I am going to start by building a 1x1 cross-shape and adding on the borders. The complete shape will be known as the cross-shape (this includes the border squares). I again used Lego bricks to make the cross-shapes. The larger diagrams for the cross-shapes in 3D are too complex to see all the squares on. The cross-shape below is 1x1 with border squares around it and you can see all but the middle square.

## Obtaining Results

I obtained my results by building this three dimensional cross-shape up as in the two dimensional investigation. The three-dimensional cross-shapes are more complex than the two-dimensional cross-shapes, due to the 3D ones having another dimensional to draw.

I used Lego bricks again, with two different colours for the border squares and the darker squares. These helped me to gather my data. This is because they give a visualisation of the problem. This is easier to understand than trying to work it out in my head.

## Analysing Results

I think that investigating five different cross-shapes will give me enough information to find a formula. This is because the information I gathered will give me enough data to analyse and utilise my results. The data I have found out is shown in the table below. I have done one less shape size in this investigation because the 3D shapes required more bricks than I had. So I had to stop at a 7x7 shape size.

Shape Sizes | Border Squares | Darker Squares | Total Squares |

Single Sq. | 0 | 1 | 1 |

1 x 1 | 6 | 1 | 7 |

3 x 3 | 18 | 7 | 25 |

5 x 5 | 38 | 25 | 63 |

7 x 7 | 66 | 63 | 129 |

Conclusion

For the third sequence, I will use the same formula, 2n-1. This time n will be the value of 3. So the formula will become 2(3)-1 which equals 5. The diagram over shows what the value is.

As the diagram shows, there are five layers. The formula 2n-1 is so far successful. I will try one more just to check the formula. I will try a 5x5 cross-shape, there should be 2n-1 layers. So by substituting the value of n for 4 (the term in the sequence is 4), the formula now reads 2(4)-1, which equals 7. So I predict that there will be 7 layers to the 5x5 cross-shape. The diagram for this shape is shown below.

This is also give the expected answer.

## Summary

Due to there being too many variables in the equationa n 3 + b n 2 + cn + d I was unable to resolve the problem in this manor. Instead I decided to use a pattern I noticed while investigating the Lego brick models. I think the new method, which I have developed, could be simplified with further development to give a much more concise formula.

There is no way to make sure that the formula will work for every sized shape but I tried to make it more reliable by showing that it works with the ones I built with Lego bricks and one other cross-shapes that I had not investigated before, which were all correct.

The only way to be certain that the formula works with every sized cross shape is by testing it on every possible sized one, which would be very difficult. This is why I feel that I have generated enough information to show that the formula that I have found out gives the expected results.

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