# Chessboard coursework

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Introduction

How many squares on a chess board

Introduction

This is my first piece of mathematics coursework. In this project I will be finding out how many squares on a 8x8 chess board. In this project I will be including labelled drawings, clear enough to allow a non mathematician to follow, clearly written work, initial drawings of each of my steps and a finished formula justified. Finally an extension on my board making it rectangular.

Plan

- Introduction

- Spider diagram

- I will be collecting data from different sized grids ranging from 2x2 to 8x8.

- Explanation of how to add up the squares on the grid

- Further calculations

- Create a table of values.

- Formula and formula justified

- Draw a graph

- Extension

- Conclusion

- Evaluation

Tree diagram

Explanation of how to add up the squares on the grid

- Count how many singular squares there are

=16

- Count how many double squares there are.

=9

Middle

+53 +2

7 140 +15

+62

8 204

Finding the formula

The cubic formula of type

3 2

Y= Ax + Bx + Cx + D

(x+1)

3 2

Y= Ax1 + Bx1 + Cx1 + D=1

A+B+C+D=4 1

(x+2)

3 2

Y= Ax2 + Bx2 + Cx2 + D=5

8A+4B+2C+D=5 2

(x+3)

3 2

Y= Ax3 + Bx3 + Cx3 + D=14

27A+9B+3C+D =14 3

(x+4)

3 2

Y= Ax4 + Bx4 + Cx4 + D=30

64A+16B+4C=D=30 4

The four equations to be used are:

1. A+B+C+D=1

2. 8A+4B+2C+D=5

3. 27A+9B+3C+D=14

4. 64A+16B+4C=D=30

4-3= 37A+7B+C=16 5

3-2= 19A+5B+C=9 6

2-1= 7A+3B+C=4 7

5-6= 18A+2B=7 8

6-7= 12A+2B=5 9

8-9= 6A=2=0.33’=1

3

Sub A= 1 in 9

3

1 A+2B=5

3

(12x1)+2B=5

3

4+2B=5

2B=5-4

2B=1

B= 1

2

Sub A and D in 7

7A+3B+C=4

(7) (1) + (3) (1) + C=4

3 2

7+3+C=4

3 2

C= 4-7-3

1 3 2

C= 4x6– 7x2 – 3x3

1 3 2

C= 24 – 14 – 9

6 6 6

C= 24-14-9

6

C= 1

6

Conclusion

7A+3B+C=6

7x 1 + 1x3+C=6

3

2 1 A+ 3B+C=6

3

6-5= 2

3

C=2

3

Sub A, B & c in 1

A+B+C+D=2

1 + 1 +2 + D =2

3 3

D=0

Conclusion

The results of my investigation has lead me to believe the following conclusion. As the size of the grid 2x2 increases to 8x8, so does the number of squares. Using my algebraic equation the formula was obtained. When tested against a known number it seemed to be working satisfactory.

Evaluation

At the beginning of my project the work seemed to be very simple but as the project progressed I found it got harder and harder. If I hade more time to do my project I would have continued on my extension and maybe, tried to work out how many squares in a triangle, and a formula for working that out. In this project I have gained much more knowledge about numbers, shapes and creating formulas.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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