# Number Grids.

Extracts from this document...

Introduction

Kathleen Smith 4449 Page 5/2/2007

Maths Coursework: Boots

Introduction:

For my coursework I will be investigating patterns in grids of different sizes. Within the grid each square has a number.

E.g.

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 |

For each boot (highlighted above) I will develop a formula to work out the total value of the numbers added up. The boot will consist of three numbers up and two across. My formula must show how to find out the added value of the boot if placed anywhere within the grid.

## Grid 5x5

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 |

I found that in grid 5x5 that the formula for the added value of the boot is 4n+26. I discovered this by:

Working out the value of boots 1 to 4.

Boot 1= 1+6+11+12=30

Boot 2= 2+7+12+13=34

Boot 3= 3+8+13+14=38

Boot 4=4+9+14+15=42

Now I have this information I can begin to work on the sequence.

30 34 38 42

4 4 4

n | 1 | 2 | 3 | 4 |

4n | 4 | 8 | 12 | 16 |

Sequence | 30 | 34 | 38 | 42 |

30-4=2634-8=2638-12=2642-16=26

These all-equal 26!

This shows that the formula is 4n+26=v

Middle

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

16

17

18

19

20

n+6

22

23

24

25

26

n+12

n+13

29

30

31

32

33

34

35

36

If I add these together n+n+6+n+12+n+13 it equals 4n+31.

This proves my formula for grid 6x6 is correct.

## Grid 7x7

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

I found that in grid 7x7 that the formula for the added value of the boot is 4n+36. I discovered this by:

Working out the value of boots 1 to 4.

Boot 1= 1+8+15+16=40

Boot 2= 2+9+16+17=44

Boot 3= 3+10+17+18=48

Boot 3= 4+11+18+19=52

Now I have this information I can begin to work on the sequence.

40 44 48 52

4 4 4

n | 1 | 2 | 3 | 4 |

4n | 4 | 8 | 12 | 16 |

Sequence | 40 | 44 | 48 | 52 |

40-4=3644-8=3648-12=3652-16=36

These all-equal 36!

This shows that the formula is 4n+36=v

n=smallest number in the boot.

v=total value of the boot.

To check my formula I will prove on another boot in the grid.

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

n = 12

4n = 4x12 = 48

60+36=84

12+19+26+27=84

To prove that 4n+36 is correct is to test it using n.

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | n | 13 | 14 |

15 | 16 | 17 | 18 | n+7 | 20 | 21 |

22 | 23 | 24 | 25 | n+14 | n+15 | 28 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

Conclusion

This proves my formula for grid 8x8 is correct.

## Linking the Formulas

Now that I have the formulas for the grids I will find a formula to link them all together.

5 x 5 grid formula: 4n + 26

6 x 6 grid formula: 4n + 31

7 x 7 grid formula: 4n + 36

8 x 8 grid formula: 4n + 41

The four before the n represents the number of squares in a boot that is why they are all four.

Now I have this information I can begin to work on the sequence for the size of the grid.

26 31 36 41

5 5 5

g = grid size

g | 5 | 6 | 7 | 8 |

5g | 25 | 30 | 35 | 40 |

Sequence | 26 | 31 | 36 | 41 |

26-1=2531-1=3036-1=3541-1=40

All need to add one to reach the sequence number!

This means the formula to work out the value of the boot and the grid size has to be 4n+5g+1.

I will test this formula on a 5x5 grid.

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 |

n=8 g=5

4x8=32 + 5x5=25 + 1

32+25+1=58

8+13+18+19=58

This proves my formula for the total value of the built and for the size of the grid is correct.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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