# number grid

Extracts from this document...

Introduction

NUMBER GRID

COURSE WORK

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 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |

71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |

81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

To investigate the patterns generated from using rules in a square grid

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally. For example take 2x2 square and multiply its corners diagonally.

25 | 26 |

35 | 36 |

25 x 36= 900 26 x 35= 910 910 - 900= 10

77 | 78 |

87 | 88 |

77 x 88 = 6776 78 x 87= 6786 6786 - 6776= 10

After a few results I observed that

Middle

61

62

63

64

31 x 64 = 1984 34 x 61=2074 2074 – 1984 = 90

n th term | SIZED | DIFFERENCE |

1 | 2 x 2 | 10 |

2 | 3 x 3 | 40 |

3 | 4 x 4 | 90 |

I put my results into table and observed that the difference is square number because the box and the grid are square shaped.

I predict that for a 5 by 5 square the difference between the products will be 160.

46 | 47 | 48 | 49 | 50 |

56 | 57 | 58 | 59 | 60 |

66 | 67 | 68 | 69 | 70 |

76 | 77 | 78 | 79 | 80 |

86 | 87 | 88 | 89 | 90 |

46 x 90 = 4140 50 x 86 = 4300 4300 - 4140 = 160

I will now attempt to prove my results algebraically.

After doing this I will check my answer using the nxn formula

2x2 square

23 | 24 |

33 | 34 |

n n+1

n+10 n+11 (24x33)- (23x34)

792-782=10

(n+1) (n+10)-n (n+11)

n2 +10n + n+10 - n2 +11n n2 +11n +10 - n2 +11n =10

In both ways the difference is

5x5 square

1 | 2 | 3 | 4 | 5 |

11 | 12 | 13 | 14 | 15 |

21 | 22 | 23 | 24 | 25 |

31 | 32 | 33 | 34 | 35 |

41 | 42 | 43 | 44 | 45 |

n n+4

n+40 n+44

n (n+44) - [(n+4) (n+40)] (5x41)-(1x45)

n2+44n - n2+40n + 4n +160 205-45= 160

n2+44n +160 - n2+44n =160

## In both ways the difference is 160

After doing this I will check if the nxn formula will work in a bigger grid size like 11x11

1 | 2 |

12 | 13 |

Conclusion

S-1 x G = D

Sized = 2 x 3 G = 10 D = 20

2-1 =1

3-1 =2

1 x 2 =2

2 x 10 =20

37 | 38 | 39 | 40 |

47 | 48 | 49 | 50 |

57 | 58 | 59 | 60 |

37 x 60 =2220 40 x 57 = 2280 2280 - 2220 = 60

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

SIZED = 3 x 4 G = 10 D = 60

3-1 = 2

4-1 = 3

2 x 3 = 6

6 x 10 = 60

I could prove that this formula can work for both rectangle and square.

53 | 54 |

63 | 64 |

53 x 64 = 3392 54 x 63 = 3402 3402 - 3392 = 10

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 2 x 2 G = 10 D = 10

2-1 = 1

2-1 = 1

1 x 1 = 1

1 x 10 = 10

FINALLY

In this project I found that not all formula can work for rectangle and square.

But I found a formula can work for rectangle and square it does not matter the size of the box and I will proof that it does not matter for grid sizes too.

28 | 29 | 30 |

40 | 41 | 42 |

52 | 53 | 54 |

28 x 54 = 1512 30 x 52 = 1560 1560 - 1512 = 48

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 3 x 3 G = 12 D = 48

3-1 = 2

3-1 = 2

2 x 2 = 4

4 x 12 = 48

If I were to extend this project further I would try and do cube in three dimensions.

ISRAA AL-RIFAEE - -

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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