# Maths Coursework - Number Grid

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Introduction

Kayleigh McCormack

Maths Coursework – Number Grid

For this task I will first be looking at a number grid from 1 to 100, like the one below :

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 |

I will start my investigation by looking at 2 by 2 squares. I will draw a square around 4 numbers, find the product of the bottom left and top right numbers and the product of the top left and bottom right numbers, then calculate the difference between the 2 products. I will see if there are any patterns and if so I will try to work it out algebraically.

I will then look at changing the size of the squares to see if there are any patterns. I will try looking at 3 by 3 squares, 4 by 4 squares and 5 by 5 squares; I will do the same with these squares as I have with the 2 by 2 squares, I will find the products of the top left and bottom right and the bottom left and top right numbers then calculate the difference between them.

Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles. I will start by looking at a 3 by 2 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern. I will then try changing the size of the rectangles and looking for patterns there. I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles.

Middle

Blue Square :

18

40

720

38

20

760

40

I have noticed that the difference of the products in each square is always forty.

I have worked out the formula for this pattern below:

N | N+2 | |

N+20 | N+22 |

(N+2)(N+20) - N(N+22) = D

N2 + 20N + 2N + 40 - N2 - 22N = D

40 = D

Difference = 10

This shows that the difference between the two products in each square is always -40; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 40 rather than -40.

I will next look at 4 by 4 squares within a 10 by 10 grid. The squares I am looking at are highlighted in the grid below. I chose these squares randomly; there is no particular reason why I have chosen them.

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 |

I am now going to calculate the products of the diagonals and the difference of these products for each square in the grid above. I will display my findings in a table to help me identify a pattern.

Top Left | Bottom | Product | Bottom | Top Right | Product | Difference | ||

Right | Left | |||||||

Yellow Square : | 1 | 34 | 34 | 31 | 4 | 124 | 90 | |

Green Square : | 26 | 59 | 1534 | 56 | 29 | 1624 | 90 | |

Purple Square : | 62 | 95 | 5890 | 92 | 65 | 5980 | 90 |

I have noticed that the difference of the products in each square is always ninety.

I have worked out the formula for this pattern below:

N | N+3 | ||

N+30 | N+33 |

N(N+33) - (N+3)(N+30) = D

N2 + 33N - N2 - 30N - 3N - 90 = D

-90 = D

Difference = 90

This shows that the difference between the two products in each square is always -90; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 90 rather than -90.

Now that I have investigated squares within a 10 by 10 grid and found algebraic formulas to explain the patterns I saw, I will move on to investigate rectangles within a 10 by 10 grid. I will first be looking at 3 by 2 rectangles. The rectangles I am looking at are highlighted in the grid below.

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 |

I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above. I will display my findings in a table to help me identify a pattern.

Top Left | Bottom | Product | Bottom | Top Right | Product | Difference | ||

Right | Left | |||||||

Yellow Rectangle : | 1 | 13 | 13 | 11 | 3 | 33 | 20 | |

Green Rectangle : | 34 | 46 | 1564 | 44 | 36 | 1584 | 20 | |

Purple Rectangle : | 58 | 70 | 4060 | 68 | 60 | 4080 | 20 | |

Blue Rectangle : | 71 | 83 | 5893 | 81 | 73 | 5913 | 20 |

I have noticed that the difference of the products in each square is always twenty.

I have worked out the formula for this pattern below:

N | N+2 | |

N+10 | N+12 |

Conclusion

Top Left | Bottom | Product | Bottom | Top Right | Product | Difference | ||

Right | Left | |||||||

Yellow Rectangle : | 1 | 25 | 25 | 21 | 5 | 105 | 80 | |

Green Rectangle : | 43 | 67 | 2881 | 63 | 47 | 2961 | 80 | |

Purple Rectangle : | 16 | 40 | 640 | 36 | 20 | 720 | 80 | |

Blue Rectangle : | 76 | 100 | 7600 | 96 | 80 | 7680 | 80 |

I have noticed that the difference of the products in each square is always eighty.

I have worked out the formula for this pattern below:

N | N+4 | |||

N+20 | N+24 |

N(N+24) - (N+4)(N+20) = D

N2 + 24N - N2 - 20N - 4N - 80 = D

-80 = D

Difference = 80

This shows that the difference between the two products in each rectangle is always -80; I have shown the difference as 80 rather than -80 as I am only interested in the number and not the sign in front of it (+/-).

I am now going to look at 4 by 5 rectangles within the same 10 by 10 grid. The rectangles I am looking at are highlighted in the grid below. I chose these rectangles randomly.

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 |

I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above. I will display my findings in a table so it will be easier to identify the pattern.

Top Left | Bottom | Product | Bottom | Top Right | Product | Difference | ||

Right | Left | |||||||

Yellow Rectangle : | 1 | 44 | 44 | 41 | 4 | 164 | 120 | |

Green Rectangle : | 16 | 59 | 944 | 56 | 19 | 1064 | 120 | |

Purple Rectangle : | 52 | 95 | 4940 | 92 | 55 | 5060 | 120 |

I have noticed that the difference of the products in each square is always one hundred twenty.

I have worked out the formula for this pattern below:

N | N+3 | ||

N+40 | N+43 |

N(N+43) - (N+3)(N+40) = D

N2 + 43N - N2 - 40N - 3N - 120 = D

-120 = D

Difference = 120

This shows that the difference between the two products in each rectangle is always -120; I have shown the difference as 120 rather than -120 as I am only interested in the number and not the sign in front of it (+/-).

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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