# Investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid

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Introduction

Naila Parveen Maths Coursework year 11

Maths Coursework-Diagonal Difference

Introduction

I am given a 10 by 10 grid. I am going to find the diagonal difference of different size grids (For e.g. 3 by 3, 4 by 4) within the 10 by 10 grid, by multiplying the opposite corners which results in two answers, we then deduct these two to get a final answer for that size.

This is the grid that I will use to help me investigate.

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 |

Aim

I am going to investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid. I will then try to find a formula which relates to the diagonal difference of each square, I will then further this investigation by trying to find the diagonal difference of an 11 by 11 grid and a 12 by 12 grid and find the formula and see if it is the same. I will also do an extension by doing a rectangle and a square and then find the diagonal difference and the formula for this.

I am going to find the formula by finding

Middle

32

33

34

41

42

43

44

11 x 44 = 484

14 x 41 = 574

Diagonal difference: 90

Now I am going to try a 5 by 5 grid.

51 | 52 | 53 | 54 | 55 |

61 | 62 | 63 | 64 | 65 |

71 | 72 | 73 | 74 | 75 |

81 | 82 | 83 | 84 | 85 |

91 | 92 | 93 | 94 | 95 |

51 x 95 = 4845

55 x 91 = 5005

Diagonal difference: 160

Now I am going to try out work out the algebraic formula for working out the diagonal differences for all squares.

So for a 6 by 6 grid I predict that the diagonal difference would be; 250.

To show this I will do a number grid and also in algebra.

The general difference formula that I predict is (n-1)2 x 10.

Now to show that it works!

45 | 46 | 47 | 48 | 49 | 50 |

55 | 56 | 57 | 58 | 59 | 60 |

65 | 66 | 67 | 68 | 69 | 70 |

75 | 76 | 77 | 78 | 79 | 80 |

85 | 86 | 87 | 88 | 89 | 90 |

95 | 96 | 97 | 98 | 99 | 100 |

45 x 100 = 4500

50 x 95 = 4750

Diagonal difference: 250

Now to show this in an algebraic form.

x | x+1 | x+2 | x+3 | x+4 | x+5 |

x+10 | x+11 | x+12 | x+13 | x+14 | x+15 |

x+20 | x+21 | x+22 | x+23 | x+24 | x+25 |

x+30 | x+31 | x+32 | x+33 | x+34 | x+35 |

x+40 | x+41 | x+42 | x+43 | x+44 | x+45 |

x+50 | x+51 | x+52 | x+53 | x+54 | x+55 |

(x+50) (x+5)-x(x+55)

= x2+50x+5x+250-(x2+55x)

= x2+55x+250-x2-55x

= 250

My formula works. So if I were to do a 7 by 7 I would do 62x10.

Now I am going to put my results in a table.

Size of square | Differences | |

2 x 2 | 10 | 12x10 |

3 x 3 | 40 | 22x10 |

4 x 4 | 90 | 32x10 |

5 x 5 | 160 | 42x10 |

6 x 6 | 250 | 52x10 |

I have shown that for a 5 by 5 square grid you will have to 42x10 to get the answer, so if I wanted to do a 7 by 7 grid I would have to do; 62x10

7 x 7 it would be (7-1)2x10

= 62 x 10

=36 x 10

=360

As I have said before that the general formula is:

(n-1)2x10

So the grid would look like this:

Now I a going to try out the rectangular grids. For this I will try to use an algebraic formula for each grid.

I am going to start of with a 2 by 3 rectangular grid.

35 | 36 | 37 |

45 | 46 | 47 |

35 x 47 = 1645

45 x 37 = 1665

Diagonal difference: 20

Now in algebra:

x | x+1 | x+2 |

x+10 | x+11 | x+12 |

Conclusion

Now I am going to try a 2 by 5 rectangular grid.

32 | 33 | 34 | 35 | 36 |

42 | 43 | 44 | 45 | 46 |

32 x 46= 1472

36 x 42= 1512

Diagonal difference: 40

x | x+1 | x+2 | x+3 | X+4 |

x+10 | x+11 | x+12 | x+13 | x+14 |

(x+4) (x+10)-x(x+14)

= x2+14x+10x+40-(x2+14x)

= x2+14x+40-x2-14x

= 40

Now I am going to put my results in a table, for the working out that is shown above.

Sizes of rectangles | |||

Width (w) | Length (L) | Differences | |

2 | 2 | 10 | 1 x 10 |

2 | 3 | 20 | 2 x 10 |

2 | 4 | 30 | 3 x 10 |

2 | 5 | 40 | 4 x 10 |

2 x L | 10 (L-1) | (L-1)x10 |

During the investigation I have discovered that my research is correct and when I observed my results using algebra the outcomes were the same as to when I used numbers.

Now I am going to change the width to 3 and keep the length the same.

Now I am going to try a 3 by 4 rectangular grid.

65 | 66 | 67 | 68 |

75 | 76 | 77 | 78 |

85 | 86 | 87 | 88 |

65 x 88 = 5720

85 x 68 = 5780

5780-5730 = 60

Now I am going to try a 3 by 5 rectangular grid

63 | 64 | 65 | 66 | 67 |

73 | 74 | 75 | 76 | 77 |

83 | 84 | 85 | 86 | 87 |

63 x 87=5481

83 x 67=5561

Diagonal difference: 80

Now I am going to try a 3 by 6 rectangular grid.

63 | 64 | 65 | 66 | 67 | 68 |

73 | 74 | 75 | 76 | 77 | 78 |

83 | 84 | 85 | 86 | 87 | 88 |

68 x 83 = 5644

63 x 88 = 5544

Diagonal difference: 100

Now I am going to do a 3 by 7 rectangular grid.

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

32 | 33 | 34 | 35 | 36 | 37 | 38 |

42 | 43 | 44 | 45 | 46 | 47 | 48 |

22 x 48 = 1056

42 x 28 = 1176

Diagonal difference: 120

Now I am going to do a table to show my results.

Sizes of rectangles | |||

Width (w) | Length (L) | Differences | |

3 | 4 | 60 | 6 x 10 |

3 | 5 | 80 | 8 x 10 |

3 | 6 | 100 | 10 x 10 |

3 | 7 | 120 | 12 x 10 |

3 x L | 10 (L-1) | (L)x10 |

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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