# Diagonal Difference.

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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 the diagonal difference of all the sizes within the 10 by 10 grids, and then try to find any patterns, which would help me in finding the formula by drawing a grid.

Middle

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.

Now I am going to do 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 |

Conclusion

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

11 | 12 | 13 | 14 | 15 | 16 |

21 | 22 | 23 | 24 | 25 | 26 |

31 | 32 | 33 | 34 | 35 | 36 |

6 x 31 = 186

1 x 36 = 36

Diagonal difference: 150

From this I can tell that my prediction was correct and now I am going to try out a 4 by 7 grid, I can tell that my prediction will be correct as the difference is going up in thirties. But just to make sure I am going to do an algebraic grid to show this.

The grid below shows a 4 by 7 rectangular grid.

x | x+1 | x+2 | x+3 | x+4 | x+5 | x+6 |

x+10 | x+11 | x+12 | x+13 | x+14 | x+15 | x+16 |

x+20 | x+21 | x+22 | x+23 | x+24 | x+25 | X+26 |

x+30 | x+31 | x+32 | x+33 | x+34 | x+35 | x+36 |

(x+30) (x+6)-x(x+36)

= x2+30x+6x+180-(x2+36x)

= x2+36x+180-x2-36x

= 180

My prediction for all this is correct. Now that I have got the formula and have noticed that it can be used for working out the diagonal difference for the squares and the rectangles, I should be able to predict any size squares or rectangles.

Now I am going to try out an 8 by 8 grid. This is my next size of the grid which I will see if my formula works.

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 |

I am going to start by a 3by 4 rectangular grid.

10 | 11 | 12 | 13 |

18 | 19 | 20 | 21 |

26 | 27 | 28 | 29 |

10 x 29 = 290

13 x 26 = 338

Diagonal difference: 48

I think that the number has ended in 8, this is because I have done an 8 by 8 grid because I did a 10 by 10 grid all the diagonal differences ended in 0.

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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