# Number grids

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Introduction

## NUMBER GRIDS

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 |

The aim of my investigation is to find the cross product difference of various grid sizes and see what I notice about this. I will try to find out a pattern from when I change the grid sizes. I will get a 10 by 10 table ranging from 1-100 and start off using squares, i.e. 2x2, 3x3, 4x4 e.c.t. I will take the two opposite corners and multiply them together doing the same on both sides, I will take the two final numbers and subtract them from one another, this will leave me with a number, which should be the same for each of the same sized grid shapes. I will place my results into a table and see if I can work out a formula for finding out all the results.

2x2

12 | 13 |

22 | 23 |

(13x22) – (12x23) = 286-276

= 10

15 | 16 |

25 | 26 |

(16x25)-(15x26) = 400-390

= 10

77 | 78 |

87 | 88 |

(78x87)-(77x88) = 6786-6776

= 10

81 | 82 |

91 | 92 |

(82x91)-(81x92) = 7462-7452

= 10

33 | 34 |

43 | 44 |

(34x43)-(33x44) = 1462-1452

= 10

In all of the small 2x2 number grids above I have found out that the product of the top left and the bottom right numbers multiplied by the top right and the bottom left numbers, subtracted away from each other had a difference of 10. This is why:

N | N+1 |

N+10 | N+11 |

N(N+11) and (N+1)(N+10)

= (N+1)(N+10)-N(N+11)

= N²+10N+1N+10-N²–11N

= 11N-11N+10

= 10

3x3

21 | 23 | |

41 | 43 |

(23x41)-(21x43) = 943-903

= 40

77 | 79 | |

97 | 99 |

(79x97)-(77x99) = 7663-7623

= 40

34 | 36 | |

54 | 56 |

(36x54)-(34x56) = 1944-1904

= 40

52 | 54 | |

72 | 74 |

(54x72)-(52x74) = 3888-3848

= 40

N | N+2 | |

N+20 | N+22 |

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

Middle

= 1100

12x12

1 | 12 | ||||||||||

133 | 144 |

(12x133)-(1x144) = 1596-144

=1452

13x13

1 | 13 | |||||||||||

157 | 169 |

(13x157)-(1x169) = 2041-169

= 1872

14x14

1 | 14 | ||||||||||||

183 | 196 |

(14x183)-(1x196) = 2562-196

= 2366

I am finally going to get the result for the 15x15 grid and compare all the results for the main grid size feature.

15x15

1 | 15 | |||||||||||||

211 | 225 |

(15x211)-(1x225) = 3315-225

= 3090

By looking at all the diagrams I have come up with a formula that works. Although this is presented on a 3x3 grid it works for any main grid size feature, i.e. 10x10, 11x11, 5x5, 15x15 e.c.t.

For example, I have looked at the results of the previous grid squares and I have transferred the information into a formula grid square. :

N | N+4 | |

N+40 | N+44 |

N | N+(g-1) | |

N+M(g-1) | N+M(g-1)+(g-1) |

CPD (Cross Product Difference):

[N+(g-1)] [N+M(g-1)] –N [N+M(g-1)+(g-1)]

= N² + M(g-1)N + N(g-1) + M(g-1)²- N²- M(g-1)N - N(g-1)

= N² + M(g-1)N + N(g-1) + M(g-1)²- N²- M(g-1)N - N(g-1)

= M(g-1)²

Therefore my formula works. I have retested this to see if this works on a 16x16 grid and it has proven that it does. In the formula the letters stand for:

N = the original number

g = grid size on side of actual grid

M= Main grid size.

16x16

1 | 16 | ||||||||||||||

241 | 256 |

(16x241)-(1x256) = 3856-256

= 3600

If you place these numbers into the formula the answer should be correct to the formula. For example:

[1+(16-1)] [1+16(16-1)] –1[1+16(16-1)+(16+1)]

= 1²+16(16-1)1 + 1(16-1) +16(16-1)² - 1² - 16(16-1)1 – 1(16-1)

Conclusion

[N+s(L-1)][N+Ms(w-1)] – N[N+Ms(w-1)+s(L-1)]

=N²+NMs(w-1) + Ns(L-1) + Ms (L-1)(w-1) – N² –NMs(w-1) -Ns(L-1)

=N² +NMs(w-1) + Ns(L-1) + Ms (L-1)(w-1) – N² –NMs(w-1) -Ns(L-1)

= Ms²(L-1)(w-1).

Know I am going to test to see if my formula is right to find the cross product difference.

The grid will have the following specifications:

Main grid size: 10X10

17 | 18 |

27 | 28 |

Step Size: 1

Length of shape: 2

Width of shape: 2

[17+1(2-1)] [17+10x1(2-1)] -17 [17+10x1(2-1)+1(2-1]

= 289 + 170 + 17 +10 – 289 – 170 – 17

= 10

If I input the numbers of the tested grid into the formula the answer should be the same as the answer 10.

Ms²(L-1)(w-1)

= 10x1²(2-1)(2-1)

=10x1(1)(1)

=10

Therefore the formula must be correct as the answer of the tested grid is the same as the answer from the formula.

I have found out many different formulas during the course of the coursework and through each feature changed the feature before is linked in someway to the next, whether it was the Main grid size or like the one above all features linked together.

M=Main grid size

s= Step size

L=Length of shape

w=width of shape.

In conclusion I have successfully completed 3 different features that could be changed. The Main grid size(M), the step size(s) and the change in shape(L,w). I feel that I productively carried out my task finding out the cross product difference not only in number work and grids but the formulas as well.

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

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