# Number Grid Product Differences Investigation

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Introduction

Maths Coursework

## Number Grid Product Differences

This assignment is concerned with product differences in different size matrices for different sized 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 |

Here is a 10x10 grid. We are trying to investigate product differences within the 10x10 grid. As you can see I have drawn a 2x2 grid inside the 10x10 matrix. To find out the product difference I do:-

The top left number x the bottom right – the top right x the bottom right.

The product difference I found was 10. This was anywhere on the grid where a 2x2 grid could be done. I can prove this by picking a 2x2 grid and using this clue to work out the product difference. Below is the result of me picking a 2x2 grid from this 10x10 grid and finding out what the product difference was.

48 | 49 |

58 | 59 |

I did (48x59) – (49x58) = 10. This shows that the product difference is the same anywhere in the grid.

Here are 2 more examples of this that prove that this works anywhere on the grid.

12 | 13 |

22 | 23 |

I did (12x23) – (13x22) = 10. This shows that the product difference is the same anywhere in the grid.

72 | 73 |

82 | 83 |

I did (72x83) – (73x82) = 10. This shows that the product difference is the same anywhere in the grid.

Middle

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 above is a 10x10 matrix with 5x5 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

33 | 34 | 35 | 36 | 37 |

43 | 44 | 45 | 46 | 47 |

53 | 54 | 55 | 56 | 57 |

63 | 64 | 65 | 66 | 67 |

73 | 74 | 75 | 76 | 77 |

I did the calculation of (33x77) – (37x73) = 160. I had figured out that the product difference was 160 in 5x5 matrices in a 10x10 matrix.

I will do another 2 examples to prove this

23 | 24 | 25 | 26 | 27 |

33 | 34 | 35 | 36 | 37 |

43 | 44 | 45 | 46 | 47 |

53 | 54 | 55 | 56 | 57 |

63 | 64 | 65 | 66 | 67 |

I did the calculation of (23x67) – (27x63) = 160.

53 | 54 | 55 | 56 | 57 |

63 | 64 | 65 | 66 | 67 |

73 | 74 | 75 | 76 | 77 |

83 | 84 | 85 | 86 | 87 |

93 | 94 | 95 | 96 | 97 |

I did the calculation of (53x97) – (57x93) = 160.

The examples prove that it works anywhere on the grid.

6x6

I did exactly the same as I have done several times.

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 above is a 10x10 matrix with 6x6 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

35 | 36 | 37 | 38 | 39 | 40 |

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 |

I did the calculation of (35x90) – (40x85) = 250. This was the product difference for 6x6 matrices in a 10x10 matrix.

I did 2 more examples to prove this.

25 | 26 | 27 | 28 | 29 | 30 |

35 | 36 | 37 | 38 | 39 | 40 |

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

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

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

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

I did the calculation of (25x80) – (30x75) = 250. This was the product difference for 6x6 matrices in a 10x10 matrix.

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

53 | 54 | 55 | 56 | 57 | 58 |

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

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

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

93 | 94 | 95 | 96 | 97 | 98 |

I did the calculation of (43x98) – (48x93) = 250.

This proves that it works anywhere on the grid.

I then decided to see if I could find a formula.

N.b The Nth term is 1 less than the size of the squares

Nth Term | 1 | 2 | 3 | 4 | 5 |

Sequence | 10 | 40 | 90 | 160 | 250 |

1st Difference | |||||

2nd Difference | |||||

10n2 | 10 | 40 | 90 | 160 | 250 |

The second difference is halved and put into the formula. This is because

Now if I wanted to work out the product difference of a 7x7 matrices in a 10x10 matrix I would use the formula of 10n2.

So my calculation would be 10x62which would = 10x 36 which would be 360. And if I know work out the long way this is what the answer would be.

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 above is a 10x10 matrix with 7x7 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

34 | 35 | 36 | 37 | 38 | 39 | 40 |

44 | 45 | 46 | 47 | 48 | 49 | 50 |

54 | 55 | 56 | 57 | 58 | 59 | 60 |

64 | 65 | 66 | 67 | 68 | 69 | 70 |

74 | 75 | 76 | 77 | 78 | 79 | 80 |

84 | 85 | 86 | 87 | 88 | 89 | 90 |

94 | 95 | 96 | 97 | 98 | 99 | 100 |

Conclusion

The above concludes that the formula 10n2 works in any matrix.

Then I decided to try if could find a rule applied which would work for 3x2 matrices in a 10x10 matrix.

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 |

Here is a 10x10 grid. We are trying to investigate product differences within the 10x10 grid. As you can see I have drawn a 3x2 grid inside the 10x10 matrix. To find out the product differences I do: -

The top left number x the bottom right – the top right x the bottom right.

Below is an example to work out

27 | 28 | 29 |

37 | 38 | 39 |

I did the calculation of (27x39) – (29x37) = 20

I tried this 2 more times to see if we had the product.

77 | 78 | 79 |

87 | 88 | 89 |

I did the calculation of (77x89) – (79x87) = 20

44 | 45 | 46 |

54 | 55 | 56 |

I did the calculation of (77x89) – (79x87) = 20

This proved that the formula worked.

Nth Term | 1 | 2 | 3 | 4 | 5 |

Sequence | 20 | 60 | 120 | 200 | 300 |

1st Difference | |||||

2nd Difference | |||||

10(L-1)(W-1) | 20 | 60 | 120 | 200 | 300 |

N.b The Nth term is 1 less than the size of the squares

I have done this to help me work out the formula.

With the information above put into a table I figured out the general formula using my knowledge gained from the previous work that the general formula is 10(L-1)(W-1). I came to this by firstly using the 2nd product difference and halving it this is because

This would for different sized matrices in different sized matrix. Examples of this would be like a 3x2 in an 8x8 grid etc.

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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