# He analysis of number patterns on various types of number grids.

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Introduction

The analysis of number patterns on various types of number grids

## The main objective of this mathematics coursework is to analyse the various number patterns that emerge from carrying out a very simple mathematical operation on the set of numbers found in a number grid of natural integers. The number grid is arranged in the form of a square grid as shown below:

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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 diagram shows the method that needs to be done to make the calculations so that the number pattern can be found out. The calculation would have to be carried out for all the other number grids and patterns so that the rule which applies to one number pattern should also apply to another number pattern.

Once all the calculations have been carried out the results are analysed to find out if any specific number pattern can be seen or

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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 |

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 |

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 |

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 |

From the above results we can make an assumption that a formula can be made which will allow us to find the answer to other sizes of boxes in other larger number grids. At first we need to carry out the same calculations for even larger number grids.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |

31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 |

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Conclusion

Let X = size of box, and N = size of number grid.

Therefore b = a + ( X – 1 ) and c = a + N ( X – 1) and d = c + ( X – 1)

Therefore, d = a + N ( X – 1) + ( X – 1)

d = a + NX – N + X – 1

According to the worksheet, we have to multiply a x d and c x b, which gives us ad and bc, followed by taking ad away from bc, as ad is smaller than bc.

Let us work out what ad and bc are in terms of X and N.

ad = a { a + NX – N + X – 1 }

= a2 + aNX – aN + aX – a

bc = { a + ( X – 1 ) }{ a + N ( X – 1) }

= (a + X – 1) ( a + NX – N )

= a2 + aNX – aN + aX + NX2 – XN – a – NX + N

= a2 + aNX – aN + aX + NX2 – 2XN – a + N

bc – ad = [a2 + aNX – aN + aX + NX2 – 2XN – a + N ] – [a2 + aNX – aN + aX – a ]

=a2 + aNX – aN + aX + NX2 – 2XN – a + N - a2 - aNX + aN - aX + a

= NX2 – 2XN + N

= N ( X2 - 2X + 1 )

= N ( X –1 )2

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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