# Number Grids

Extracts from this document...

Introduction

Introduction

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 |

We were given this number grid and shown that if a two–by-two square was drawn anywhere on the grid, encompassing four numbers, the difference between the product of the top left number and the bottom right number and the top left number.

For example (13x22) - (12x23) = 10

Also (16x25) - (15x26) = 10

We were then told to investigate further.

To give us an idea of where we were headed, our teacher told us we should be aiming at least to find a formula that would allow a person to work out the difference between the products of the corners of any rectangle on such a grid of any length.

The Investigation

The fact that the difference came out to ten no matter where the square was drawn made sense, it could be easily demonstrated algebraically.

Let’s say the top left number was ‘n’. If followed along the grid, the other numbers would come to be: (bottom right) n+11, (top right) n+1 and (bottom left) n+10.

So if worked out, the sum would look like:

After this, I tried squares of different sizes around the grid.

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 |

I found that the differences in all the 3x3 squares came out to 40

Eg.

(14x32) - (12x34) = 40

(18x36)

Middle

17

18

19

20

21

22

23

24

25

26

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31

32

33

34

35

36

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121

2x2:

(2x12) - (1x13) = 11

(5x15) – (4x16) = 11

3x3:

(36x56) – (34x58) = 44

(40x60) – (38x62) = 44

4x4:

(81x111) – (78x114) = 99

(86x116) – (83x119) = 99

The pattern came out as:

2x2 | 3x3 | 4x4 | 5x5 |

11 | 44 | 99 | 176 |

So,

11 44 99 176

33 55 77 1st Difference

22 22 2nd Difference

, determining the coefficient to be used for n2.

n | 2 | 3 | 4 | 5 |

pattern | 11 | 44 | 99 | 176 |

11n2 | 44 | 99 | 176 | 275 |

pattern – 11n2 | -33 | -55 | -77 | -99 |

Formula:

-22 -22 -22 Formula for linear part:

Put together:

(Simplifying down to this)

Indeed the only part of the end formula that changed, was the part hypothesized to be directly linked to the length of the grid. With the grid length 10, the formula came out to be, with the grid length as 11, it came out to be. As this part of the formula appeared to be linked to the grid length, it could now be replaced with an expression representing the grid length so it would work for any grid. Thus the formula now became. If the assumption that one of the s were in fact the length, and the other the width continued, the formula then became.

To prove this formula, I looked at the situation algebraically and came up with a list of expressions to represent the real life values.

Conclusion

Simplifying down to

This was my first 3D formula.

Now for the second set, the calculation was -. When the respective expressions were substituted for these values, it came out to:

Simplifying down to

This was my second 3D formula.

I tested the both of these to check that everything was ok with them.

The example I came up for the test was:

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 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 |

109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 |

118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 |

127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 |

In the example, , , , , , , and . Other than that, , , , , .

For the first formula,

If done with the formula instead,

For the second formula,

If done with the formula instead,

In both cases, the answer came out the same, both formulas worked. One slight problem was that somewhere in the process I managed to mix up the order I subtracted in so that my first formula always gave a negative answer (as I subtracted a larger value from a smaller one) and my second formula always gave a positive answer (as I subtracted a smaller value from a larger one, like I was supposed to). This isn’t that much of a problem as what the actual number was would always came out the same, the sign in front of it solely depends on the order in which the numbers were subtracted from one another. As the investigation asks for a mere difference, the sign could be ignored altogether.

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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