# Number Grids

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Introduction

Number Grid Maths Coursework

Introduction

In this investigation I will look at different number grids and different sized rectangles within these grids and try to explain the patterns and give algebraic equations for the results that are found during this investigation. In a grid a rectangle is drawn and then the product of the opposite corners is found and the difference is calculated. From this, and further investigation I will find these patterns and work out equations which can then be proven by numerical examples.

Squares in a 10 x 10 grid

In this first section of the investigation I will investigate what patterns can be found with 2x2, 3x3, 4x4 etc. squares in a 10x10 grid.

Numerical Examples

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 |

Square | Difference |

12,13,22,23 (2x2) | 10 |

84,85,94,95 (2x2) | 10 |

41,43,61,63 (3x3) | 40 |

78,80,98,100 (3x3) | 40 |

44,47,74,77 (4x4) | 90 |

7,10,37,40 (4x4) | 90 |

Results

From the number grid and table above I can therefore say that with every 2x2 square in a 10x10 grid that the difference is 10. I can also say that for all 3x3 squares that the difference is 40 and for all 4x4 squares the difference is 90 in 10x10 grids. From this I can therefore say that all squares of the same size have the same difference when the top right and bottom

Middle

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Rectangle | Difference |

1,3,11,13 (3x2) | 20 |

56,57,76,77 (2x3) | 20 |

7,10,17,20 (4x2) | 30 |

32,35,52,55 (4x3) | 60 |

48,50,88,90 (3x5) | 80 |

71,75,91,95 (5x3) | 80 |

Results

From these results of the numerical examples we can again notice that every rectangle, like the squares, of the same size has the same difference. This can be seen by the results of the 5x3, with a difference of 80, and 2x3 rectangles, with a difference of 20 each time. Because of this pattern I will now make the size of the rectangle variable so the width=W and length=L and see if there is a general rule that applies to all rectangles and their differences each time.

An LxW rectangle in a 10x10 Grid

Here the rectangle investigation will carry on and I will try to find and explain an algebraic rule for the differences of all rectangles.

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 |

Rectangle | Difference |

1,3,11,13 (3x2) | 20 |

7,10,17,20 (4x2) | 30 |

32,35,52,55 (4x3) | 60 |

71,75,91,95 (5x3) | 80 |

57,60,97,100 (4x5) | 120 |

For each of these rectangle sizes an individual algebraic formula can be created

For a 3(L)x2(W) rectangle:

n | n+2 |

n+10 | n+12 |

For a 4x2 rectangle:

n | n+3 |

n+10 | n+13 |

For a 4x3 rectangle:

n | n+3 |

n+20 | n+23 |

For a 5x3 rectangle:

n | n+4 |

n+20 | n+24 |

For a 5x4 rectangle:

n | n+4 |

n+30 | n+34 |

But as these only work 4 each rectangle they do not give a general rule for all rectangles. However the patterns found in each of these corners of the different rectangles leads to a rule being able to be produced.

In each case whatever W is, the number in the top right corner is always n + (W-1) which is one step to finding the formula.

For the bottom left corner of any rectangle to make it in terms of L and W the number is always n+10(W-1), which works for every rectangle.

For the bottom right corner to make it in terms of L and W for all rectangles the number is always n+10(L-1)+(W-1).

So this then gives us a rule that works for any rectangle on a 10x10 grid.

n | n+(L-1) |

n+10(W-1) | n+10(W-1)+(L-1) |

L

From this algebraic form of all the corners of any rectangle an equation like with the numerical examples is made to find the difference.

n[n+10(W-1)+(L-1)]=n2+10n(W-1)+n(L-1)

And

[n+(L-1)]x[n+10(W-1)]=n2+10n(W-1)+n(L-1)(W-1)

Then when the two equations are subtracted the difference of the product between the opposite corners for any rectangle in a 10x10 grid is given.

This difference (D)=10(L-1)(W-1)

## Conclusion

So in this section I have found out that the difference between the products of the opposite corners in a 10x10 grid is always 10(L-1)(W-1). Also I have found that all rectangles of the same size have the same difference regardless of whether the length or width is the largest. I will work out some of the rectangle differences to prove that my algebraic theory is correct.

## In a 3x2 rectangle

D= 10(L-1)(W-1)

D= 10x2x1

D=20

## In a 4x5 rectangle

D= 10(L-1)(W-1)

### D= 10x3x4

D= 120

Both of these rectangle differences are correct which proves that my algebraic formula and working out is correct.

## An LxL square or an LxW rectangle in a GxG grid

In this investigation I have so far discovered and proved the formulas for any square or rectangle in a 10x10 grid. In this section I aim to find the formula, by simple number manipulation, of any square or rectangle in any sized grid (GxG). First of all I will investigate any size squares (LxL) in any sized grids (GxG).

## Numerical examples

In a 6x6 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 |

Conclusion

n2gnx(W-1)+nx(L-1)+gx2(L-1)(W-1) - n2 x gnx(W-1)+nx(L-1) = gx2 (L-1)(W-1)

Then when these are subtracted the total difference is obtained

D= gx2 (L-1)(W-1)

Here is an example to check the formula

A 3x4 rectangle in a 6x6 grid with a grid difference (x) of 5

D= gx2 (L-1)(W-1)

D= 6x52(4-1)(3-1)

D= 6x25x3x2

D=900

Conclusion

So overall in this investigation I started small and worked up. From a square in a 10x10 grid with simple algebraic and number manipulation I finally ended up with the formula for finding the differences in any square or rectangle, in any size, in any sized grid, with any differences between the numbers in that grid. Here are the 4 rules that I have found

- For any square- D=10(L-1) 2
- For any rectangle- D=10(L-1)(W-1)
- In any sized grid- D= g(L-1)(W-1)
- With any differences between the numbers in the grid- D= gx2 (L-1)(W-1)

Each of these rules uses elements from the one before and the each equation is a step up from the one before resulting finally in being able to find the difference between any square or rectangle of any size in any sized grid with any difference between the numbers in that grid. And all of these have been demonstrated and proved to be true.

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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