# "Multiply the figures in opposite corners of the square and find the difference between the two products. Try this for more 2 by 2 squares what do you notice?"

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Introduction

Gurprit Singh Khela

Opposite Corners-coursework

“Multiply the figures in opposite corners of the square and find the difference between the two products.

Try this for more 2 by 2 squares what do you notice?”

Investigate!

In this investigation I will research various squares and rectangles within selected grids such as 10 by 10, 11 by 11 and so forth. I will find patterns between the differences and the squares and rectangles within the grids. By the end of this investigation my aim is to achieve a formula that will connect and link the shape within the grid whether it is a square, rectangle etc. to the size of the grid itself. If I have time I may possibly go into 3 Dimensions and investigate the addition of the next dimension and see how this will affect the 2D work so far and how it can be linked.

To start off with I will take a 2 by 2 square out of the top left hand corner of my grid which for now is 10 by 10 size.

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 |

2 x 11 =22

1 x 12 =12

I have multiplied the opposite corners and now will subtract the smaller number from the larger one as explained in the question above.

22 – 12 = 10

By subtracting the 2 totals I have found the difference which is 10.I wonder if this difference of 10 will remain constant if I change the position of the 2 by 2 square on the grid. To test this I will move the square along a place and see what the difference is.

2 | 3 |

12 | 13 |

3 x 12 = 36

Middle

(x + 3)(x + 30) = x2 + 33x + 90

(x2 + 33x + 90) – (x2 + 33x) = 90

The difference is 90 and I have proven this with the use of algebra and numbers therefore my prediction was wrong and now I shall collate the differences so far and see what I can conclude on.

Here are my results so far:

Square Size | Difference | First Difference | Second Difference |

2 x 2 | 10 | ||

3 x 3 | 40 | +30 | |

4 x 4 | 90 | +50 | +20 |

5 x 5 | ???? | ???? | ????? |

This is interesting as the first difference isn’t constant but the second difference is 20.Looking at the sequence I predict that the 20 will remain the same for the second difference and the main difference for the 5 by 5 square will be 160.This is because I will add the 20 to the already increasing first difference of 50 which will be 70.Then I will add the 70 to the 4 by 4 main difference of 90 which will make 160.Therefore a 5 by 5 square in my opinion should have a difference of 160.Now I will test my new prediction.

1 | 2 | 3 | 4 | 5 |

11 | 12 | 13 | 14 | 15 |

21 | 22 | 23 | 24 | 25 |

31 | 32 | 33 | 34 | 35 |

41 | 42 | 43 | 44 | 45 |

1 x 45 = 45

5 x 41 = 205

205 – 45 = 160

x | x + 1 | x + 2 | x + 3 | x + 4 |

x + 10 | x + 11 | x + 12 | x + 13 | x + 14 |

x + 20 | x + 21 | x + 22 | x + 23 | x + 24 |

x + 30 | x + 31 | x + 32 | x + 33 | x + 34 |

x + 40 | x + 41 | x + 42 | x + 43 | x + 44 |

x(x + 44) = x2 + 44x

(x + 4)(x + 40) = x2 + 44x + 160

(x2 + 44x + 160) – (x2 + 44x) = 160

My theory is correct and the difference has been proven to be 160.I have now found a pattern and have enough results to come up with a formula that will give me the results for any size square on a 10 by 10 size grid.

Conclusion

n= one side of the square

I have expanded my formula because now it will give me the difference from any size square within any size grid. To test my formula I will choose a 3 by 3 square on a 12 by 12 firstly work it out with the formula then the full way and see if the answers match.

With the formula:

x(n – 1) 2

12(3 – 1)2

The difference in theory is 48

I will now test the difference to see if it remains the same:

1 | 2 | 3 |

13 | 14 | 15 |

25 | 26 | 27 |

3 x 25 = 75

1 x 27 = 27

75 – 27 = 48

The difference is 48

This proves that my formula will work with any grid and any size square within it at any place on the grid. I have now reached the limit of how squares within a 2 dimensional grid can be investigated. To add another variable into my work I am going to investigate rectangles. This will add a further variable because as with squares both the length and width are represented by 1 number whereas rectangles can have 2 completely different numbers. Therefore I am expecting to work out a much different formula from the one above with the inclusion of another variable. This formula should provide me with the following:

- Difference to any size square or rectangle on a size grid in any location on the grid.

I will work out the difference to a 2 by 3 rectangle, all the work from this point will remain on a 10 by 10 grid until stated otherwise.

1 | 2 | 3 |

11 | 12 | 13 |

1 x 13 = 13

3 x 11 = 33

Difference is 20

x | x + 1 | x + 2 |

x +10 | x + 11 | x + 12 |

(x + 2)(x + 10) = x2 + 12x + 20

x(x + 12) = x2 + 12x

Difference is 20

2 x 4:

1 | 2 | 3 | 4 |

11 | 12 | 13 | 14 |

4 x 11 = 44

1 x 14 = 14

Difference is 30

x | x + 1 | x + 2 | x + 3 |

x +10 | x + 11 | x + 12 | x + 13 |

Page

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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