# I am going to investigate taking a square of numbers from a grid, multiplying the opposite corners and finding the difference of these two results

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Introduction

Maths Coursework

Corners

I am going to investigate taking a square of numbers from a grid, multiplying the opposite corners and finding the difference of these two results.

To start I used a 5x5 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 |

Then I took 2x2 squares out of this grid and multiplied the opposite corners, to find the difference:

1 | 2 |

6 | 7 |

1 x 7 = 7

2 x 6 = 12

12 – 7 = 5

So the difference between the answers is 5. Next I took another 2x2 square from the same grid.

19 | 20 |

24 | 25 |

19 x 25 = 475

20 x 24 = 480

480 – 475 = 5

The difference is 5 again. I thought this could mean that all 2x2 squares in a 5x5 grid would come out with a difference of 5. To check this, I took another 2x2 square out of the grid to check.

12 | 13 |

17 | 18 |

12 x 18 = 216

13 x 17 = 221

221 – 216 = 5

So this shows that my prediction was right and every 2x2 square in a 5x5 grid should come out with a difference of 5.

## Any 2x2 square in a 5x5 grid = Difference of 5

Now I am going to start taking 2x2 squares out of 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 |

1 | 2 |

7 | 8 |

1 x 8 = 8

2 x 7 = 14

14 – 8 = 6

The difference is 6.

21 | 22 |

27 | 28 |

21 x 28 = 588

22 x 27 = 594

594 – 588 = 6

Again the difference is 6. So I can see like in the 5x5 grid there is a pattern. If I am right every 2x2 square in a 6x6 grid should have a difference of 6. To check if I am right I will take one more square out of the grid.

16 | 17 |

22 | 23 |

16 x 23 = 368

17 x 22 = 374

374 – 368 = 6

This shows that I am right and every 2x2 square in a 6x6 grid will have a difference of 6.

## Any 2x2 square in a 6x6 grid = Difference of 6

I now know the differences of a 2x2 square in a 5x5 and 6x6 grid:

5x5 grid = Difference of 5

6x6 grid = Difference of 6

Looking at this I can see another pattern. The difference of a 2x2 square in a grid seems to be the number of the grid size (E.

Middle

x+2

x+2g

x+2g+2

Then like the normal squares we just multiply the corners.

x(x+2g+2) = x²+x2g+2x

(x+2)(x+2g) = x²+x2g+2x+4g

Then take the answers away from each other to get the expression for the difference.

(x²+x2g+2x+4g) - (x²+x2g+2x) = 4g

So this means for a 3x3 square in any grid, the difference can be worked out by using this algebraic expression:

-d = 4g-

Now I have this I am going to try it on one of the 3x3 squares just to make sure it is right. I will use the equation to calculate the difference of a 3x3 square in a 9x9 grid, then I will actually use a real 3x3 square from a 9x9 grid to check if the equation is correct.

1 | 2 | 3 |

10 | 11 | 12 |

19 | 20 | 21 |

d = 4x9 = 36

So if the equation is right the difference will be 36.

1 x 21 = 21

3 x 19 = 57

57 – 21 = 36

So the difference is 36 and the equation worked it out correctly. Now I know that I can use this equation for any 3x3 square in any square grid.

A 3x3 square in a:

22x22 grid: d = 4 x 22 = 88

13x13 grid: d = 4 x 13 = 52

45x45 grid: d = 4 x 45 = 180

123x123 grid: d = 4 x 123 = 492

1005x1005 grid: d = 4 x 1005 = 4020

Now I am going to look at 4x4 squares.

Again we can call the first corner ‘x’.

(Please remember there are numbers between these corners but we do not need to use them)

x | - |

- | - |

This time I don’t really need to look at an example to find the top right expression, instead I can just work it out by thinking about it.

If this is the top line of a 4x4 square then we can just work out how much the top right corner will be more than x.

x | - | - | - |

So the top right corner is just x+3

Conclusion

(x) . (x+g((n-1)+(n-1))) = x(x+g((n-1)+(n-1)))

= x²+gx((n-1)+(n-1))

= x²+gx(n-1)+gx(n-1)

= x²+2gx(n-1)

(x+g(n-1)-(n-1)) . (x+g(n-1)+(n-1))

= x(x+g(n-1)-(n-1))

+ g(n-1)(x+g(n-1)-(n-1))

+ (n-1)(x+g(n-1)-(n-1))

= x²+gx(n-1)-x(n-1)+gx(n-1)+g²(n-1)²-g(n-1)²+x(n-1)+g(n-1)²

-(n-1)²

= x²+2gx(n-1)+g²(n-1)²-(n-1)²

Now I will subtract the answers from each other.

(x²+2gx(n-1)+g²(n-1)²-(n-1)²) – (x²+2gx(n-1))

= g²(n-1)²-(n-1)²

So the equation we can use to find the difference of any rhombus in any square grid is:

-d = g²(n-1)²-(n-1)²-

To do a final check on this I am going to use the 3x3 rhombus we used from the 8x8 grid earlier and see if the equation can work out the difference correctly.

4 | ||

18 | 22 | |

36 |

d = 8²(3-1)²-(3-1)² = 64x2²-2² = 64x4-4 = 252

So if the equation is right the difference will be 252.

4 x 36 = 144

18 x 22 = 396

396 – 144 = 252

The difference is 252, so the equation is right.

I have now got the equations to work out:

Any Square in any square grid: -d = g(n-1)²-

Any Rectangle in any square grid: -d = g(l-1)(w-1)-

Any Rhombus in any square grid: -d = g²(n-1)²-(n-1)²-

If I had more time I could go on to find the equation for any Square, Rectangle or Rhombus in different shaped grids. I could then make the numbers in the grids different like a grid of odd numbers, or equal numbers. There are lots of things you could do to extend the task, and using the information I have already found it should be easier to work out the equations for any square, rectangle or rhombus on different grids, because there will most likely be some similarities.

William Ellis 10T

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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