# Maths number grid

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Introduction

Number Grid Coursework

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

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 9 x 20 = 180 |

21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 19 x 10 = 190 |

31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | d = 10 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 40 | |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 89 x 100 = 8900 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 99 x 90 = 8910 |

71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | d = 10 |

81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

I have noticed that there is a difference of ten between the answers when multiplying the diagonally opposite corners of a 2 x 2 grid. I have also noticed that the 2 x 2 grids I have used are the corners of the 100 square grid. I will try using a 2 x 2 grid located more towards the centre of the 100 square grid to see if it is just the corner 2x2 grids that give a difference of 10.

34 x 45 = 1530

44 x 35 = 1540

d = 10

As you can see by my calculations it is not just a 2 x 2 grid in the corner that gives a difference of 10 when the diagonally opposite corners are multiplied. The position of the 2 x 2 grid on a 100 square grid does not change the difference.

I am now going to investigate if the difference is the same in a 2 x 3 grid when multiplying diagonally opposite corners.

2 x 3 grid

63 | 64 | 65 | 63 x 75 = 4725 |

73 | 74 | 75 | 65 x 73 = 4745 |

d = 20

39 | 39 | 40 | 38 x 50 = 1900 |

48 | 49 | 50 | 48 x 40 = 1920 |

d = 20

11 | 12 | 13 | 11 x 23 = 253 |

21 | 22 | 23 | 21 x 13 = 273 |

d = 20

Middle

2 high boxes

Height | Width | Difference |

2 | 2 | 10 |

2 | 3 | 20 |

2 | 4 | 30 |

2 | 5 | 40 |

As you can see in my table, I feel I could now use my findings to predict the difference for a 2 x 5 grid. As I have already predicted, the difference will be 40.

2 x 5 grid

1 | 2 | 3 | 4 | 5 | 1 x 15 = 15 |

11 | 12 | 13 | 14 | 15 | 11 x 5 = 55 |

d = 40

My prediction was correct in saying that the difference in a 2 x 5 grid would be 40.

I have noticed that it does not make any difference to the outcome wherever the grid is taken from in a 100 square grid.

From now on I only need to try one box of each size as I know that position has no effect on the difference.

2 x w Algebra (Any width)

n | n+w | n+w-1 | ||

n+10 | n+10+w-1 | (n+10+w-1) = n+w+9 |

(n) (n+w+9) = n2 + nw +9n

(n+10) (n+w-1) = n2 + nw – 1n +10n + 10w -10

(n2 +nw+9n) – (n2+nw+9n+10w-10) = d

10w-10= d

Conclusion

n | n+w-1 | |

g(h-1)+n | g(h-1)+n+(w-1) |

gh-g+n gh-g+n+gw-1

n(gh-g+n+w-1)

=ngh-ng+(n2)+nw-n

(n+w-1)(gh-g+n)

=ngh-ng+(n2)+wgh-wg+wn-gh+g-n

d=(ngh-ng+(n2)+nw-n)-(ngh-ng+(n2)+wgh-wg+wn-gh+g-n)

d=wgh-wg-gh+g

I will now demonstrate how my formula works:

If g=10 h=4 w=3 n=5:

d= wgh-wg-gh+g

d=(3X10X4)-(3x10)-(10x4)+10

d=120-30-40+10

d=60

As you can see below, this is the answer I got for a 4x3 grid.

5 | 6 | 7 | 5 x 37 = 185 |

15 | 16 | 17 | 7 x 35 = 245 |

25 | 26 | 27 | d = 60 |

35 | 36 | 37 |

By Philip Redpath

## Algebra for h x w box on a 10x10 grid

n | n+w-1 | |

10(h-1)+n | 10(h-1)+n+(w-1) |

10h-10+n 10h-10+n+w-1

n(10h-10+n+w-1)

=10hn-10n+(n2)+wn-n

(n+w-1)(10h-10+n)

=10hn-10n+(n2)+10wh-10w+nw-10h+10-n

d=(10hn-10n+(n2)+wn-n)-( 10hn-10n+(n2)+10wh-10w+nw-10h+10-n)

d=10wh-10w-10h+10

I will now demonstrate how my formula works:

h=3 w=2 n=6

d=10wh-10w-10h+10

d=(10x2x3)-(10x2)-(10x3)+10

d=60-20-30+10

d=20

As you can see below, this is the same answer I got for a 3x2 grid:

3 | 4 | 3 x 24 = 72 |

13 | 14 | 23 x 4 = 92 |

23 | 24 | d = 20 |

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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