# Number Grid

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Introduction

Maths Coursework - Number Grid

Friday 13th July 2007

Number Grid Coursework

For this piece of coursework, I will investigate the difference when 2x2, 3x3, 4x4, 5x5 and rectangle snapshots are taken from a 10x10 number grid and have their corners multiplied and the difference worked out. For the first part, I will use 2x2 snapshots.

2x2 Boxes

Box 1

2 | 3 |

12 | 13 |

2x13=26

→ 36-26=10

3x12=36

Box 2

32 | 33 |

42 | 43 |

32x43=1376

→ 1386-1376=10

33x42=1386

Box 3

6 | 7 |

16 | 17 |

6x17=102

→ 112-102=10

7x16=112

Box 4

5 | 6 |

15 | 16 |

5x16=80

→ 90-80=10

6x15=90

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 10. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 2x2 snapshot, the difference I will find will be 10.

Box 5

25 | 26 |

35 | 36 |

25x36=900

→ 910-900=10

26x35=910

My predictions that I made earlier about ‘Box 5’ have turned out to be correct as when I multiplied the numbers in the 2x2 box and worked out the difference, I was left with 10.

Middle

59

60

67

68

69

70

77

78

79

80

47x80=3760

→ 3850-3760=90

50x77=3850

My predictions that I made earlier about ‘Box 4’ have turned out to be correct as when I multiplied the numbers in the 4x4 box and worked out the difference, I was left with 90.

I should now try the same method but with boxes of 5x5 dimensions.

5x5 Boxes

Box 1

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 |

1x45=45

→ 205-45=160

5x41=160

Box 2

6 | 7 | 8 | 9 | 10 |

16 | 17 | 18 | 19 | 20 |

26 | 27 | 28 | 29 | 30 |

36 | 37 | 38 | 39 | 40 |

46 | 47 | 48 | 49 | 50 |

6x50=300

→ 460-300=160

10x46=460

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 160. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 5x5 snapshot, the difference I will find will be 160.

Box 3

23 | 24 | 25 | 26 | 27 |

33 | 34 | 35 | 36 | 37 |

43 | 44 | 45 | 46 | 47 |

53 | 54 | 55 | 56 | 57 |

63 | 64 | 65 | 66 | 67 |

23x67=1541

→ 1701-1541=160

27x63=1701

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 5x5 box and worked out the difference, I was left with 160.

I should now try the same method but with rectangles.

Rectangles

Rectangle 1

1 | 2 | 3 | 4 |

11 | 12 | 13 | 14 |

21 | 22 | 23 | 24 |

1x24=24

→ 84-24=60

4x21=84

Rectangle 2

7 | 8 | 9 | 10 |

17 | 18 | 19 | 20 |

27 | 28 | 29 | 30 |

7x30=210

→ 270-210=60

10x17=270

I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 60. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 4x3 snapshot, the difference I will find will be 60.

Rectangle 3

22 | 23 | 24 | 25 |

32 | 33 | 34 | 35 |

42 | 43 | 44 | 45 |

22x45=990

→ 1050-990=60

25x42=1050

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 4x3 box and worked out the difference, I was left with 60.

Algebra Variables

For this specific investigation, I have decided to use algebra to represent and then make formulae for each snapshot I have taken from the 1-100 grid. I will use the letter H to represent the height of each snapshot and use the letter W for the width of each snapshot, no matter of size and dimensions. In this, S is the top left corner in each snapshot. I will now prove how 2x2 boxes always leave you with a difference of 10 through algebra.

Equation

S | S+1 |

S+10 | S+10+1 |

Conclusion

= S2-20W-11S+11WS+10W2+10

Next, I will show how to find the formula for any Rectangle.

Finding the Formula Which Can Be Used On Any Rectangle

S+10L-10xS+(W-1)=

S2+10LS-10S(W-1)S+10(L1)(W-1)

SxS+10(L-1)+(W-1)=

S2+10LS-10S(W-1)S+10(L-1)(W-1)-(S2+10LS-10S+(W-1)S)

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

I have now found the general rule to work out the formula for any rectangle on a 10x10 grid. The rule is 10(L-1)(W-1).

Expanding the Brackets

s(s+11)

S | +11 | |

S | S2 | +11 |

s2+11s

(s+10)(s+1)

s | +10 | |

s | s2 | +10s |

+1 | +s | +10 |

s2+11s+10-(s2+11s)

s2+11s+10-(s2-11s)

=+10

Algebra

2x2 Box

s | s +1 |

s+10 | s+10+1 |

s(s+11)(s+1)-s(s+11)

3x3 Box

s | s+1 | S+1+1 |

S+10 | s+10+1 | S+10+1+1 |

s+10+10 | s+10+10+1 | S+10+10+1+1 |

s(s+10+10+1+1)-(s+1+1)

(s+10+10)

s+20

4x4 Box

S | S+1 | S+1+1 | S+1+1+1 |

S+10 | S+10+1 | S+10+1+1 | S+10+1+1+1 |

S+10+10 | S+10+10+1 | S+10+10+1+1 | S+10+10+1+1+1 |

s+10+10+10 | s+10+10+10+1 | s+10+10+10+1+1 | S+10+10+10+1+1+1 |

s(s+33)-(s+30)(s+3)

5x5 Box

s | s+1 | s+1+1 | s+1+1+1 | s+1+1+1+1 |

s+10 | S+10+1 | S+10+1+1 | S+10+1+1+1 | S+10+1+1+1+1 |

S+10+10 | S+10+10+1 | S+10+10+1+1 | S+10+10+1+1+1 | S+10+10+1+1+1+1 |

S+10+10+10 | S+10+10+10+1 | S+10+10+10+1+1 | S+10+10+10+1+1+1 | S+10+10+10+1+1+1+1 |

s+10+10+10+10 | s+10+10+10+10+1 | s+10+10+10+10+1+1 | s+10+10+10+10+1+1+1 | s+10+10+10+10+1+1+1+1 |

s(s+44)-(s+40)(s+4)

DIFFERENCE IS 10 WHEN YOU GO DOWN A BOX.

Shiva Dhunna

11S/Ma2

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