# opposite corners

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Introduction

Maths Coursework:

Opposite Corners

Opposite Corners:

Introduction:

My algebra coursework is about opposite corners in a square in a number grid. The top right and the bottom left numbers are multiplied and the same is done with the top left and bottom right numbers. The difference is calculated between the 2 products and the answer is used to find a pattern. The size of the square will be changed, 2x2, 3x3 and 4x4, to see whether the answers left will help to determine whether or not there is a pattern.

10x10 Grid:

I am starting off by using a 10x10 and within this grid I will outline 2x2 squares, 3x3 squares and 4x4 squares. With these squares I will work out the opposite corners in order to see whether or not there is a pattern.

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 |

2x2 Squares:

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 |

This is the smallest of the squares I will be working out the differences to. I am multiplying the top left number by the bottom right and multiplying the top right by the bottom left in order to find the differences of the two products.

12 | 13 |

22 | 23 |

12x23= 276

13x22= 286

Difference= 286- 276= 10

I am doing more than one example in order to check the accuracy of my work, if my work is not accurate the pattern will not be and I will therefore not be able to find a formula.

87 | 88 |

97 | 98 |

87x98= 8526

88x97= 8536

Difference= 8536 – 8526= 10

47 | 48 |

57 | 58 |

47x48= 2726

57x48= 2736

Difference= 2736 – 2726= 10

So far my results have all been 10; this is showing that there may be a pattern. However I will have to work out the differences for other sized squares in order to see whether there is a pattern for a 10x10 grid.

3X3 Squares:

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 |

Middle

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4 | 5 | 6 |

16 | 17 | 18 |

28 | 29 | 30 |

4x30= 120

6x28= 168

Difference=168 – 120=48

99 | 100 | 101 |

111 | 112 | 113 |

123 | 124 | 125 |

99x125= 12375

101x123= 12423

Difference= 12423 – 12375= 48

56 | 57 | 58 |

68 | 69 | 70 |

80 | 81 | 82 |

56x82= 4592

58x80= 4640

Difference= 4640 – 4592= 48

I have found that my theory for all 3x3 squares having a difference of 2² multiplied by the grid size is correct as all my results are the same and are in fact 2²(4) x 12.

4x4 Squares:

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 | 59 | 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 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 |

109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 |

121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 | 130 | 131 | 132 |

133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 |

18 | 19 | 20 | 21 |

30 | 31 | 32 | 33 |

42 | 43 | 44 | 45 |

54 | 55 | 56 | 57 |

18x57= 1026

21x54= 1134

Difference= 1026 – 1134= 108

81 | 82 | 83 | 84 |

93 | 94 | 95 | 96 |

105 | 106 | 107 | 108 |

117 | 118 | 119 | 120 |

81x120= 9720

84x117= 9828

Difference= 9828 – 9720= 108

86 | 87 | 88 | 89 |

98 | 99 | 100 | 101 |

110 | 111 | 112 | 113 |

122 | 123 | 124 | 125 |

86x125= 10750

89x122= 10858

Difference= 10858 – 10750= 108

Again, all the results are the same and they coincide with the pattern 3² x size of the grid, 9 multiplied by 12 equals 108, the answer I got with every difference.

Table of Results:

Square size (s) | Difference (d) | Pattern | |

2x2 | 12 | 1x12 | 1²x12 |

3x3 | 48 | 4x12 | 2²x12 |

4x4 | 108 | 9x12 | 3²x12 |

[5x5 | 192 | 16x12 | 4²x12] |

[Prediction]

The pattern is the same as before, the same square numbers (1, 4 and 9...) are being multiplied by the size of the grid, in this case 12. If I were to do a 6x6 grid the differences would be half of the results I acquired here.

Prediction:

I predicted that the 5x5 square would have a difference of 192 by multiplying 16 (the next square number up from 9) by 12, the size of the grid. I did this because it follows on from my pattern before, the square numbers multiplied by the size of the grid.

I will now do an example to prove that my prediction of 192 is correct.

19 | 20 | 21 | 22 | 23 |

31 | 32 | 33 | 34 | 35 |

43 | 44 | 45 | 46 | 47 |

55 | 56 | 57 | 58 | 59 |

67 | 68 | 69 | 70 | 71 |

79 | 80 | 81 | 82 | 83 |

91 | 92 | 93 | 94 | 95 |

103 | 104 | 105 | 106 | 107 |

19 | 20 | 21 | 22 | 23 |

31 | 32 | 33 | 34 | 35 |

43 | 44 | 45 | 46 | 47 |

55 | 56 | 57 | 58 | 59 |

67 | 68 | 69 | 70 | 71 |

19x71= 1349

23x67= 1541

Difference= 1541 – 1349= 192

My prediction was correct. I knew the answer would be 192 as I followed the same pattern with all of the squares inside the grids, difference= the next square number x grid size.

Algebra:

Although I have found a pattern with all of the square sizes and I am sure it is correct, I need to use algebra to prove that it is correct. I will still be using the same size squares within the grid, 2x2, 3x3, and 4x4, and I will still be multiplying the top left by the bottom right and doing the same with the top right and bottom left and finding the difference. The only difference with what I will be doing is I am using letters instead of numbers to prove that our working is right.

10x10 Grids:

If I take a portion of a 10x10 grid:

14 | 15 | 16 | 17 | 18 | 19 |

24 | 25 | 26 | 27 | 28 | 29 |

34 | 35 | 36 | 37 | 38 | 39 |

44 | 45 | 46 | 47 | 48 | 49 |

54 | 55 | 56 | 57 | 58 | 59 |

64 | 65 | 66 | 67 | 68 | 69 |

74 | 75 | 76 | 77 | 78 | 79 |

And I draw a 4x4 square within this grid and work out the opposite corners.

36 | 37 | 38 | 39 |

46 | 47 | 48 | 49 |

56 | 57 | 58 | 59 |

66 | 67 | 68 | 69 |

36x69= 2484

39x66= 2574

Difference= 2574 – 2484= 90

I found 90 in the previous results but to prove that the theory is correct I can use algebra.

X | X+1 | X+2 | X+3 |

X+10 | X+11 | X+12 | X+13 |

X+20 | X+21 | X+22 | X+23 |

X+30 | X+31 | X+32 | X+23 |

X x X+33= X(X+33)= X²+33X

X+3 x X+30= (X+3)(X+30)= X²+33X+90

F= XxX= X²

O= Xx30= 30X

I= 3xX= 3X

L= 3x30= 90

[30x+3X= 33X]

Difference= X²+33X – X²+33X+90= 90

This proves that the difference of the opposite corners multiplied of4x4 square in a 10x10 grid is definitely 90.

To check the accuracy of my theory I will do two more examples using a 2x2 square and a 3x3 square,

2x2 Squares:

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 |

6 | 7 |

16 | 17 |

6x17= 102

16x7= 112

Difference= 112 – 102= 10

X | X+1 |

X+10 | X+11 |

XxX+11= X(X+11) = X²+11X

X+1xX+10= (X+10)(X+1)= X²+11X+10

F= XxX= X²

O= Xx1= 1X

I= 10xX= 10X

L= 1x10= 10

[1X+10X= 11X]

Difference= X²+11X – X²+11X+10= 10

This proves that in a 10x10 grid, a 2x2 square will all have opposite corners with differences of 10 when multiplied together.

3x3 Squares:

...read more.Conclusion

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 |

X | X+1 | X+2 |

X+10 | X+11 | X+12 |

And still have a difference of 20 when the opposite corners were multiplied, like so:

XxX+12= X(X+12)= X²+12X

X+2xX+10= (X+2)(X+10)= X²+12X+20X

F= XxX= X²

O= Xx10=10X

I= 2xX= 2X

L= 2x10= 20X

[10X+2X=12X]

Difference= X²+12X – X²+12X+20= 20

And a 4x3 rectangle would look like this and still have a difference of 60:

X | X+1 | X+2 | X+3 |

X+10 | X+11 | X+12 | X+13 |

X+20 | X+21 | X+22 | X+23 |

XxX+23= X(X+23)= X²+23X

X+3xX+20= (X+3)(X+20)= X²+23X+60

F= XxX= X²

O= Xx20= 20X

I= 3xX= 3X

L= 3x20= 60

[20X+3X=23X]

Difference= X²+23X – X²+23X+60= 60

And a 5x4 rectangle could not contain the numbers, but algebra and the opposite corners will still have a difference of 120 when multiplied together:

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 |

XxX+34= X(X+34)= X²+34X

X+4xX+30= (X+4)(X+30)= X²+34X+120

F= XxX= X²

O= Xx30=30X

I= 4xX= 4X

L= 30x4= 120

[30X+4X+34X]

Difference= X²+34X – X²+34X+120= 120

I also managed to find a pattern, during this miniature investigation. Although the differences were not as concise as the differences for the squares, I managed to find a pattern using the rectangles in a 10x10 grid.

The pattern I have found is that as the number of rows in the rectangle increases the multiplier increases accordingly. I assume the multiplier increases accordingly with different sized grids.

Also I found that as the multiplier increases you can find the pattern by taking away the lower multiplier from the higher one and you are left with a pattern. So…

- As the first multiplier of 20 is one you are unable to subtract that from anything and the next multiplier is 3 so take 1 from that you are left with 2. The next multiplier up is 6, subtract 3 from that and you are left with 3. A row of consecutive numbers like so:

1

3-1= 2

6-3= 3

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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## Here's what a teacher thought of this essay

This is a good investigation into the difference between the products of opposite corners in a grid. It is well structured and supported by algebra. There are good patterns identified. It is limited however by the inaccurate nature of the algebra to find the difference. Terms have been written in the incorrect order and without brackets. With errors in the algebra this report can achieve no more than three stars.

Marked by teacher Cornelia Bruce 18/04/2013