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Introduction

Tom Gowing 11A GCSE Maths Coursework

## Investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a 100 square.

During the investigation my aim is to find a formula to work out the difference of these products for any size rectangle on any size grid.

I will start off by working out the difference on different shapes and looking for a basic pattern. I will try moving the shapes around the grid to see how this affects the difference.

I am going to use the following 5 shapes during this part of the investigation:

A 2x2 square

A 3x3 square

A 2x3 rectangle

A 3x5 rectangle

## A 2x2 Square

 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

For the 2x2 I have decided to try 3 different positions along the horizontal,

3 along the vertical and 3 along the diagonal this should tell me if the difference alters depending on which way you move the shape around the grid. I have done 3 along each axis to provide accurate results by highlighting any anomalous ones. The only way to get perfectly accurate results would be to test every position the shape could be on the grid, but this is a waste of time when you can take 3 measurements from each axis and see any anomalous results. I will also do one test at a random site on the grid to make sure that results are as close to perfect as possible.

I predict that the difference will increase as the shape moves across or down the grid.

 1 2 11 12

1 x 12 = 12

2 x 11 = 22

22 – 12 = 10

 5 6 15 16

5 x 16 = 80

5 x 15 = 90

90 – 80 = 10

 9 10 19 20

9 x 20 = 180

10 x 19 = 190

190 – 180 = 10

From this I can see that for a 2x2 square on a 100 grid the difference is equal no matter where the shape is on the horizontal axis. This disproves my theory.

Middle

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For the 2x5 tests I predict that the difference will remain the same for the shape in any position on the grid.

 1 2 3 4 5 11 12 13 14 15

1 x 15 = 15

5 x 11 = 55

55 – 15 = 40

 6 7 8 9 10 16 17 18 19 20

6 x 20 = 120

10 x 16 = 160

160 – 120 = 40

This proves that for a 2x5 the difference remains the same when you move it across the grid in a horizontal direction.

I will now test this vertically.

 1 2 3 4 5 11 12 13 14 15

1 x 15 = 15

5 x 11 = 55

55 – 15 = 40

 81 82 83 84 85 91 92 93 94 95

81 x 95 = 7695

85 x 91 = 7735

7735 – 7695 = 40

This proves that for a 2x5 the difference remains the same when you move it down the grid in a vertical direction.

I will now test this diagonally.

 1 2 3 4 5 11 12 13 14 15

1 x 15 = 15

5 x 11 = 55

55 – 15 = 40

 34 35 36 37 38 44 45 46 47 48

34 x 48 = 1632

38 x 44 = 1672

1672 – 1632 = 40

This proves that for a 2x5 the difference remains the same when you move it across the grid in a diagonal direction.

I will now test a random shape to ensure accurate results.

 14 15 16 17 18 24 25 26 27 28

14 x 28 = 392

18 x 24 = 432

432 – 392 = 40

This confirms my predictions that no matter where a 2x5 rectangle is on the grid the differences are always the same. It is starting to look like all identical rectangles have the same difference no matter where they are positioned, as with the squares. I will do one more test to confirm this, a 5x3.

 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

For the 3x5 tests I predict that the difference will remain the same for the shape in any position on the grid.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25

1 x 25 = 25

5 x 21 = 105

105 – 25 = 80

 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30

6 x 30 = 180

10 x 26 = 260

260 – 180 = 80

This proves that for a 3x5 the difference remains the same when you move it across the grid in a horizontal direction.

I will now test this vertically.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25

1 x 25 = 25

5 x 21 = 105

105 – 25 = 80

 71 72 73 74 75 81 82 83 84 85 91 92 93 94 95

71 x 95 = 6745

75 x 91 = 6825

6825 – 6745 = 80

This proves that for a 3x5 the difference remains the same when you move it down the grid in a vertical direction.

I will now test this diagonally.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25

1 x 25 = 25

5 x 21 = 105

105 – 25 = 80

 45 46 47 48 49 55 56 57 58 59 65 66 67 68 69

45 x 69 = 3105

49 x 65 = 3185

3185 – 3105 = 80

This proves that for a 3x5 the difference remains the same when you move it across the grid in a diagonal direction.

I will now test a random shape to ensure accurate results.

 13 14 15 16 17 23 24 25 26 27 33 34 35 36 37

13 x 37 = 481

17 x 33 = 561

561 – 481 = 80

From all my tests so far I can deduce that as long as the rectangle is the same size and on the same angle that no matter where it is on the grid, the difference of the products is always the same.

But does the difference remain the same if the shape is rotated 90 degrees? Technically speaking this is a completely different shape e.g. a 2x3 is not the same shape as a 3x2, but it would be interesting to see if the differences are the same. Obviously this will work for squares because they are the same shape no matter what angle they are on, so I will try rotating the last 3 rectangles that I tested, a 2x3, a 2x5 and a 3x5. These will therefore become a 3x2, a 5x2 and a 5x3. From my previous tests I know that it doesn’t matter where I position them on the grid and I only need to do 1 for each test.

 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

For all of these shapes I would expect the differences to be different from the non-rotated ones.

 1 2 11 12 21 22

1 x 22 = 22

2 x 21 = 42

42 –22 = 20

This is the same difference as found on the 2x3.

 24 25 26 34 35 36 44 45 46 54 55 56 64 65 66

24 x 66 = 1584

26 x 64 = 1664

1664 – 1584 = 80

This is the same difference as found on the 3x5.

 58 59 68 69 78 79 88 89 98 99

Conclusion

But what happens on a rectangular grid? I expect this to need a different formula, because the X and Y-axes are different, as they were when I changed to square shapes to rectangle.

I will now try the formula out on 2 different sized rectangular grids, a 4x5 and a 9x7. For each of these I will test with 3 different shapes to prove my argument and to show up any anomalous results.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 1 2 5 6

1 x 6 = 6

2 x 5 = 10

10 – 6 = 4

 7 8 11 12 15 16

7 x 16 = 112

8 x 15 = 120

120 – 112 = 8

 13 14 15 16 17 18 19 20

13 x 20 = 260

16 x 17 = 272

272 – 260 = 12

The formula for this should be 4(χ-1)(γ-1)

The difference for the 2x2 should be 4

4(2-1)(2-1) = 4

This is correct

The difference for the 3x2 should be 8

4(3-1)(2-1) = 8

This is correct

The difference for the 2x4 should be 12

4(2-1)(4-1) = 12

This is correct

This has disproved my first theory about needing to change the formula for a rectangle because it has a different sized X-axis than Y-axis.

It has told me that on a 4x5 grid that you simply replace the number at the beginning of the formula with the number of squares going across the top of the grid (the x-axis of the grid).

I will now test this on a 7x9 grid to prove this, and to make sure that it is definitely the X-axis that needs inserting into the formula.

 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
 1 2 8 9

1 x 9 = 9

2 x 8 = 16

16 – 9 = 7

 17 18 19 20 21 24 25 26 27 28

17 x 28 = 476

21 x 24 = 504

504 – 476 = 28

 50 51 52 53 54 55 56 57 58 59 60 61 62 63

50 x 63 = 3150

56 x 57 = 3192

3192 – 3150 = 42

If my theory is correct the formula for these should be 7(χ-1)(γ-1).

The difference for the 2x2 should be 7

7(2-1)(2-1) = 7

This is correct

The difference for the 2x5 should be 28

7(2-1)(5-1) = 28

This is correct

The difference for the 2x7 should be 42

7(2-1)(7-1) = 42

This is correct

This has proved my theory.

The final formula, for finding the difference between the products of the numbers in the opposite corners of any rectangle, on any size grid is:

α(χ - 1) (γ - 1)

Where: α = The horizontal width (χ-axis) of the grid

χ = The horizontal width (χ-axis) of the shape

γ = The vertical depth (γ-axis) of the shape

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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