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Introduction

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Number Grids Investigation Coursework

In this coursework, I am going to investigate patterns in number grids as far as I can.

It will be based around this number 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 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

There is a 2 x 2 square highlighted:

 13 14 23 24

I shall work out the difference between the products of opposite corners, i.e. this is the calculation I will use:

(top right x bottom left) – (top left x bottom right)

For example in the square highlighted the difference between the products in the opposite corners is:

14 x 23 – 13 x 24 = 10

2 x 2 Squares

The above shows that, in one particular square, the difference between the products of the opposite corners is 10, so I shall now investigate some other examples to see if this is consistent in all 2 x 2 grids.

 1 2 11 12

(top right x bottom left) – (top left x bottom right)

= 2 x 11 – 1 x 12

= 22 – 12

= 10

 67 68 77 78

(top right x bottom left) – (top left x bottom right)

= 68 x 77 – 67 x 78

= 5236 – 5226

= 10

 49 50 59 60

(top right x bottom left) – (top left x bottom right)

= 50 x 59 – 49 x 60

= 2950 – 2940

= 10

The examples above are consistent with the original example. I shall now use algebra to try and prove that this is the case for all 2 x 2 squares in this grid:

Let the top left number equal a, and therefore;

 a a+1 a+10 a+11

Therefore, if I put this into the calculation I have been using, the difference between the products of opposite corners would be:

(top right x bottom left) – (top left x bottom right)

= (a + 1) (a + 10) – a (a + 11)

= a2 + 10a + a +10 – a2 – 11a

= a2 + 11a +10 – a2 – 11a

= (a2 – a2) + (11a – 11a) + 10

= 10

So I have proved that, in a 2 x 2 square, the difference between the products of the opposite corners will always equal 10 because the expression will cancel down to 10.

3 x 3 Squares

Middle

76

77

78

79

80

81

31

32

40

41

Taking the highlighted 2 x 2 square, I will work out the difference between the products of opposite corners of the grid in the same way as I have been doing.

(top right x bottom left)–(top left x bottom right)

= 32 x 40 – 31 x 41

= 1280 – 1271

= 9

I will do another example of this to see if the difference between the products of opposite corners still equals 9.

 60 61 69 70

(top right x bottom left) – (top left x bottom right)

= 61 x 69 – 60 x 70

= 4209 – 4200

= 9

As it is 9 once more in this example, I will now use algebra to try and prove that 9 is the difference between the products of opposite corners in all 2 x 2 squares within 9 x 9 grids, by substituting letters in for the numbers.

Let the top left number in a square equal a, and therefore:

 a a+1 a+9 a+10

So I can work out the difference between the products of opposite corners by:

(top right x bottom left) – (top left x bottom right)

= (a + 1) (a + 9) – a (a + 10)

= a2 + a + 9a + 9 – a2 – 10a

= a2 + 10a + 9 – a2 – 10a

= (a2 – a2) + (10a – 10a) + 9

= 9

This algebra has thus proved the difference between the products of opposite corners in a 2 x 2 square in 9 x 9 grids will always equal 9.

I will go on to investigate the difference between the products of opposite corners of 3 x 3 squares inside 9 x 9 grids:

 15 16 17 24 25 26 33 34 35

(top right x bottom left) – (top left x bottom right)

= 17 x 33 – 15 x 35

= 561 – 525

= 36

I will do another example of this to see if the difference between the products of opposite corners is 36 again:

 47 48 49 56 57 58 65 66 67

(top right x bottom left) – (top left x bottom right)

= 49 x 65 – 47 x 67

= 3185 – 3149

= 36

I will prove this using algebra, to show that all 3 x 3 squares in 9 x 9 grids have a difference between the products of opposite corners of 36.

Let the top left number in a square equal a, and therefore:

 a a+1 a+2 a+9 a+10 a+11

Conclusion

= wp2 (m – 1) (n – 1)

As my formula for the difference between the products of opposite corners in a square with m and n lengths within a w sized grid with differences between the numbers of p was

D = wp2 (m – 1) (n – 1) and my algebraic expression for the difference between the products of opposite corners cancelled down to wp2 (m – 1) (n – 1), I have successfully proved that my formula will work in any rectangle in any grid.

Validity and Conclusion

Validity – This formula is only valid in grids in which the numbers will fit into the grid alignment, that is numbers that go up by consecutive amounts – because otherwise it would be impossible to have a difference between the numbers in the grid, which is p in the formula; therefore it would not be possible to complete the formula. The numbers in the grid do not have to be integers though, as long as the numbers go up consecutively, as this example shows:

D = wp2 (m – 1) (n – 1)                        (w = 10) (m = 2) (n = 3) (p = 0.5)

= 10 x 0.52 (2 – 1) (3 – 1)

= 10 x 0.25 x 1 x 2 = 5

If I then put this in an example of a grid and work out the difference between the products of opposite corners, I can check that the formula works in this example in which the numbers in the grid are not all integers:

 0.5 1 1.5 5.5 6 6.5

(top right x bottom left) – (top left x bottom right)

= 1.5 x 5.5 – 0.5 x 6.5

= 8.25 – 3.25

= 5

So my formula is valid with any numbers, integers or not, in the grid, so long as they go up by consecutive amounts

In conclusion, I have investigated the difference between the products of opposite corners in rectangles such that, from starting with the information that the difference between the products of opposite corners in a 2 x 2 square in a 10 x 10 number grid equals 10, I have come up with a formula with 4 variables for the difference between the products of opposite corners in rectangles within number grids that is valid in number grids that go up consecutively. This formula is: D = wp2 (m – 1) (n – 1)

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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