Number Grids Investigation Coursework

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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:

There is a 2 x 2 square highlighted:

                                                         

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.

           

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

        = 2 x 11 – 1 x 12

        = 22 – 12

        = 10

           

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

        = 68 x 77 – 67 x 78

        = 5236 – 5226

        = 10

           

           (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;

           

        

        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

        Now that I have proved that the difference between the products of the opposite corners will always equal 10 in 2 x 2 squares in this grid, I shall investigate the difference between the products of the opposite corners in 3 x 3 squares:

           

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

        = 15 x 33 – 13 x 35

        = 495 – 455

        = 40

So in this square, the difference between the products of the opposite corners equals 40. I will now try some more examples of this to make sure this is the case in all 3 x 3 squares in this number grid.

           

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

        = 77 x 95 – 75 x 97

        = 7315 – 7275

        = 40

           

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

        = 20 x 38 – 18 x 40

        = 760 – 720

        = 40

        So I can see that in these examples of 3 x 3 squares, the difference between the products of opposite corners equals 40. I will now use algebra to prove that this is the case for all 3 x 3 squares:

        Let the top left number equal a, and therefore;

           

        

        

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 + 2) (a + 20) – a (a + 22)

                = a2 + 20a + 2a + 40 – a2 – 22a

                = a2 + 22a + 40 – a2 – 22a

                = (a2 – a2) + (22a – 22a) + 40

                = 40

Therefore I have proved that the difference between the products of opposite corners in all 3 x 3 squares equals 40 because the algebraic expression for the difference between the products of opposite corners will cancel down to 40.

4 x 4 Squares

Now I will do the same thing with 4 x 4 squares, to see if I can find a pattern between the size of the grid and the difference between the products of opposite corners.

           

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

        = 16 x 43 – 13 x 46

        = 688 – 598

        = 90

        In this example, the difference between the products of opposite corners equals 90, so I will work out the difference between the products of opposite corners of some more 4 x 4 squares, to see if this is the same in all 4 x 4 squares in this grid.

           

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

        = 10 x 37 – 7 x 40

        = 370 – 280

        = 90

           

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

        = 66 x 93 – 63 x 96

        = 6138 – 6048

        = 90

        In the above examples the difference between the products of opposite corners equals 90, so I will formulate an algebraic expression for the difference between the products of opposite corners in a 4 x 4 square to prove that the difference between the products of opposite corners in all 4 x 4 squares equals 90:

        Let the top left number equal a, and therefore;

           

        

        

        So the algebraic expression for the difference between the products of opposite corners would be:

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

                = (a + 3) (a + 30) – a (a + 33)

                = a2 + 30a + 3a + 90 – a2 – 33a

                = a2 + 33a + 90 – a2 – 33a

                = (a2 – a2) + (33a – 33a) + 90

                = 90

        Therefore I have proved that the difference between the products of opposite corners in 4 x 4 squares must always equal 90 because the algebraic expression for the difference between the products of opposite corners in 4 x 4 squares will simplify to 90.

Table

        Now I will put my results for the difference between the products of opposite corners in different sized squares in a table, to see if there is a pattern between the 2 x 2, 3 x 3 and 4 x 4 squares.

        So from this table it is clear that the difference between the products of opposite corners is increasing by one square number for every increase of 1 in the length and width of the square, multiplied by 10.

        I can therefore predict that a 5 x 5 square will have a difference between the products of opposite corners of 160, as that is the next square number multiplied by 10 after 90:

Join now!

           

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

        = 36 x 72 – 32 x 76

        = 2592 – 2432

        = 160

        So I successfully managed to predict the difference between the products of opposite corners in a 5 x 5 square, and therefore I was correct that the difference between the products of opposite corners goes up in square numbers multiplied by 10 for ever increase in size of the grid

Formula

        If I go back to my table, I can work ...

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