Investigating the number of patterns in a certain grid.

COURSEWORK (PAGE 1)

Introduction

This piece of coursework is about investigating the number of patterns in a certain grid. The grid I am starting off with is 10x10. The rule or pattern I have to investigate is whether when you multiply the opposite corners in a 2x2 box, you can find a pattern or link between the two answers.

I will also investigate to see if there are any patterns when I multiply the opposite corners of a box but using a larger grid (increase the row length) or increase the size of the square (3x3etc.).

I will always use algebra to check if my assumptions are correct.

I will also see what happens if I do not use a square but use other shapes such as a rectangle or a T-shape. I will see whether the rule still applies and I will generalise the rule by using algebra.

Finally I will invent my own rule and if it is successful I will generalise using algebra

Here is a list, in order, of the plans of my investigation:

1 check if the rules are universal

Do the arithmetic

2 generalise using y         y+1

Do the algebra    y +7     y+8

3 change the size of the grid (e.g. row 1-10)

Do arithmetic

Do algebra (row 1-n)

4 change the size of the square y         y+1     y+2

Do the arithmetic                          y+7     y+8     y+9

Do the algebra                              y+14   y+15   y+16

5 generalise the square y………………y+r

Do the algebra (nxn)        y+5

Square

6 extensions

Rules for different shapes

Change pattern in grid

1 3 5 7 etc.

This is the number grid I am going to be investigating:

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

Example 1

1. 4

13   14

Example 2

27    28

37    38

Example 3

1. 53

62   63

The green boxes indicate the examples I have chosen and will be using in this investigation. I am going to find the product of the top left number and the bottom right number in each box and see if I can find a pattern or a general rule connecting them, when you calculate the difference between the two answers.

Example 1

3 x 14 =14                  difference =10

13 x 4 =52

Example 2

27 x 38 =1036            difference =10

37 x 28 =1036

Example 3

52 x 63 =3276            difference =10

62 x 53 =3286

From these examples I have discovered that the difference always appears to be 10.

Because I have only used three examples however, I need to prove that these are not just anomalous results by using algebra.

Y              y+1

Y+10        y+11

This is the generalised version of the 2x2 grids

Y(y+11)                    = y^2+11y

(y+1)(y+10)              = y^2+11y+10

After the two equations cancel each other out, the difference left is 10.

This proves that the rule is correct for 2x2 grids, and that the difference is always 10.

Now I will see what changes when 3x3 grids are used instead.

3    4   5                        27    28   29                       52   53   54

13  14 15                      37    38   39                       62   63   64

23  24 25                       47   48   49                   ...