Investigating the number of patterns in a certain grid.

3    4   5                        27    28   29                       52   53   54

13  14 15                      37    38   39                       62   63   64

23  24 25                       47   48   49                       72   73   74

3x25 =75                       27x49=1323                    52x74 =3848

5x23 =115                     29x47 =1363                   72x54=3888

difference =40               difference =40                 difference =40

These examples show that the difference seems to be 40 for 3x3 grids.

As before I will check my results with algebra to make sure they are correct.

Y          y+1      y+2

Y+10    y+11    y+12

Y+20    y+21    y+22

This is the generalised version of the 3x3 grids.

Y(y+22)          = y^2+ 22y

(y+2)(y+20)    = y^2 + 22y + 40

After the equations cancel each other out, the only difference left is 40.

This proves that for the 3x3 grids, the difference will always be 40.

I am now going to investigate any changes when 4x4 grids are used.

3    4    5    6                          27   28   29   30                     52   53   54   55

13  14  15  16                        37   38   39   40                     62   63   64   65

23  24  25  26                        47   48   49   50                     72   73   74   75

33  34  35  36                        57   58   59   60                     82   83   84   85

3x38 =108                             27x60 =1620                         52x85 =4420

6x33 =198                             57x30 =1710                         82x55 =4510

The difference in every case is +90.

These examples show that the difference seems to be 90 for every 4x4 grid.

As before I will check my results with algebra to make sure these were not just freak results.

Y          y+1       y+2       y+3

Y+10    y+11    y+12     y+13  This is the generalised version of the 4x4 grids

Y+20    y+21    y+22     y+23                            

Y+30    y+31    y+32     y+33

Y(y+33)         =y^2+33y

(y+3)(y+30)   =y^2+33y+90

After the equations cancel each other out, the only difference left is +90.

This proves that for the 4x4 grids, the difference will always be +90.

Tabulating my results:

Size:           Difference:

2x2             10 / 1x1x10

3x3             40 / 2x2x10

4x4             90 / 3x3x10

5x5             160 / 4x4x10

6x6             250 / 5x5x10

nxn             (n-n)^2 x10

This table shows the difference between the two results you get when you multiply the opposite corners, but it also shows the difference on different size squares. As you can see from the table there is a distinct pattern.

By recognising that the differences are the square numbers multiplied by 10 I was able to deduce what a 5x5 and 6x6 square would give.

If you want to work out a formula for any square (nxn) then you have to realise that it is the side length (n) minus 1 then you square it and finally multiply by 10. You are then left with the formula (n-n)^2 x10.

Now I will change the length and height of the grid to generalise the formula more. The grid is now 13 by 8.

1     2     3     4     5     6     7     8     9     10     11     12     13

14  

27

40

53

66

79

92

These are the examples I am going to use:

14   15                    44   45                  37   38

27   28                    57   58                  50   51

14x28 =392           44x58 =2552        37x51 =1887

15x27 =405            57x45 =2565       50x38 =1900

The difference is 13 every time.  

I am now going to use algebra to check my answer.

Y         y+1                 This is the generalised version of the 2x2 grids.

Y+13   y+14

Y(y+14)         =y +14y

(y+13)(y+1)   =y +14y+13

The algebra proves that the difference is 13.

I will now investigate 3x3 squares.

14    15    16                 44    45    46                37    38    39

27    28    29                 57    58    59                50    51    52

40    41    42                 70    71    72                63    64    65

14x42 =588                  44x72 =3168               37x65 =2405

16x40 =640                  70x46 =3220               39x63 =2457

Difference in every case =52

I am going to use algebra to check my answer.

Y           y+1        y+2                             y(y+28) =y +28y

Y+13     y+14      y+15                           (y+26)(y+2) =y +28y+52

Y+26     y+27      y+28                                                    

After the equations cancel each other out the only difference left is +52.

This proves that for 3x3 squares in the new grid the difference is 52.

I am now going to see what happens when a 4x4 square is used.

I have only used two examples this time as my third from my three original ones went off the end of the grid.

14    15    16    17              44    45    46    47

27    28    29    30              57    58    59    60

40    41    42    43              70    71    72    73

53    54    55    56              83    84    85    86

14x56 =784                       44x86 =3784

53x17 =901                       83x47 =3901

difference in both cases =117

I will use algebra to see if this is correct.

Y         y+1      y+2     y+3

Y+13   y+14   y+15   y+16

Y+26   y+27   y+28   y+29

Y+39   y+40   y+41   y+42

Y(y+42) =y +42

(y+39)(y+3)=y +42+117                  difference=+117

After the equations cancel each other out the only difference is 117.

This proves that for 4x4 squares the difference will always be 117.

This table displays my results:

Size of square:               Difference:               Size of square:             Difference:

(13x10)                                                           (10x10)

2x2                                 13                             2x2                               10

3x3                                 52                             3x3                               40

4x4                                 117                           4x4                               90

nxn                                 (n-1)^2x13               nxn                               (n-1)^2x10

I now believe I have sufficient examples to say that the difference is (n-1)^2 x R when the row length is R.

I will try to prove this with algebra.

R= Row length of the grid

                                                        N

              Y                                                                              y+(n-1)

N

             Y+(n-1)R                                                                  y+(n-1)R+(n-1)

Y x y + (n-1)R + (n-1) = y^2ynR – yR + yn-y

Y + (n-1)Rxy + (n-1) = y^2 + yn – y + ynR – yR + (n-1)^2R

Difference =(n-1)^2R

Testing the formula:

Y=3  n=3   R=3

3 x 21 =63                              Difference =32

5 x 19 =95            

3 – 1 =2 x 2 =4 x 8 =32

The formula works.  

To further investigate I will now test rectangles and see if the rule still applies and if the formula is changed or stays the same. I will check rectangle shapes on the original grid (10x10). As I have already discovered it does not make a difference to the formula what the row length is so I have continued to use the 10x10 grid.

2x3 examples:

1. 2                         27    28                     52    53

11   12                       37    38                     62    63

21   22                       47    48                     72    73

1x22 =22                   27x48 =1296           52x73 =3796

21x2 =42                   47x28 =1316           72x53 =3816

Difference =20 every time.

I will now test some 2x4 rectangles to see if a pattern emerges.

2x4 examples:

2     3                      17    18

12   13                    27    28

22   23                    37    38

32   33                    47    48

2x33 =66                17x48 =816

32x3 =96                47x18 =846

Difference =30 both times

I will test some 2x5 rectangles.

2x5 examples:

15     16                 45     46

25     26                 55     56

35     36                 65     66

45     46                 75     76

55     56                 85     86

15x56 =840          45x86 =3870

55x16 =880          85x46 =3910

Difference =40 every time.

I  have noticed that the differences seem to go up by 10 every time so I am now going to use algebra to see if there is a formula using dimensions N,M in a grid length R.

                                                        N

      Y                                                -                                       Y+(N-1)

        -                                                -                                            -

       Y+(M-1)R                                 -                                        Y+(M-1)R+(N-1)

M            

R =Row length of grid.

YxY+(M-1)R+(N-1) = Y^2 +MRY+RY+YN-Y

(Y+MR-R)x(Y+N-1) =Y^2 +MRY+NY+RMN-Y-RM+R

Difference =RMN-RN-RM+R =R(MN-N-M+1) =R(M-1)(N-1)

This formula R(M-1)(N-1) will give you the difference between the difference of the two products when they are found in this way for any size rectangle in any size grid, when N and M are the dimensions of the rectangle and R is the row length of the overall grid.

I can now display all of results for rectangles:

Size of rectangle:                   Difference:

2x3                                         20 =1x2x10

2x4                                         30 =1x3x10

2x5                                         40 =1x4x10

NxM                                       R(M-1)(N-1)

I have now completed the original problem set in this algebra investigation and I have generalised as far as possible and have found different rules connecting my results.

I am now going to do some extension work to see if I can find any rules for different shapes or when my original grid goes up by different numbers.

2       4        6        8       10      12     14     16     18     20

22     24     26      28      30      32     34     36     38     40

42     44     46      48      50      52     54     56     58     60

62     64     66      68      70      72     74     76     78     80

82     84     86      88      90      92     94     96     98     100

102   104   106    108    110    112   114   116   118   120

122   124   126    128    130    132   134   136   138   140

142   144   146    148    150    152   154   156   158   160

162   164   166    168    170    172   174   176   178   180

182   184   186    188    190    192   194   196   198   200

This new grid I have chosen to investigate. It is similar to the other grid except it goes up by two every time.

I will test the original rule and attempt to achieve similar results. My examples include one 2x2, one 3x3 and one 4x4 square.

2x24 =48                    30x74 =2220                  124x190 =23560

22x4 =88                    70x34 =2380                  184x130 =23920

difference =40           difference =160               difference =360

I will now use algebra to see if there is a connection.

y            y+2

y+2R     y+2R+2

yxy+2R+2       =y^2 +2Ry +2y

y+2r x y+2      =y^2 2Ry+4R+2y

Difference =4R

R=10           (4R=40)

Y           y+4

Y+4R    y+4R+4

Yxy+4R+4     =y^2 4Ry +4y

Y+4R x y+4   =y^2 4Ry +16R+4y

Difference =16R

R=10           (16R=160)

Y                  y+6

Y+6R           y+6R+6

Y x y+6R+6      =y^2+6Ry+6y

Y+6R x y +6     =y^2+6Ry+6y+36R

Difference =36R

R =10          (36R =360)

My results in a table:

Size of square:                                        Difference:(when the numbers go up by two and R =row length of grid)

2x2                                                          4R

3x3                                                          16R

4x4                                                          36R

NxN                                                         R((N-1)^2 x4)

 

I will now attempt to find a formula for any rectangle where the numbers go up by two.

                                               N

              Y                               -                         y+2(N-1)

M           -                                -                             -

              Y+2(M-1)R              -                          y+2(M-1)R+2(N-1)

Y x y+(2(M-1)R+3(N-1))         =y^2 +2MRY-2RY+2YN-2Y

(y+2MR-2R) x (y+2N-2)          =y^2 +2MRY+2NY-2RY-2Y+4RMN-4RM-4RN+4R

Difference =4RMN-4RN-4RM+4R =R(MN-N-M+1)x4 =4(R(M-1)(N-1))

The number two represents how much the numbers go up each time. It van be replaced by the letter H IN the formula to give the complete formula to the problem as:                    H^2(R(M-1)(N-1))

K =number the numbers go up by each time

R =row length of the total grid

M+N =dimensions of the rectangle inside the grid

I will test this formula:

K =3                 R=5               M=2             N=3

3                             6

18                           21

33                           36

                                     

Because K was 3 that meant the numbers went up by three every time.

3x36 =108

6x33 =198                             difference =90

K^2(R(M-1)(N-1)) =9(5(1)(2)) = 9x10, =90

The formula appears to work.

I have now completed the investigation trying as many different things as possible and ending up with a final formula using numerical examples and algebra throughout to check what I have done. Of course after I have generalised for any square or rectangle it is only relevant to my peticular grid. That is why at one stage I used 3 examples instead of two as my third example went off the grid as I testes 4x4 squares.

I believe I have carried out a successful investigation.