Number grids. In this investigation I have been attempting to work out a formula that will find the difference between the products of the top left and bottom right of a number grid and the top right and bottom left of a number grid.

35x46=1610

                        Difference=10

36x45=1620

The difference equals 10. So what would happen if you tried the same thing with a 3x3 box?

  37x19=703

                            Difference=40

  17x39=663

The difference equals 40. So if you try it with a 4x4 box…

61x94=5734

             Difference=90

91x64=5824

The difference equals 90. Lets just try one more:

55x99=5445

            Difference=160

95x59=5605

We now have enough results.

A 9x9 number grid

If you choose any 2x2 box on a 9x9 number grid then the difference should equal 9…

30x40=1200

                            Difference=9

39x31=1209

The difference equals 9. So what would happen if you tried the same thing with a 3x3 box?

 24x44=1056

                          Difference=36

 42x26=1092

 

The difference equals 36. So if you try it with a 4x4 box…

46x76=3496

            Difference=81

73x49=3577

The difference equals 81. Lets just try one more:

41x81=3321

             Difference=144

77x45=3465

These figures now give me enough

information to work out the formula:

                          9(S-1)2

To take this investigation a step further I decided that I needed to work out the algebraic expression. To do this I needed to each corner as a letter instead of a number. Then I should add up the diagonal values, expand the result and then find the answer.

For a 2x2 box

        

We then repeat this for a 3x3 box.

For a 3x3 box

        N(N+20)=N2+20N

        (N+2)(N+18)=N2+20N+36

        As before the difference equals                              

        20.

From these results we can work out that, if S is the size of the square, then these are the results.

For a SxS box

        N[N+10(S-1)]=N2+10(S-1)N

        [N+(S-1)][N+9(S-1)]

                                             

                                             The difference as an expression                                                                                                          

        would be:

                                             

                                              =N2+9(S-1)N+(S-1)N+9(S-1)2

        =N2+10(S-1)+9(S-1)2

        

                                             Difference=9(S-1)2

N[N+(G+1)x(S-1)=N2+(G+1)x(S-1)N

        

[N+(S-1)][N+G(S-1)]

=N2+(S-1)G(S-1)+NG(S-1)+N(S-1)

=N2+  (S-1)2   +N(S-1)(9+1)

 When G = the grid size and S = square size

The formula is G(S-1)2

Rectangle

This rectangle is of length L and of width W. For a grid of 10x10.

N[N+(L-1)+10(W-1)]=N2+(L-1)N+10(W-1)N

[N+(L-1)+10(W-1)] [N+(L-1)]

N2+10(W-1)N+(L-1)N+

Difference=10(L-1)(W-1)

Rectangular grid= LxW

Difference =G(L-1)(W-1)

Finally we can work out how the change of multiples in the grid would affect the formula.

In this formula L=Length, W=width, M=multiple and G=grid size.

N[N+M(L-1)+MG(W-1)]

=N2+N(l-1)M+NMG(w-1)N

[N+M(L-1)][N+GM(W-1)]

N2+NGM(W-1)+NM(L-1)+

This formula is the final part of this investigation. I cannot think of another way to extend my investigation. This investigation has enlightened me to the real-life ways in which math’s can be applied. Previously I was not aware of how complicated and interesting number grids can be!