Number grids

16x34=544

544-504=40

45x67=3015

47x65=3055

3055-3015=40

The answer is 40 each time I have done it.

There looks as though there is a pattern going on here the answer will always be 40.

Now I am going to choose a general position of my square, the top right will be “M”.

Mx(M+22)= M2+22M

(M+2)x(M+20)= M2+22M+40

(M2+22M+40) – (M2+22M)= 40

This has proven that any 3 by 3 square on this number grid, I choose and take the products away the answer will always be 40.

Now I am going to investigate the same thing but with a 4 by 4 box as I think there might be a pattern to follow.

25x58=1450

28x55=1540

1540-1450=90

44x77=3388

47x74=3478

3478-3388=90

The answer is always 90

There looks as though there is a pattern going on here the answer will always be 90.

Now I am going to choose a general position of my square, the top right will be “M”.

Mx(M+33)= M2+33M

(M+3)x(M+33)= M2+33M+90

(M2+33M+90) – (M2+33M)= 90

This has proven that any 4 by 4 square on this number grid, I choose and take the products away the answer will always be 90.

This is the following which I have found out:

I have decided to work out a formula for this:

In this formula “N” will be the box size and “M” is the number of a square in the top right hand corner.

M                M+N-1

M+10N-10                M+10(N-1)+N-1

=M+10(N-1)        =M+11(N-1)

M x M+11(N-1)= M2+11(N-1)

M+10(N-1) x M+N-1 = M2+11(N-1)+10(N-1) 2

M2+11(N-1)+10(N-1) 2 - M2+11(N-1) = 10(N-1) 2

This proves that the following formula is correct and working for squared numbers.  10(N-1) 2

Now I have done this I am going to work out the formula when you use a number grid which is M x N.

This formula is simple to work out as 10(N-1) 2  is for squared numbers you can multiply this out so its 10((N-1) x (N-1)).

In this “N” is the grid size but we can change this so it is length and width instead of always being the same number.

So the formula is 10((L-1)(W-1))

To prove this…….

M                M+L-1

M+10W-10                M+10(W-1)+L-1

=M+10(W-1)        

M x M+10(W-1)+(L-1)= M2+10M(W-1)+M(L-1)

M+10(W-1) x M+L-1 = M2+10M(W-1)+M(L-1)+10((L-1)(W-1))

M2+10M(W-1)+M(L-1)+10((L-1)(W-1)) - M2+10M(W-1)+M(L-1)= 10((L-1)(W-1))

So this has been proven.

Now I have decided to change the size of the rows in the grid.

I am going to work out the formula which is for squared numbers. MxM

As before “M” will be the number in the top right.

“N” will be for the size of the box.

“S” will be for the size of one column in the grid.

 

M                M+N-1

M+S(N-1)                M+S(N-1)+N-1

        

M x M+S(N-1)+N-1= M2+SM(N-1)+M(N-1)

M+S(N-1) x M+N-1 = M2+SM(N-1)+M(N-1)+S(N-1) 2

M2+SM(N-1)+M(N-1)+S(N-1) 2 - M2+SM(N-1)+M(N-1)= S(N-1) 2

This proves that this formula will give you the differences of squared grid numbers when find the product of the top right and bottom left, and take the product of the top left and bottom right.  This will work for any grid size.

        Like before now I need to find the formula for which I can work out the differences of boxes (M x N) which are rectangles rather than squares.

M                M+L-1

M+S(W-1)                M+S(W-1)+L-1

M x M+S(W-1)+L-1= M2+SM(W-1)+M(L-1)

M+S(W-1) x M+L-1 = M2+SM(W-1)+M(L-1)+S((W-1)(L-1))

M2+SM(W-1)+M(L-1)+S((W-1)(L-1)) - M2+SM(W-1)+M(L-1) = S((W-1)(L-1))

So this proves that this formula will give you the differences of any box size (M x N) when find the product of the top right and bottom left, and take the product of the top left and bottom right.  This will work for any grid size.

Now I have completed the formulas for them I have decided that I want to find out what the differences will be when I change the step size of the grid.  So a grid could look like this……

To work out a square in this grid a simple formula would be simply:

As before “M” will be the number in the top right.

“N” will be for the size of the box.

“T” will be for the step size in this square it is “2”

M                M+TN-2

M+5(TN-2)                M+5(TN-2)+TN-2

M x M+6(N-2)= M2+6(N-2)

M+5(N-2) x M+N-2 = M2+6(N-2)+5(N-2) 2

M2+6(N-2)+5(N-2) 2 - M2+6(N-2) = 5(N-2) 2

This would be the formula to work out a square on this grid.

To work out a rectangle (MxN) on this grid the formula would be:

M                M+TL-2

M+5(TW-2)                M+5(TW-2)+TL-2

M x M+5(TW-2)+(TL-2)= M2+5(TW-2)+M(TW-2)

M+5(TW-2) x M+TL-2 = M2+5M(W-2)+M(TL-2)+5((TW-2)(TL-2))

M2+5M(W-2)+M(TL-2)+5((TW-2)(TL-2)) - M2+5(TW-2)+M(TW-2) = 5((TW-2)(TL-2))

This proves that for this grid this will give you differences to any rectangle (MxN).

        I now want to find out a general formula which would let me work out the difference.  The formula will work out the difference even if the all the following variables are changed:

• Step size
• Changing the length of the grid.
• Any size of rectangle (M x N)

This formula will be my general formula for working out all the variables with one formula.

As before “M” will be the number in the top right.

“L” is the length

“W” is the width

“T” will be for the step size

“S” will be for the size of one column in the grid.

M                M+(TL-T)

M+S(TW-T)                M+S(TW-T)+(TL-T)

M x M+S(TW-T)+(TL-T) = M2+SM(TW-T)+M(TL-T)

M+(TL-T) x M+S(TW-T) = M2+SM(TW-T)+M(TL-T)+S((TW-T)(TL-T))

M2+SM(TW-T)+M(TL-T)+S(TW-T)(TL-T) - M2+SM(TW-T)+M(TL-T) = S((TW-T)(TL-T))

This is the general rule for working out the differences when finding the product of the top right and bottom left, and take the product of the top left and bottom right from it. The formula is for all the variables which I have been working out through out this piece of work.

S((TW-T)(TL-T))

“M” will be the number in the top right.

“L” this is the length of the box

“W” this is the width of the box

“T” will be for the step size

“S” will be for the size of one column in the grid.

By SAAD ZAHID