He analysis of number patterns on various types of number grids.

        

        

        

From the above results we can make an assumption that a formula can be made which will allow us to find the answer to other sizes of boxes in other larger number grids. At first we need to carry out the same calculations for even larger number grids.

        

The method of obtaining the formula for the number grids problem can be found out as follows.

The number grids as drawn above are made out series of whole numbers, which are arranged in a square grid, called the grid of numbers. The grid has the same number of whole numbers on each row as on each column, so making it a square grid of numbers. Therefore we call this the N by N grid, where N stands for the size of the square grid of numbers.

The following diagram shows the labels, which would be used to show where the numbers are for each box that is drawn on the number grid. The box stands for the size of the square figure that we put around the chosen numbers, where X be the size of box. The diagram below shows an example where we have drawn different size boxes in a 10 by 10 grid. Here the number N = 10, hence 10 by 10 number grid.

                            4 by 4 box  (here X = 4)                                           3 by 3 box (here X = 3)

        

                                                                                                   2 by 2 box  (here X = 2)

                                                     

                                         5 by 5 box  (here X = 5)

The objective is now to find out the formula for obtaining the total value of the difference between the products for any size of number grids and any size of box inside the number grid. The next calculations will be made to prove that how the answers are calculated by trying to obtain a simple formula connecting the size of the number grid (called N) and the size of the box inside the grid (called X).

 The figure below shows the general box in any type of number grid.

        

        

The letters a, b c, and d stand for the four different numbers which are found in the four corners of the general box in any number grid.

We can now connect the four letters (that is the numbers) in a specific way as detailed below:

Let X = size of box, and N = size of number grid.

Therefore b = a + ( X – 1 ) and c = a + N ( X – 1) and d = c + ( X – 1)

Therefore, d =  a + N ( X – 1) + ( X – 1)

                           

                 d  =  a + NX – N +  X – 1

According to the worksheet, we have to multiply  a x d and c x b, which gives us ad and bc, followed by taking ad away from bc, as ad is smaller than bc.

Let us work out what ad and bc are in terms of X and N.

ad  =  a { a + NX – N +  X – 1 }

 

      =   a2   +  aNX – aN + aX – a

bc  =  { a + ( X – 1 ) }{ a + N ( X – 1) }

       =  (a + X – 1) ( a + NX – N )

       =  a2 + aNX – aN + aX + NX2 – XN – a – NX + N

       =   a2 + aNX – aN + aX + NX2 – 2XN – a + N

bc – ad =  [a2 + aNX – aN + aX + NX2 – 2XN – a + N ] – [a2   +  aNX – aN + aX – a ]

             =  a2 + aNX – aN + aX + NX2 – 2XN – a + N    - a2   -  aNX +  aN -  aX +  a

                =    NX2 – 2XN + N

                =    N ( X2 - 2X + 1 )

                =    N ( X –1 )2