number grid

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Number Grid

For this task I will first be looking at a number grid from 1 to 100, like the one below:

I will start my investigation by looking at 2 by 2 squares. I will draw a square around 4 numbers, find the product of the top left and bottom right numbers and the product of the bottom left and top right numbers, then calculate the difference the between the 2 products. I will see if there are any patterns and if so I will try to work them out algebraically.

I will then look at changing the size of the squares to see if there are any patterns.  I will try looking at 3 by 3 squares, 4 by 4 squares and 5 by 5 squares; I will do the same with these squares as I have with the 2 by 2 squares, I will find the products of the top left and bottom right and the bottom left and top right numbers then calculate the difference between them.

Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles.  I will start by looking at a 2 by 3 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern.  I will then try changing the size of the rectangles and looking for patterns there.  I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles.

        I will also look at changing the size of the number grids to see if this has an affect on the patterns.  I will look at a 9 by 9 grid, an 11 by 11 grid and a 5 by 5 grid.  I will be looking for patterns in 2 by 2 squares within the different size grids and trying to find an algebraic formula to explain my findings.

2 X 2 Grid

 I have chosen the top left number of the square randomly. I have done this by using the random number function on my calculator. In my investigation I am going to find the product of the top left number and the bottom right number. Also I am going to find the product of the top right number and the bottom left number. When I have found these I will find the difference between the two numbers.

77 x 88 = 6776

78 x 87 = 6786      

        _6786

          6776

              10

Now I am going to repeat my investigation so my results are more reliable and so I can create a table with them.

_22

     12  .

  10

For this 2 X 2 grid I have done the exact same thing as I did for the first one.

I am now going to repeat my investigation again so that my results are more reliable and so I can create a table with them.

     _3640

       3630

            10

For this 2 X 2 grid I have done the exact same thing as I did for the first one.

I am now going to repeat my investigation again so that my results are more reliable and so I can create a table with them.

       _190

         180

           10

For this 2 X 2 grid I have done the exact same thing as I did for the first one.

I am now going to repeat my investigation again so that my results are more reliable and so I can create a table with them.

 

           _2236

             2226

                 10

For this 2 X 2 grid I have done the exact same thing as I did for the first one.

        

I have now taken 5 2 X 2 grids from a 10 X 10 grid and found the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for each of them. After that I found the difference between the two numbers that I found. Now I am going to put my results into a table to help me analyse them and to find a formula for the difference in a 2 X 2 grid.

Results

Here I have created a table to show the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number.

After looking at my table I have found out that the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number is always 10 in a 2 X 2 grid. Now I will test out my theory by picking out another 2 X 2 and finding the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number. If my theory is right the difference will be 10.

     

_3886

                     3876

                         10

I have tested my theory by using a different 2 X 2 grid. The difference is again 10 so therefore it looks like my theory is right.

The Difference for any 2 X 2 grid is always 10 inside a 10 X 10 grid.

3 X 3 Grid

I have now found the difference for any 2 X 2 grid inside a 10 X 10 grid, now I will try to find out the difference for any 3 X 3 grid inside a 10 X 10 grid. But this time instead of drawing all the grids I will try to work is out algebraically. I will do this because it will save time and get to the point quicker.

Now I will pick out a 3 X 3 grid from a 10 X 10 and examine it to see if there is anyway to work out the difference algebraically.

After looking at this grid I have found out that the top right number is two numbers bigger than the top left number, the bottom left number is twenty numbers bigger that the top left and that the bottom right number is 22 numbers bigger than the top left. This should be the case anywhere I put my 3 X 3 grid.

Now I am going produce a grid in algebra form.

I have used the letter ‘a’ for the top left number and added how much bigger the other key numbers are away from it.

In my investigation I have to find the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number. So therefore I will multiply ‘a’ by ‘a+22’ and also I will multiply ‘a+2’ by ‘a+20’ and find the difference.

Therefore:

a(a+22) = a² + 22a

Join now!

(a+2)(a+20) = a² + 22a + 40

(a² + 22a + 40) – (a² + 22a) = 40

According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 3 grid inside a 10 X 10 grid will always be 40.

Now I will draw a 3 X 3 grid to test if my theory is correct.

41 x 63 = 2583

43 x 61 = 2623

2623 – 2583 = ...

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