# Number Grid Product Differences Investigation

Maths Coursework

## Number Grid Product Differences

This assignment is concerned with product differences in different size matrices for different sized grids.

Here is a 10x10 grid. We are trying to investigate product differences within the 10x10 grid. As you can see I have drawn a 2x2 grid inside the 10x10 matrix. To find out the product difference I do:-

The top left number x the bottom right – the top right x the bottom right.

The product difference I found was 10. This was anywhere on the grid where a 2x2 grid could be done. I can prove this by picking a 2x2 grid and using this clue to work out the product difference. Below is the result of me picking a 2x2 grid from this 10x10 grid and finding out what the product difference was.

I did (48x59) – (49x58) = 10. This shows that the product difference is the same anywhere in the grid.

Here are 2 more examples of this that prove that this works anywhere on the grid.

I did (12x23) – (13x22) = 10. This shows that the product difference is the same anywhere in the grid.

I did (72x83) – (73x82) = 10. This shows that the product difference is the same anywhere in the grid.

Then we were told that we had to find a general formula to work out the product difference in 10x10 grid but with a 3x3 or 5x5 matrices inside. I firstly found out the product difference in a 3x3, 4x4, 5x5 and a 6x6 grid. Below are the differences of the matrices.

Here is another 2 more examples to prove that the formula above works

I did (74x85) – (75x84) = 10. This shows that the product difference is the same anywhere in the grid.

I did (59x70) – (60x69) = 10. This shows that the product difference is the same anywhere in the grid.

The examples prove that this does work anywhere on the grid.

The way I found out the product differences for the other size squares was as follows: -

3x3

Above is a 10x10 matrix with 3x3 matrices inside it. To work out the product difference I do:-

The top left x the bottom right – the top right x the bottom left.

I took an example from the grid to find out the product difference.

So the calculation that I did was (75x97) – (77x95) = 40. With this calculation I figured out that the product for a 3x3 was 40.

I will do another 2 examples to prove this

I did (13x35) – (15x33) = 40. This shows that the product ...