# Number Grid Investigation

Number Grid Investigation

Aim: My aim is to find out how the number of rows and columns in a square, on a certain sized grid will affect the difference between the product of the top left number and the bottom right number in the square subtracted from the top right number and the bottom left number.

Prediction: I predict that when you've got numbers in a square box in a 10 by 10 grid, the difference will always be a square number because the box is shaped as a square.

Method: Firstly, I'm going to work out the difference, in a 2 by 2 square in a 10 by 10 grid, between the two products that I get from multiplying the top left number and the bottom right number in the square by the top right number and the bottom left number. I am then going to repeat this another 3 times but these times I will work out a 3 by 3 square, 4 by 4 square and a 5 by 5 square. After doing this, I will try to find a formula that can find the difference in any sized square. I will then decrease the size of the grid so that it becomes 9 by 9 and will then do exactly the same method as I did before and then I will do this again but with a 5 by 5 grid. I will then work out formulas for both of these grids, to find the difference. I will then use algebra to prove that my formulas are able to work out the correct difference. To investigate further I will do the whole investigation again but with rectangles instead of squares.

This is a table to show the differences in squares in a 10 by 10 grid.

No. of Rows (r)

No. of Columns (c)

Difference

Formula

2

2

0

0(r-1)²

3

3

40

0(r-1)²

4

4

90

0(r-1)²

5

5

60

0(r-1)²

Firstly, I noticed that if you took the zeros off the difference, then you would get square numbers for 1, 2, 3 and 4. So I tried to find a way where you get the square numbers:

2-1=1 x10= 10

The formula would be 10(r-1)

So I tried using this formula for a 3 by 3 square:

0(3-1) = 20

...