Maths coursework

Investigating diagonal difference

Aim:

For this course work my task is to investigate the diagonal difference of different size cutouts on different size grids.

E.g. this is a 2x2 cutout on a 10x10 grid.

Finding the diagonal difference of a cutout is achieved by finding the product of the bottom left corner and the top right corner, then the product of the bottom right corner and the top left corner and finally calculating the difference between these two products.

I will investigate the diagonal differences for cutouts anywhere on a grid and use algebra to prove any rules I may discover. I will log my results in a table to help me to find any formulas that may become apparent as I proceed. As I investigate this problem I will make appropriate predictions from rules I may discover from the table of results I produce, and from any patterns that can be seen.

Method:

I will start by investigating the diagonal difference of 2x2 cutouts on 10x10 grids.

2x2 cutouts

What is the diagonal difference of a 2x2 cutout on a 10x10 grid?

22 x 13 = 286

- 23 x 12 = 276
- 286 –276 = 10

The diagonal difference is 10

Is the diagonal difference of a 2x2 cutout the same anywhere on the 10x10 grid?

68 x 59 = 4012

- 69 x 58 = 4002
- 4012 – 4002 = 10

The diagonal difference is 10

This seems to be the case. I notice that for a 2x2 cutout on a 10x10 grid the diagonal difference is 10, so to prove that this is common in all 2 x 2 cutouts on a 10 x 10 grid, I will calculate the diagonal difference of a further 2x2 cutout.

What have I noticed?

From these cutouts I have noticed that the diagonal difference of a

2 x 2 cutout is 10 and that the grid length is 10. I will keep this in mind, as this may become relevant for later on in my investigation.

I have noticed a general pattern for a 2x2 cutout anywhere on the

10x10 grid from the 2x2 cutouts I have chosen. The top left number and the top right number have a difference of 1, the top left number and the bottom left number have a difference of 10 and the top left number and the bottom right number have a difference of 11.

E.G 62 – 61 = 1

From this I can find a solution for a 2x2 cutout anywhere on the 10x10 grid by implementing the use of simple algebra. I can call the top left number in the cutout n, the top right number n + 1, the bottom left number n + 10 and the bottom right n + 11, as this is a trend for all of the 2x2 cutouts in a 10x10 grid. From this I will be able to calculate the diagonal difference of a cutout anywhere on the grid, as n is independent of where the cutout is on the grid.

I will now calculate the diagonal difference of a 2 x 2 cutout on a 10 x 10 using n.

(N = first number in cutout)

Since calculating the diagonal difference of the 2x2 cutout (above) using n, I have noticed that the bottom left number added to n is the same as the length of grid.

This number is the length of the grid

I will keep this in mind when calculating the diagonal difference for different size cutouts from different size grids as this may be of use to me when trying to find a general formula.

I will now look at 2x2 cutouts in different size grids.

What is the diagonal difference of a 2 x 2 cutout on an

8 x 8 grid?

Earlier on in the investigation I noticed that the diagonal difference of a 2x2 cutout on a 10x10 grid is 10, and that 10 is the length of the grid, so I predict that diagonal difference of a 2 x 2 cutout in an 8 x 8 grid will be 8

Calculating the diagonal difference of a 2x2 cut out on an 8 x 8 grid using n

I have noticed that for 2 x 2 cutouts on an 8 x 8 grid there is a similar pattern as 2 x 2 cutouts on a 10 x 10 grid when using n. The top two numbers can still be represented as n and n + 1, because the top right number is always 1 more than the top left number.

But the bottom values are dependant on the length of the grid, so in the case of a 2 x 2 cutout on an 8 x 8 grid the length of the grid is 8. The bottom left value when using n is n + 8 and the bottom right number is always 1 more so its value will be n + 9. I predict that using this method will work for any 2x2 out on any size grid.

I will now calculate the diagonal difference using n

I am now confident that by using n I will find the correct diagonal difference of a 2 x 2 cutout on any size grid, so I will continue my investigation with using only algebra.

To check that my algebraic solutions work for any size grid I will investigate the diagonal difference of a further 2 x 2 cutout from a 7 x 7 grid

Calculating the diagonal difference of a 2x2 cut out on a 7 x 7 grid using n

I predict that the diagonal difference will be 7

I will now produce a table showing the corner values of 2 x 2 cutouts on different size grids, when n is included

From this table I can predict the values of each square on a 2 x 2 cutout on any size grid and find the diagonal difference e.g. a 6 x 6 grid

This is what the squares on the cutout should look like and the diagonal difference should equal 6.

To test my prediction I will calculate the diagonal difference of a random 2 x 2 cutout on a 6 x 6 grid using n.

I made this prediction by analysing my table of results. I noticed that the diagonal difference of a 2 x 2 cutout is length of the grid and that it must be the same on a 6 x 6 grid. I will now look at the table I produced and see if there are any other realtionships and patterns.

Further analysis of table of results

From a further analysis of my table of results I have noticed a relationship between the length of the grid and the bottom left corner in a 2 x 2 cutout. If we subtract n from the bottom left corner we are left with the same number as the grid length.