E.g. a 2 x 2 cutout from a 6 x 6 grid. When n is subtracted from the bottom left corner we are left with the length of the grid, or grid length i.e. 6.
From noticing that the length of the grid is the number that we are left with in the bottom left corner when we subtract n, I can now implement further use of algebra. I will call the number added to n in the bottom left corner G and this will stand for grid length, so I will now call the bottom left corner in a 2 x 2 cutout n + G, and as the value of the number to the right is always 1 more than the value to the left I will call the bottom right corner in a 2 x 2 cutout n + G + 1. This solution should work for a 2 x 2 cutout on any size grid. I will now calculate the diagonal difference of a 2x2 cutout using my n and G solutions.
The diagonal difference of a 2 x 2 cutout on any size grid equals G. I predict that for a 2x2 cutout on a 5 x 5 grid, the diagonal difference will be 5.
.
Calculating the diagonal difference of a 2x2 cut out on any size grid using n and G solutions.
If we calculate diagonal difference of a 2 x 2 cutout using n and g solutions we can find the diagonal difference of any 2x2 cutout on any size grid, by first solving the equation (n + G) (n+1) – n(n + G + 1), then replacing the final simplified solution of G with the length of the grid that the 2x2 cutout originated from. So now that I‘ve found a solution for 2 x 2 cutouts, I will now try to investigate different size cutouts and try to apply n and G solutions. To do this I will need to compare previous investigated cutouts and look at the similarities and differences.
I will begin by calculating the diagonal difference of the cutout numerically. And then by using only n, as n relates to every corner on the cutout. I will then see if G is applicable.
The diagonal difference is 40
Is the diagonal difference of a 3 x 3 cutout the same anywhere on the 10x10 grid?
Investigating further cutouts
I will now calculate the diagonal difference of a 3 x 3 cutout on a 10 x 10 grid using n.
What I have noticed
From calculating the diagonal difference of a 3 x 3 cutout on a 10 x 10 grid I have noticed that the number added to n in the bottom left corner is twice the length of the grid, from noticing this I can now implement G into the equation, only if the number added to n in the bottom left corner is always the size of the grid that the 3 x 3 cutout originated from. So I will now look at 3x3 cutouts on a different size grid.
In the case of a 5 x 5 grid G = 5
I will now check if this is a common trend in 3 x 3 cutouts by calculating the diagonal difference of a further cutout.
In the case of a 4 x 4 grid G = 4
I will now convert the cutout into its
Algebraic form
Converting the cutout into its algebraic form.
Step 1 Step 2 Step 3
I will now calculate the diagonal difference of a 4 x 4 cutout. Once I have looked at 4 x 4 cutouts I will compare all my examples of different size cutouts and look to see if there are any patterns or relationships between them. Hopefully by the end of this analysis I will have found a general formula for which I can work out the diagonal difference for any size, square cutout on any size, square grid.
I notice that a 4 x 4 cutout is 1 more in length than a 3 x 3 cutout and 10 more in height. From this I can convert the cutout straight into its algebraic form using my previous knowledge of 3 x 3 and other cutouts.
I will now convert the cutout into its algebraic form
Converting the cutout into its algebraic form.
Step 1 Step 2 Step 3
I have now found solutions for 2 x 2, 3 x 3 and 4 x 4 cutouts. I will now analyse these cutouts and see if I notice any patterns. My main aim will be to try and find a general solution/formula for any square, size cutout on any square, size grid.
Since analysing the different size, square cutouts I have noticed a pattern. The value added to n in the top right corner is 1 less than the length of the cutout. For example a 4 x 4 cutouts top right corner value would be n + 3 as shown above. This value seems to be common in the bottom two corners of a square cutout. It seems to relate to the number of G’s added to n in both bottom corners and what number must be added to the n and G solutions bottom right corner. For example on a 2 x 2 cutout the top right corner is n + 2, the bottom left is n + 2G, and the bottom right is n + 2G + 2. So now that I have discovered this common factor in the cutouts I can now implement further algebra, but before I do this I must see weather or not my findings are true by looking at the corner values of a further cutout. I will convert a 5 x 5 cutout into its algebraic form.
Step 1 Step 2 Step 3
I will now do the same for a further cutout to make sure my findings are a trend in different size cutouts.
Step 1 Step 2 Step 3
I now have sufficient evidence that the number added to n in the top right corner of a cutout is 1 less than the actual size of the cutout, and that this number is common in the bottom two corners, to be able to implement the use of algebra. I shall call this number X – 1, since this number is the length of the cutout (value across the grid) (X), takeaway 1 (X - 1) (see diagram below). Because I have now used total algebra to represent the corner values of a cutout I feel confident that the product of the diagonal difference of a cutout of total algebra will be my final algebraic formulae for any size, square cutout on any size, square grid. I will now produce the solution.
To check that the algebraic solutions in each corner of the above cutout work, I will substitute all algebra with the values from a 2 x 2 cutout from a 10 x 10 grid and see if the end products are correct
The solutions give the correct final products and the corner values are the same as the corner values of a 2 x 2 cutout on a 10 x 10 grid
Calculating the diagonal difference of the algebraic cutout.
I will first calculate the top right corner multiplied by the bottom left corner. I will then calculate the top left corner multiplied by the bottom right corner. Once I have done this I will takeaway both products and find the final formulae.
(X – 1)² G
(X – 1)² G could be the formula for working out the diagonal difference for any square, size cutout. To check this, I will now calculate the diagonal difference of a 2 x 2 cutout on a 10 x 10 grid using the formula. I know from previous examples that the diagonal difference should be 10.
X = 2
G = 10
The formula works for 2x2 cutouts
To make sure this formula works for other size cutouts I will use it to work out the diagonal difference of a 4 x 4 cutout on a 6 x 6 grid. I know from previous examples that the diagonal difference should be 54.
The formula works!
(X – 1)² G is the universal formula for all square cutouts on square grids.
Now that I have found a solution for all square cutouts on square grids, I will now look at square cutouts on rectangular grids.
Calculating the diagonal differences square cutouts on rectangular grids.
Diagonal differences of square cutouts on an 8 x 7 grid
Since calculating the diagonal difference for a 2 x 2 cutout on an 8 x 7 grid I have noticed that the diagonal difference is 8, and that the diagonal difference of a 2 x 2 cutout on an 8 x 8 grid is also 8. I will now compare an 8 x 7 grid and an 8 x 8 grid to find out why this.
From analysing and comparing both grids I have noticed that an 8 x 7 grid is the similar to an 8 x 8 grid. The only difference is that the 8 x 7 grid has a less row below. So in the case of an 8 x 7 grid, the grid stops at 56 as it is not a perfect square and this the product of 8 x 7, but in the case of an 8 x 8 grid, the grid stops at 64 as this is the product of 8 x 8 and it is a perfect square. From discovering this I think it’s safe to assume that all coinciding square size cutouts anywhere on an 8 x 7 grid have the same diagonal differences of coinciding square size cutouts anywhere on an 8 x 8, as I have discovered earlier on in my investigation, that a cutout of the same size anywhere on a square grid has the same diagonal difference as that of a cutout elsewhere on the grid. E.g. a 4 x 4 cutout anywhere on an 8 x 7 grid has the same diagonal difference as a 4 x 4 cutout of the same size anywhere on an 8 x 8 grid.
8 x 7 grid
So from recognising this I know that when calculating the diagonal difference of a cutout on a rectangular grid, we can use the same formula used when calculating square cutouts on square grids as the cutouts of a rectangular grid have identical corner values to that of square grids when looking at grids with the same length. To check that my assumptions are correct I will calculate the diagonal difference of a 4 x 4 cutout numerically and then by using my formula.
The formula works for the above cutout and I can safely say that it works for all other square cutouts on rectangular grids as it did for all square cuitouts on square grids, (bearing in mind the relationship between square grids and rectangular grids). But will this formula work when the length of the grid is less than the height? I will now look at a 7 x 8 grid and a 7 x 7 grid to investigate this further as there may be a relationship between the two as with 8 x 7 grids and 8 x 8 grids.
7 x 8 grid 7 x 7 grid
From analysing and comparing both grids I have noticed that a 7 x 8 grid is the similar to a 7 x 7 grid. The only difference is that the 7 x 8 grid has an extra row below. So in the case of a 7 x 8 grid, the grid stops at 56 as it is not a perfect square, and this the product of 7 x 8, but in the case of a 7 x 7 grid, the grid stops at 64 as this is the product of 7 x 7 and it is a perfect square. From discovering this I think it’s safe to assume that all coinciding square size cutouts anywhere on a 7 x 8 grid have the same diagonal differences of coinciding square size cutouts anywhere on a 7 x 7, as this was the case with 8 x 7 and 8 x 8 grids.
So from recognising this I know that when calculating the diagonal difference of a cutout on a rectangular grid, we can use the same formula used when calculating square cutouts on square grids as the cutouts of a rectangular grid have identical corner values to that of square grids when looking at grids with the same length. To check that my assumptions are correct I will calculate the diagonal difference of a 3 x 3 cutout from the 7 x 8 grid numerically and then by using my formula. I predict that the formula will work and that the diagonal difference will be 4G i.e. 28.
From investigating square cutouts on rectangular grids I have found that my previous formula for square cutouts on square grids works for square cutouts on rectangular grids. The formula works on any size rectangle i.e. when the height is bigger that length or when the length is bigger than the height. This is because of the relationship between square and rectangular grids. The values of the squares on the grid are not affected as the length of the grid, or G, is not affected by the height of the grid meaning that any square size cutout from either grid has the same corner values as each other and they both give the same diagonal difference. For example a 2 x 2 cutout from a 7 x 7 grid has the same corner values as a 2 x 2 cutout, thus meaning the same diagonal difference. To check that it works on any size grid I will now calculate the diagonal difference of a 2 x 2 cutout on a 9 x 5 grid and a 9 x 7grid. The diagonal differences should be the same.
I have investigated diagonal difference of square cutouts as much as possible. I feel I have achieved what I aimed to achieve by investigating square cutouts on different size grids, which was to find a general formula for any size square cutout on any size grid. One last thing I noticed is that the length of the cutout can never be greater than the length of the grid it can either be equal to the length of the grid or smaller than the length of the grid.i.e. G ≥ X
Investigating diagonal difference of rectangular cutouts
Now that I have proved my formula works for square cutouts on square and rectangular grids, I will now investigate weather or not my formula works for rectangular cutouts. I will start by investigating a 2 x 3 cutout on a 10 x 10 grid, as it is impossible to work out the diagonal difference of a 2 x 1 cutout as there isn’t value for each individual corner.
But will my previous formula work?
(X – 1) ²G
My formula does not work for rectangle cutouts.
Since calculating the diagonal difference of a 2 x 3 cutout I have noticed that the diagonal difference is 40. but is the diagonal difference 40 when the cutout is aligned so that height is greater than the length, i.e. a 3 x 2 cutout.
From analysing the differently aligned cutouts I have noticed that both cutouts have the same area and both cutouts have the same diagonal differences of 20, meaning that so long as the area of the of the cuouts is the same, the diagonal difference stays the same. I also notice that the difference between the top left number and the top right number is 2, the difference between the top left number and the bottom right number is 12, the difference between the top left number and the bottom left number is 10 when the length of the grid is greater than the height. From noticing this I can implement the use of n as I did earlier with square cutouts. So I will now represent a 2 x 3 cutout anywhere on a 10 x 10 grid as -
This should give me the correct diagonal difference of 20
(n + 10) (n + 2) – n(n + 12)
=> n² + 10n + 2n + 20 - n² - 12n
=>n² + 12n + 20 - n² - 12n
=>20
Using the cutout formula of n gives the correct diagonal difference of 20. But can I apply g to the equation as well?
Earlier on in the investigation G represented the length of the grid and it was common in the bottom 2 corners. In the case of a 2 x 3 cutout on a 10 x 10 grid G would be equal to 10. I will now analyse the cutout and see if G can be applied.
From analysing the cutout I have noticed that G can be applied as the bottom left corner is n + 10 and 10 is the length of the grid. so from noticing this I can implement the use of G. the bottom left corner should read n + G, and the bottom right corner should read n + G + 2 (12-10 =2).
But can G be applied to a vertically aligned cutout?
A vertically aligned cutout containing only n expressions reads –
And from analysing this cutout we can see a clear relation between g and the number added to n in the bottom two corners. In the case of the above cutout the bottom left corner should read n + 3G and the bottom right hand corner should read n + 3G + 1 (31 - 30 = 1).
I will now investigate a series of 2 x X cutouts on a 10 x 10 grid to see if I can spot any relationships or patterns that will help me to apply further algebra. At this point in my investigation I believe it’s safe to assume that all cutouts of the same area from the same size grid have the same diagonal difference. So I will proceed with investigating rectangles by only giving one example of each size cutout from a 10 x 10 grid. I will convert every cutout into its general form by using n and g solutions.
2 x 4 cutouts
Since the diagonal differences are the same for differently aligned cutouts with the same area, I will continue my investigation by only calculating the diagonal difference of my initial example for each size cutout i.e. the horizontally aligned cutout
2 x 5 cutouts
I theorise that every time I increase the width by one, the difference increases by 10.
I predict that the 2 x 6 rectangle will have a diagonal difference of 50.
2 x 6 cutouts
From calculating the diagonal difference of these 2 x X cutouts I can now produce a table of results. I will start by producing a table of results for the horizontally aligned cutouts, and then I will produce a further table of results for vertically aligned cutouts.
I will also include the results of a 2 x 2 cutout, as a square is a special form of a rectangle.
Table of results for horizontally aligned cutouts
Table of results for vertically aligned cutouts
What Do I Notice?
Horizontal cutouts
I notice that the diagonal difference increases in increments of 10, so from this I’m able to make a correct prediction of the next cutout in the series. I also notice that there is a consistent pattern of one G added to n in the bottom left corner and the bottom right corner.
As with square cutouts I also notice that there is a relationship between the length of the cutout, the top right corner, and the bottom right corner. The number added needed to make up the value of the top right and bottom right corner is always 1 less than the length of the cutout. I can now relate my previous solution of X – 1 to the equation of the cutout. So far the cutout should now read; n in the top left corner, n + (X – 1) in the top right corner, n + G in the bottom left corner and n + G + (X – 1).
This cutout solution should give the correct diagonal difference of a horizontally aligned 2 x X cutout. I’ll try it for a 2 x 4 cutout. I predict that the diagonal difference will be 30
So in the case of a 2 x 4 cutout the diagonal difference = (4-1)10 = 30
My prediction was correct
But this solution will not work for a vertically aligned cutout as there is more than one G. so I will now analyse my table of results for vertically aligned cutouts and see if I can find a solution to this problem.
Table of results for vertically aligned cutouts
From analysing my table of results for vertically aligned cutouts I have noticed a relationship between the height of the cutout and the amount of G’s needed in the bottom two corners of a cutout. The height of the cutout takeaway 1 defines the amount of G’s needed in the bottom two corners, and since this is a trend for all the different size cutouts I can now substitute this number with an algebraic expression. As we are dealing with height I shall call this number Y-1. This is also common in horizontally aligned cutouts, but it is not as apparent, as there is always only one G and the height is always 2. The same applies to vertically aligned cutouts when we look at X – 1, as the length of the cutout is always 2 and the difference between the corners next to each other is always 1, which makes it harder to notice. Because I have now used total algebra to represent the corner values of a rectangular cutout and from previous knowledge know that a square is a special type of rectangle, I feel confident that the product of the diagonal difference of a cutout of total algebra will be my final algebraic formulae for any size cutout on any size grid. I will now produce the solution cutout.
To check that the algebraic solutions in each corner of the above cutout work, I will substitute all algebra with the values from a 2 x 3 cutout from a 10 x 10 grid and see if the end products are correct.
The solutions give the correct final products and the corner values are the same as the corner values of a 2 x 3 cutout on a 10 x 10 grid.
I will now calculate the diagonal difference of the algebraic cutout
Calculating the diagonal difference of the algebraic cutout.
I will first calculate the top right corner multiplied by the bottom left corner. I will then calculate the top left corner multiplied by the bottom right corner. Once I have done this I will takeaway both products and I hopefully find the final general formulae for any size cutout on any size grid.
D = (X – 1)(Y – 1)G could be the general formula for working out the diagonal difference for any size cutout. To check this I will now calculate the diagonal difference of a 3 x 5 cutout on a 10 x 10 grid using the formula.
21 x 5 = 105
=> 25 x 1 = 25
=> 105 – 25 = 80
=> 80
The diagonal difference is 80
I will now work out the diagonal difference using my general formula
(X - 1)(Y – 1)G
=> 5 – 1 = 4
=> 3 – 1 = 2
=> 4 x 2 = 8
=> 8 x 10 = 80
=> 80
The General formula works !
Conclusion
The universal general formula (X – 1)(Y – 1)G gives the correct diagonal difference of any size cutout weather it be a square or a rectangle. When considering the height and length of a square, where X is equal to Y, the expression (X – 1)(Y – 1)G simplifies to my earlier formula (X – 1)² G. I think it’s safe to assume that a rectangle cutout would have no change in its diagonal difference on a rectangle grid, by looking at my previous investigations.
Evaluation
I believe I have reached the targets and aims I set myself in the beginning. Since starting this coursework I have found a general formula that works on any size cutout on any size grid through progressive investigation and through using my algebraic methods.
I believe I have taken this coursework to its limit in the time I have been given. I am happy with the work that I have produced and cannot think of anything I could have carried out better or improved upon.
Maths Coursework
X
Y
INVESTIGATION OF DIAGONAL DIFFERENCE
Khalil Sayed –Hossen 9B
