To get the next corner top right I am going to look at a 2x2 square taken from a grid:
From this I have seen that the top right corner is one more than the top left corner. This is the same in any other 2x2 square. So using the top left corner, x, we can say that the top right corner is x + 1.
We now have the top two expressions:
Again to find like the top right corner, we are going to have to look at some example 2x2 squares to find the bottom left corners expression.
A 2x2 square from:
A 5x5 grid =
A 6x6 grid =
A 8x8 grid =
Looking at these 2x2 squares I can see that the bottom left corner, is the top left corner plus the grid size. If we call the grid size ‘g’ we can use it in our expressions, and we know that the bottom left corner is the top left corner plus the grid size, so we can just say that the bottom left corner is x + g.
To get the last expression all we have to do is to add the same addition to the bottom left as we did to the top left to get the top right. As the top right corner is just one more than the top left, the bottom right is going to be just one more than the bottom left, and as the bottom left is the top left plus the grid size, then the bottom right corner is just going to be x + g + 1.
Now we have the expressions of a 2x2 square in any grid, to find the expression for the difference we just have to use the square like we have normal squares. So we multiply the corners:
x(x+g+1) = x²+gx+x
(x+1)(x+g) = x²+x+gx+g
(x²+x+gx+g) – (x²+gx+x) = g
If we use the letter d to stand for difference, we can now just use this simple expression to find the difference of any 2x2 square.
d = g
This just stands for what we worked out earlier, that the difference of the 2x2 square would be the number of the grid size. But now we can just use this algebraic expression instead.
Now I have looked at 2x2 squares I am going to look at 3x3 squares. This time instead of looking at the patterns then going on to the algebra, I am going to just do the algebra to find an expression to find the difference in a 3x3 square in any grid size.
So like the 2x2 square we are going to give the corners of a 3x3 square expressions to stand for the numbers that would be there in any grid.
Straight away we can call the top left corner ‘x’.
In this 3x3 square the only numbers in the grid we need to use are the corner numbers, so when drawing the 3x3 squares, with expressions in, from now on I will leave those middle numbers out.
To find the top right corner I will look at a 3x3 square taken from a 5x5 grid.
From this I have seen that the top right corner, is two more than the top left corner. This is the same in any other 3x3 square in any grid. So using the top left corner, x, we can say that the top right corner is x + 2.
Now I’ll look at some 3x3 squares in different grids to find the expression for the bottom left corner.
3x3 square in a
5x5 grid:
Looking at this example you can see that the bottom left corner is, 2 times the grid size more, than the top left corner. So it is 2g more than the top left corner. Meaning we can give the bottom left corner the expression, x + 2g.
Then to find the last one like in the 2x2 square we add the same increment to the bottom left as we did to the top left to get the top right.
So the bottom right expression will be x + 2g + 2.
Then like the normal squares we just multiply the corners.
x(x+2g+2) = x²+x2g+2x
(x+2)(x+2g) = x²+x2g+2x+4g
Then take the answers away from each other to get the expression for the difference.
(x²+x2g+2x+4g) - (x²+x2g+2x) = 4g
So this means for a 3x3 square in any grid, the difference can be worked out by using this algebraic expression:
d = 4g
Now I have this I am going to try it on one of the 3x3 squares just to make sure it is right. I will use the equation to calculate the difference of a 3x3 square in a 9x9 grid, then I will actually use a real 3x3 square from a 9x9 grid to check if the equation is correct.
d = 4x9 = 36
So if the equation is right the difference will be 36.
1 x 21 = 21
3 x 19 = 57
57 – 21 = 36
So the difference is 36 and the equation worked it out correctly. Now I know that I can use this equation for any 3x3 square in any square grid.
Now I am going to look at 4x4 squares.
Again we can call the first corner ‘x’.
If this is the top line of a 4x4 square then we can just work out how much the top right corner will be more than x.
So the top right corner is just x+3
To find the bottom left corners expression I am going to have to look at some examples.
4x4 square in a:
5x5 grid:
Looking at this example you can see that the bottom left corner is, 3 times the grid size more, than the top left corner. So it is 3g more than the top left corner. Meaning we can give the bottom left corner the expression, x + 3g.
Then the last corners expression is just, x + 3g + 3.
Now we multiply the corners.
x(x+3g+3) = x²+x3g+3x
(x+3)(x+3g) = x²+x3g+3x+9g
Then take away the answers from each other.
(x²+x3g+3x+9g) – (x²+x3g+3x) = 9g
So this means for a 4x4 square in any grid, the difference can be worked out by using this algebraic expression:
d = 9g
Now I have this equation I just need to check it. I will do this by working out the difference of a 4x4 square in a 6x6 grid, with the equation, then working it out properly to see if the equation works it out correctly.
d = 9x6 = 54
So if the equation is correct the difference will be 54.
1 x 22 = 22
4 x 19 = 76
76 – 22 = 54
The difference is 54, so the equation is correct and can be used to work out the difference for any 4x4 square in any square grid.
So I now have an expression for a, 2x2, 3x3 and 4x4 square in any grid.
2x2 square in any grid: d = g
3x3 square in any grid: d = 4g
4x4 square in any grid: d = 9g
To work out the difference of any size square in any size grid I will have to get expressions for the four corners of any square in any grid.
Again we can start by calling the top left corner ‘x’.
To find the top left expression I am going to look back at what it was when working out the expressions for the 2x2, 3x3 and 4x4.
2x2 top right corner = x + 1
3x3 top right corner = x + 2
4x4 top right corner = x + 3
Looking at these you can see that the number added onto x is, the square size minus one. If we call the square size ‘n’ then we can use it in our expressions. (n in a 7x7 square would be 7). We know that the number added onto x is the square size minus one, so the expression will just be, x + (n-1).
Now we need to look at the expressions in the bottom left corner of the squares.
2x2 bottom left corner = x + g
3x3 bottom left corner = x + 2g
4x4 bottom left corner = x + 3g
When we found out the top right corner expression we saw that the numbers added onto x where n-1. These numbers again are being used in these bottom left corners. In a 2x2 square n-1 is 1, and 1 x g is g, which is being added onto x. In a 3x3 square n-1 is 2, and 2 x g is 2g, which is being added onto x, and so on.
So the expression we can use in the bottom left corner is x + g(n-1).
Then like when we found expressions for the squares like 2x2 and 3x3. To get the bottom right we add the same increment to the bottom left as we did to the top left to get the top right.
So the bottom right expression will be x+g(n-1)+(n-1).
Now we just multiply the corners.
x(x+g(n-1)+(n-1)) = x²+gx(n-1)+x(n-1)
(x+(n-1))(x+g(n-1)) = x²+gx(n-1)+x(n-1)+g(n-1)²
(x²+gx(n-1)+x(n-1)) - (x²+gx(n-1)+x(n-1)+g(n-1)²) = g(n-1)²
Now we have an algebraic expression that can be used on any sized square in any sized square grid, to work out the difference of the answers, if the corners were multiplied.
d = g(n-1)
To check this equation I am just going to take a 5x5 square from an 8x8 grid, work out its difference with the equation, then check it by working it out properly.
So if the equation is correct the difference will be 128.
11 x 47 = 517
15 x 43 = 645
645 – 517 = 128
So the difference is 128, meaning the equation is correct and it can be used to work out any square in any square grid.
Task 2
Now that I have worked out how to get the difference from any square in any square grid, I am going to try and do the same for any rectangle in any square grid.
As a rectangle isn’t that different to a square, instead of working out the expression the same way as I did for the square, I am going to look at the final expression for any square in any square grid and try to change it to work for a rectangle in any square grid.
This is the expression we got for any square in any square grid:
d = g(n-1)²
This is the same as:
d = g(n-1)(n-1)
I am going to work out the difference for a 4x2 rectangle in a 6x6 square grid. First I will work it out using the above equation then work it out by actually taking the numbers and multiplying them out then getting the difference.
D = 6(4-1)(2-1)
D = 6 x 3
D = 18
So if the equation is right then the difference of the 4x2 rectangle in a 6x6 square grid should be 18.
Now I will take a 4x2 rectangle out of a 6x6 square grid.
8 x 17 = 136
11 x 14 = 154
154 – 136 = 18
The difference was 18. So the equation worked it out correctly.
To check if the equation is really correct I will take one more rectangle and test the equation with it.
A 6x3 rectangle from the above 6x6 square grid.
D = 6(6-1)(3-1)
D = 6 x 10
D = 60
1 x 18 = 18
6 x 13 = 78
78 – 18 = 60
So the difference is 60 like the equation worked out.
Now I have double checked it, I am quite sure that the equation is correct. So to work out the difference of any rectangle in any square grid you just need to use this equation:
d = g (1-1)(w-1)
Just to check this equation I will take a 4x3 rectangle from an 8x8 grid, work out its difference using the equation, then checking it by doing it properly.
d = 8(4-1)(3-1) = 8 x 3 x 2 = 48
So if the equation is correct then the difference will be 48.
12 x 31 = 372
15 x 28 = 420
420 – 372 = 48
The difference is 48 so the equation is correct and we can use it to work out the difference for any rectangle in any square grid.
