(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.
A 2x2 square in a:
60x60 grid: d = 60
17x17 grid: d = 17
204x204 grid: d = 204
134x134 grid: d = 134
2345x2345 grid: d = 2345
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:
6x6 grid:
9x9 grid:
Looking at these 3 examples 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.
A 3x3 square in a:
22x22 grid: d = 4 x 22 = 88
13x13 grid: d = 4 x 13 = 52
45x45 grid: d = 4 x 45 = 180
123x123 grid: d = 4 x 123 = 492
1005x1005 grid: d = 4 x 1005 = 4020
Now I am going to look at 4x4 squares.
Again we can call the first corner ‘x’.
(Please remember there are numbers between these corners but we do not need to use them)
This time I don’t really need to look at an example to find the top right expression, instead I can just work it out by thinking about it.
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:
6x6 grid:
12x12 grid:
Looking at these 3 examples 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.
A 4x4 square in a:
9x9 grid: d = 9 x 9 = 81
43x43 grid: d = 9 x 43 = 387
79x79 grid: d = 9 x 79 = 711
354x354 grid: d = 9 x 354 = 3186
5343x5343 grid: d = 9 x 5343 = 48087
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-
But what if I want to find the difference of any sized square, in any sized grid? To work out this 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)²-
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.
d = 8(5-1)² = 8x4² = 8x16 = 128
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.
The difference of a:
7x7 Square in a 16x16 grid = 16(7-1)² = 576
23x23 Square in a 300x300 grid = 300(23-1)² = 145,200
400x400 square in a 1000x1000 grid = 1000(400-1)² = 159,201,000
Extension 1
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)
This expression works on any square in any square grid. So if you look at a labelled diagram of any square in any square grid we can look at how we can change it to work for a rectangle.
Now to help lets look at this diagram but with real numbers instead of the letters.
The 3 in the above diagram is ‘n’. In the expression u have to take 1 away from n. So 3-1 is 2. If you do that to the top 3 and the left 3, then times them (2x2) you get 4. Then the 4 times the grid size, 5 is 20. Which is correct for the difference of a 3x3 square in a 5x5 square grid.
Now if we look at a diagram like the above but instead of a square taken from the grid, we will look at a rectangle.
I have named the top side of the rectangle ‘l’ standing for length, and the left side of the rectangle ‘w’ standing for width.
Now if the rectangle works the same as the square, the top side of the rectangle (‘l’) minus 1, and the left side of the rectangle (‘w’) minus one, multiplied together then multiplied by the grid size (‘g’), we should get the difference for the rectangle in the square grid.
So the expression for a rectangle should be this, if it works the same way as the square does:
-d = g(l-1)(w-1)-
Now I just need to test it to see if it works.
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 the an actually 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(l-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.
The difference of a:
4x3 rectangle in a 5x5 square grid = 5(4-1)(3-1) = 30
7x5 rectangle in a 12x12 square grid = 12(7-1)(5-1) = 288
12x4 rectangle in a 20x20 square grid = 20(12-1)(4-1) = 660
Extension 2
Now I have equations to work out:
- The difference of any square in any square grid
- The difference of any rectangle in any square grid
Next I am going to work out an equation to find the difference of a Rhombus when its corners are multiplied then the answers subtracted from each other.
To do this I am going to look at different sized rhombus’s like a 2x2, 3x3 and a 4x4, then work out use them to find the equations for the corners of any rhombus. I will then multiply them out and subtract the answers from each other, to get a final equation which I can use to get the difference of any rhombus in any square grid.
A 2x2 Rhombus:
If we look at the above 2x2 rhombus we can get the expressions for each of the corners.
We call the first one ‘x’ like we did when finding out the square and the rectangle.
In the above rhombus 11 is 7 more than 4.
So 11 is ‘x+7’ but we need something that will work with any 2x2 rhombus. The grid size is 8 so to get to 7 we need to take away one. So the expression for the left corner could be ‘x+g-1’. If you look at any other 2x2 rhombus in the 8x8 grid then you will see that, that does work, so we can use it as the equation for the left corner.
We can work out the last two corners quite easily.
If you add ‘g’ to ‘x’ then it would be the number in the centre of the 2x2 rhombus. You subtract one to get to the left corner, so to get to the right corner all you have to do is add one. This means the equation for the right corner is ‘x+g+1’.
Finally if by adding ‘g’ to ‘x’ you get to the centre of the 2x2 rhombus you just need to add ‘2g’ to get to the bottom. So the bottom corners equation is ‘x+2g’.
Now we have the four corners expressions for a 2x2 rhombus.
A 3x3 Rhombus:
*Remember the above is shown like a 2x2 rhombus but there are numbers between the four corners. Only the corners are shown because we don’t need the other numbers.
Again we will start by calling the top corner ‘x’.
In the example 3x3 rhombus we have taken from the 8x8 square grid, the left corner is 18 and the top corner is 4. 18 is 14 more than four. Two times the grid size is 16, and if you take away two from 16 you get 14. So the equation for the left corner could be ‘x+2g-2’. If you look back at the 8x8 grid you can see that if you look at any 3x3 rhombus in it that equation will work for the left corner.
If you add ‘2g’ to ‘x’ you will get into the middle of the 3x3 rhombus. You take away two from that to get to the left corner so you just add 2 to get to the right corner. So the equation for the right corner is ‘x+2g+2’.
If you are in the middle when you add ‘2g’ to ‘x’ then you just add another ‘2g’ to get to the bottom corner. So the bottom corners equation is ‘x+4g’
Now we have the equations for the four corners of a 2x2 rhombus and a 3x3 rhombus.
A 4x4 Rhombus:
*Remember the above is shown like a 2x2 rhombus but there are numbers between the four corners. Only the corners are shown because we don’t need the other numbers.
The equation will be ‘x’ like in the 2x2 and 3x3 rhombus’s.
The example 4x4 rhombus’s left corner is 25 and the top corner is 4. 25 is 21 more than 4. ‘3g’ is 24, then if you minus 3 from 24 you get 21. So the equation for the left corner could be ‘x+3g-3’. Looking at the 8x8 grid, you can see that any 4x4 rhombus’s left corner’s can be found by using that equation. So it is correct.
To get to the centre of the 4x4 rhombus you just add ‘3g’ to ‘x’. To get from the centre to the left corner you subtract 3, so to get to the right corner you just add 3. So the equation for the right corner is ‘x+3g+3’.
From the centre you just need to add another ‘3g’ to get to the bottom corner. So the equation for the bottom corner is ‘x+6g’
Now I have the equations for the four corners of a 2x2, 3x3, and 4x4 rhombus’s.
To find the equations for the corners of any sized rhombus I am going to look at the 2x2, 3x3, and 4x4’s equations.
We already know one corner equation for any sized rhombus and that is the top corner which is just ‘x’.
The left corner for a:
2x2 rhombus = x+g-1
3x3 rhombus = x+2g-2
4x4 rhombus = x+3g-3
Like when we did the squares, in these rhombus equations the number multiplied by ‘g’ is just ‘n-1’. Then at the end of these equations the same numbers that are multiplied by g are subtracted. So the equation for the left corner of any rhombus is ‘x+g(n-1)-(n-1)’.
The right corner is just like the left corner in the 2x2, 3x3 and 4x4 rhombus’s except that the end number is added instead of subtracted. So we just need to change the symbol at the end to get the right equation.
The bottom corner for a:
2x2 rhombus = x+2g
3x3 rhombus = x+4g
4x4 rhombus = x+6g
To find this bottom equation we will try to use ‘n-1’ because it is used in 2 of the other equation’s. ‘n-1’ in a 2x2 rhombus is 1, so ‘n-1’ plus ‘n-1’ is 2 which is then multiplied by ‘g’. In a 3x3 rhombus ‘n-1’ plus ‘n-1’ is 4 which is the number multiplied by ‘g’. ‘n-1’ plus ‘n-1’ in a 4x4 rhombus is 6 which is the number that is multiplied by ‘g’. So looking at this we can work out the bottom equation for any sized rhombus is ‘x+g((n-1)+(n-1))’
So now we need to multiply the corners, then subtract the answers, to have the final equation to work out the difference for any rhombus in any square grid.
(x) . (x+g((n-1)+(n-1))) = x(x+g((n-1)+(n-1)))
= x²+gx((n-1)+(n-1))
= x²+gx(n-1)+gx(n-1)
= x²+2gx(n-1)
(x+g(n-1)-(n-1)) . (x+g(n-1)+(n-1))
= x(x+g(n-1)-(n-1))
+ g(n-1)(x+g(n-1)-(n-1))
+ (n-1)(x+g(n-1)-(n-1))
= x²+gx(n-1)-x(n-1)+gx(n-1)+g²(n-1)²-g(n-1)²+x(n-1)+g(n-1)²
-(n-1)²
= x²+2gx(n-1)+g²(n-1)²-(n-1)²
Now I will subtract the answers from each other.
(x²+2gx(n-1)+g²(n-1)²-(n-1)²) – (x²+2gx(n-1))
= g²(n-1)²-(n-1)²
So the equation we can use to find the difference of any rhombus in any square grid is:
-d = g²(n-1)²-(n-1)²-
To do a final check on this I am going to use the 3x3 rhombus we used from the 8x8 grid earlier and see if the equation can work out the difference correctly.
d = 8²(3-1)²-(3-1)² = 64x2²-2² = 64x4-4 = 252
So if the equation is right the difference will be 252.
4 x 36 = 144
18 x 22 = 396
396 – 144 = 252
The difference is 252, so the equation is right.
I have now got the equations to work out:
Any Square in any square grid: -d = g(n-1)²-
Any Rectangle in any square grid: -d = g(l-1)(w-1)-
Any Rhombus in any square grid: -d = g²(n-1)²-(n-1)²-
If I had more time I could go on to find the equation for any Square, Rectangle or Rhombus in different shaped grids. I could then make the numbers in the grids different like a grid of odd numbers, or equal numbers. There are lots of things you could do to extend the task, and using the information I have already found it should be easier to work out the equations for any square, rectangle or rhombus on different grids, because there will most likely be some similarities.
