These are the corners for a square of any length sides i.e. an LxL square
The corners are then multiplied as with the numerical examples which leaves us with two algebraic equations that need subtracting to find the difference. These are.
n x [n+11(L-1)]= n2 +11n(L-1)
And
[n+(L-1)]x[n+10(L-1)]= n2 +10n(L-1)+1n(L-1)+10(L-1) 2
n2 +10n(L-1)+1n(L-1)+10(L-1) 2 – [n2 +11n(L-1)]= 10(L-1) 2
When these two equations are then subtracted we are left with the difference for any square. So D=10(L-1) 2 .
Conclusion
From the equations and numerical examples above I can therefore conclude that the difference (D) for any square in a 10x10 grid is 10(L-1) 2 . These results and steps shown above all make certain that this is proof and I will demonstrate this with some examples of different sized squares in a 10x10 grid.
For a 2x2 square
D= 10(L-1) 2
D= 10(2-1) 2
D= 10x 12
D= 10
For a 4x4 square
D= 10(L-1) 2
D= 10(4-1) 2
D= 10x 32
D= 90
This therefore proves that the algebraic formula to find out the difference for any LxL square is correct.
Rectangles in a 10x10 Grid
In this section the investigation carries on to see what differences are made to the formula and patterns if rectangles are used instead of squares in a 10x10 grid.
Numerical examples
Results
From these results of the numerical examples we can again notice that every rectangle, like the squares, of the same size has the same difference. This can be seen by the results of the 5x3, with a difference of 80, and 2x3 rectangles, with a difference of 20 each time. Because of this pattern I will now make the size of the rectangle variable so the width=W and length=L and see if there is a general rule that applies to all rectangles and their differences each time.
An LxW rectangle in a 10x10 Grid
Here the rectangle investigation will carry on and I will try to find and explain an algebraic rule for the differences of all rectangles.
For each of these rectangle sizes an individual algebraic formula can be created
For a 3(L)x2(W) rectangle:
For a 4x2 rectangle:
For a 4x3 rectangle:
For a 5x3 rectangle:
For a 5x4 rectangle:
But as these only work 4 each rectangle they do not give a general rule for all rectangles. However the patterns found in each of these corners of the different rectangles leads to a rule being able to be produced.
In each case whatever W is, the number in the top right corner is always n + (W-1) which is one step to finding the formula.
For the bottom left corner of any rectangle to make it in terms of L and W the number is always n+10(W-1), which works for every rectangle.
For the bottom right corner to make it in terms of L and W for all rectangles the number is always n+10(L-1)+(W-1).
So this then gives us a rule that works for any rectangle on a 10x10 grid.
L
From this algebraic form of all the corners of any rectangle an equation like with the numerical examples is made to find the difference.
n[n+10(W-1)+(L-1)]=n2+10n(W-1)+n(L-1)
And
[n+(L-1)]x[n+10(W-1)]=n2+10n(W-1)+n(L-1)(W-1)
Then when the two equations are subtracted the difference of the product between the opposite corners for any rectangle in a 10x10 grid is given.
This difference (D)=10(L-1)(W-1)
Conclusion
So in this section I have found out that the difference between the products of the opposite corners in a 10x10 grid is always 10(L-1)(W-1). Also I have found that all rectangles of the same size have the same difference regardless of whether the length or width is the largest. I will work out some of the rectangle differences to prove that my algebraic theory is correct.
In a 3x2 rectangle
D= 10(L-1)(W-1)
D= 10x2x1
D=20
In a 4x5 rectangle
D= 10(L-1)(W-1)
D= 10x3x4
D= 120
Both of these rectangle differences are correct which proves that my algebraic formula and working out is correct.
An LxL square or an LxW rectangle in a GxG grid
In this investigation I have so far discovered and proved the formulas for any square or rectangle in a 10x10 grid. In this section I aim to find the formula, by simple number manipulation, of any square or rectangle in any sized grid (GxG). First of all I will investigate any size squares (LxL) in any sized grids (GxG).
Numerical examples
In a 6x6 grid:
In a 12x12 grid:
Results
From these results above it is easy to see that with a different sized grid comes a different difference with the same shape. For instance, a 2x2 square in a 6x6 grid has a difference of 6 whereas a 2x2 square in a 12x12 grid is 12. This is the same for all sizes of rectangles and squares as shown in the tables above.
This is because of a simple difference in the numbers. In a larger table the numbers in the bottom corners are larger thus making a larger difference when multiplied and the difference is found. This can be shown in the algebraic squares.
For a 2x2 square in a 6x6 grid:
Whereas in a 2x2 square in a 12x12 grid:
And when these are multiplied they give us different differences because of the varying grid sizes. However there is a rule that can be found if grid size and length and width of the shape are entered into the equation.
This works as a general rule for all rectangles and squares in any sized grids to find the numbers in each corner. If this is multiplied and the difference is found for this square we get the algebraic equation to find the difference of any shape in any grid.
The two equations that need to be worked out then subtracted are:
n[n+g(W-1)(L-1)]= n2+gn(W-1)+n(L-1)
And
[n+(L-1)]x[n+g(W-1)]= n2+gn(L-1)+n(L-1)(W-1)
That then subtracts and leaves the difference
D= g(L-1)(W-1)
Conclusion
Now in this section the investigation has become more thorough. Instead of finding specifically squares or rectangles in a 10x10grid the grid has also become a variable factor and so the algebraic rule has been discovered to find the difference between the corners in any square or rectangle in any sized grid.
Here are some equations related to the numerical examples above to prove the formula works.
A 3x2 rectangle in a 12x12 grid
D=g(L-1)(W-1)
D=12(3-1)(2-1)
D=12x2x1
D=24
A 3x3 square in 6x6 grid
D=g(L-1)(W-1)
D=6(3-1)(3-1)
D=6(3-1)2
D=24
An LxL square or an LxW rectangle in a GxG grid without consecutive numbers (x)
In this final section of the investigation I will continue to manipulate the formula gradually. Since any rectangle or square can be found in any sized grid I will now investigate if there is an equation and a general algebraic rule that can be used to find these things in a grid without consecutive numbers. Instead of going 1,2,3,4,5 I will see if there is a rule when the grid goes 2,4,6,8,10 or 3,6,9,12,15 where x is the difference between the numbers on the grid.
Numerical Examples
Here is a 6x6 grid with a difference of 2 between the numbers
Here is a 6x6 grid with a difference of 5 between the numbers:
Results
I have used the same sized grid and shapes each time so that the only variable is the difference between the numbers in the grid. From these numerical examples it is clear to see that with a larger difference between the grid numbers results in a larger difference altogether. The grid size still plays a part in this too and with a larger grid comes a larger difference.
If like before I look at the individual squares for the corners and not the overall pattern it is possible to gain only the individual algebraic equation. Here are some examples:
A 2x3 rectangle in a 6x6 grid with a difference of 2
Whereas in a 2x3 rectangle in a 6x6 grid with a difference of 5
But if this rule is generalized and number difference along with all the other different variables (any shape, grid size,) are put into the equation a rule for any shape in any grid with any number difference in the grid can be achieved.
Here is that square
Here are the two equations obtained from this square:
[n x n+gx(W-1)]+[x(L-1)]= n2gnx(W-1)+nx(L-1)+gx2(L-1)(W-1)
And
[n + x(L-1)] x [n+gx(W-1)]= n2 x gnx(W-1)+nx(L-1)
n2gnx(W-1)+nx(L-1)+gx2(L-1)(W-1) - n2 x gnx(W-1)+nx(L-1) = gx2 (L-1)(W-1)
Then when these are subtracted the total difference is obtained
D= gx2 (L-1)(W-1)
Here is an example to check the formula
A 3x4 rectangle in a 6x6 grid with a grid difference (x) of 5
D= gx2 (L-1)(W-1)
D= 6x52(4-1)(3-1)
D= 6x25x3x2
D=900
Conclusion
So overall in this investigation I started small and worked up. From a square in a 10x10 grid with simple algebraic and number manipulation I finally ended up with the formula for finding the differences in any square or rectangle, in any size, in any sized grid, with any differences between the numbers in that grid. Here are the 4 rules that I have found
-
For any square- D=10(L-1) 2
-
For any rectangle- D=10(L-1)(W-1)
-
In any sized grid- D= g(L-1)(W-1)
-
With any differences between the numbers in the grid- D= gx2 (L-1)(W-1)
Each of these rules uses elements from the one before and the each equation is a step up from the one before resulting finally in being able to find the difference between any square or rectangle of any size in any sized grid with any difference between the numbers in that grid. And all of these have been demonstrated and proved to be true.
