I can now confirm that for any square on a 10 x 10 grid it doesn’t matter where it is vertically positioned, the difference remains the same.
I will now test this diagonally.
1 x 23 = 23
3 x 21 = 63
63 – 23 = 40
45 x 67 = 3015
47 x 65 = 3055
3055 – 3015 = 40
I will now try a random position on the grid to finalise my results so far.
77 x 99 = 7623
79 x 97 = 7663
7663 – 7623 = 40
I can now confirm that for any square on a 10 x 10 grid it doesn’t matter where it is positioned, the difference remains the same. I predict that this is the same for any rectangle on a grid. I will now try out several other shapes to test this theory.
A will test a 2x3 shape, a 2x5 shape and a 3x5 shape. For each of these I will use the same tests as I used on the 3x3 square – 2 vertical tests, 2 horizontal tests and 2 diagonal tests, as well as a random test to confirm results and show up any anomalous results.
For the 2x3 tests I predict that the difference will remain the same for the shape in any position on the grid.
1 x 13 = 13
3 x 11 = 33
33 – 13 = 20
8 x 20 = 160
10 x 18 = 180
180 – 160 = 20
1 x 13 = 13
3 x 11 = 33
33 – 13 = 20
81 x 93 = 7533
83 x 91 = 7553
7553 – 7533 = 20
This proves that for a 2x3 the difference remains the same when you move it down the grid in a vertical direction.
I will now test this diagonally.
1 x 13 = 13
3 x 11 = 33
33 – 13 = 20
45 x 57 = 2565
47 x 55 = 2585
2585 – 2565 = 20
This proves that for a 2x3 the difference remains the same when you move it across the grid in a diagonal direction.
I will now test a random shape to ensure accurate results.
63 x 75 = 4725
65 x 73 = 4745
4745 – 4725 = 20
This confirms my predictions that no matter where a 2x3 rectangle is on the grid the differences are always the same.
I will now try this for a 2x5 rectangle
For the 2x5 tests I predict that the difference will remain the same for the shape in any position on the grid.
1 x 15 = 15
5 x 11 = 55
55 – 15 = 40
6 x 20 = 120
10 x 16 = 160
160 – 120 = 40
This proves that for a 2x5 the difference remains the same when you move it across the grid in a horizontal direction.
I will now test this vertically.
1 x 15 = 15
5 x 11 = 55
55 – 15 = 40
81 x 95 = 7695
85 x 91 = 7735
7735 – 7695 = 40
This proves that for a 2x5 the difference remains the same when you move it down the grid in a vertical direction.
I will now test this diagonally.
1 x 15 = 15
5 x 11 = 55
55 – 15 = 40
34 x 48 = 1632
38 x 44 = 1672
1672 – 1632 = 40
This proves that for a 2x5 the difference remains the same when you move it across the grid in a diagonal direction.
I will now test a random shape to ensure accurate results.
14 x 28 = 392
18 x 24 = 432
432 – 392 = 40
This confirms my predictions that no matter where a 2x5 rectangle is on the grid the differences are always the same. It is starting to look like all identical rectangles have the same difference no matter where they are positioned, as with the squares. I will do one more test to confirm this, a 5x3.
For the 3x5 tests I predict that the difference will remain the same for the shape in any position on the grid.
1 x 25 = 25
5 x 21 = 105
105 – 25 = 80
6 x 30 = 180
10 x 26 = 260
260 – 180 = 80
This proves that for a 3x5 the difference remains the same when you move it across the grid in a horizontal direction.
I will now test this vertically.
1 x 25 = 25
5 x 21 = 105
105 – 25 = 80
71 x 95 = 6745
75 x 91 = 6825
6825 – 6745 = 80
This proves that for a 3x5 the difference remains the same when you move it down the grid in a vertical direction.
I will now test this diagonally.
1 x 25 = 25
5 x 21 = 105
105 – 25 = 80
45 x 69 = 3105
49 x 65 = 3185
3185 – 3105 = 80
This proves that for a 3x5 the difference remains the same when you move it across the grid in a diagonal direction.
I will now test a random shape to ensure accurate results.
13 x 37 = 481
17 x 33 = 561
561 – 481 = 80
From all my tests so far I can deduce that as long as the rectangle is the same size and on the same angle that no matter where it is on the grid, the difference of the products is always the same.
But does the difference remain the same if the shape is rotated 90 degrees? Technically speaking this is a completely different shape e.g. a 2x3 is not the same shape as a 3x2, but it would be interesting to see if the differences are the same. Obviously this will work for squares because they are the same shape no matter what angle they are on, so I will try rotating the last 3 rectangles that I tested, a 2x3, a 2x5 and a 3x5. These will therefore become a 3x2, a 5x2 and a 5x3. From my previous tests I know that it doesn’t matter where I position them on the grid and I only need to do 1 for each test.
For all of these shapes I would expect the differences to be different from the non-rotated ones.
1 x 22 = 22
2 x 21 = 42
42 –22 = 20
This is the same difference as found on the 2x3.
24 x 66 = 1584
26 x 64 = 1664
1664 – 1584 = 80
This is the same difference as found on the 3x5.
58 x 99 = 5742
59 x 98 = 5782
5782 – 5742 = 40
This is the same difference as found on the 2x5.
All of my investigations so far have shown me that position and rotation are not factors that affect the differences. This will make it much easier to find a formula that works for any size rectangle, as it means fewer variables in the final formula.
The only way to achieve a formula is to convert my working so far into algebra.
Algebra
I will now convert the calculations for the shapes that I have investigated so far into algebraic calculations.
A 2x2 square –
(χ) (χ + 11) = χ² + 11χ
(χ + 1) (χ+10) = χ² + 10 + 1χ + 10χ
= χ² + 11χ +10
χ² + 11χ +10
- χ² + 11χ
= 10
A 3x3 square –
A 2x3 Rectangle
A 2x5 Rectangle
(χ) (χ + 14) = χ² +14χ
(χ + 4) (χ + 10) = χ² +40 +4χ +10χ
= χ² +14χ + 40
χ² + 14χ + 40
- χ² + 14χ
= 40
A 3x5 Rectangle
(χ) (χ + 24) = χ² +24χ
(χ + 4) (χ + 20) = χ² +80 +4χ +20χ
= χ² +24χ + 80
χ² + 24χ + 80
- χ² + 24χ
= 80
These algebraic equations prove all of my previous work. All the answers calculated from the initial tests and the algebra are identical.
Now that I have proved my answers and theories correct I can move on to try and calculate a formula.
Calculating a formula
Just by looking at the results table a few similarities in the results arise. All the differences are multiples of 10. This could have something to do with the 10x10 grid the tests were all performed on. I think there has to be a reference to this in the final formula. I also think that there has to be a reference to X (the width of the shape) and/or Y (the depth of the shape) in the formula.
I will start off trying to find a formula for a square. This should be simpler than that of a rectangle because both X and Y are identical.
From the table I can see that the difference is always bigger than X or Y.
The first thing I will try is to multiply the X figure by 10 to make it bigger.
This doesn’t give the correct answer for either the 2x2 or the 3x3, so this can’t be right.
Next I will try dividing the difference by 10 and see what happens.
Just looking at the 2x2 square there appears to be a difference of 1 between the χ figure and the difference divided by 10.
In other words the difference divided by 10 = χ-1
So if I times χ-1 by 10 I should get the difference for a 2x2 square (10).
10 x (2 – 1) = 10
This appears to be correct.
This would mean that for a 2x2 square the formula is 10(χ-1). As I predicted this contains both a reference to 10 and a reference to χ.
I will now try this formula out on a 3x3 square. The answer should be 40 and if this works then I know this is the formula for working out the difference in a square.
10 x (3 – 1) = 20
This is incorrect. The answer is half what it should be. If the formula were changed to 2(10 x (χ-1)) then this would give the correct answer for a 3x3, but if this were used on a 2x2 –
2(10 x (2 – 1)) = 20
It gives the wrong answer, so this cannot be the formula either. However if I square the χ - 1 it should give the correct answer for both.
10 x (2 – 1)² = 10
10 x (3 – 1)² = 40
This worked, so the formula for finding the difference between the products of the opposite corners of a square on a 10x10 grid must be:
10(χ-1) ²
To prove this I will work out the differences of 3 squares (2x2, 3x3 and 4x4) manually, and then using the formula. If the answers match then I will have proved by formula for squares correct. I will use 3 different squares to give a good average and to show up any anomalous results.
I have already manually worked out the answers for 2x2 and 3x3 earlier in the investigation. 2x2 gives a difference of 10, and 3x3 gives a difference of 40.
A 4x4 =
34 x 67 = 2278
37 x 64 = 2368
2368 – 2278 = 90
So the answer for a 4x4 should equal 90.
A 2x2 = 10 x (2 – 1)² = 10
A 3x3 = 10 x (3 – 1)² = 40
A 4x4 = 10 x (4 – 1)² = 90
These are all correct and prove my formula to be correct for all squares.
I now need to find out if this same formula works for rectangles.
A 2x3 rectangle should give a difference of 20:
10 x (2 – 1) ² = 10
This is wrong. It is half the correct answer. I could try multiplying this by two to give the correct answer.
2x(10 x (2 – 1) ²) = 20
This is correct. But does it work for different sizes of rectangles. I will now try it for a 2x5; this should give a difference of 40.
2x(10 x (2 – 1) ²) = 20
This is incorrect, so dividing by two doesn’t always give the right answer for a rectangle. Looking at the formula for the squares, 10(χ - 1) ², I can see that this doesn’t contain any references to the Y-axis (depth) of the shape. In a square this wouldn’t matter because the X-axis is identical to the Y-axis, but in a rectangle the Y is different to the X.
If the formula for a square = 10 x (χ - 1) x (χ - 1)
Then a rectangle could = 10 x (χ - 1) x (γ - 1)
A 2x3 should give a difference of 20
10 x (2-1) x (3-1) = 20
This is correct. I will now try this out on all the rectangles that were used in the initial tests for this investigation (2x3, 2x5, 3x5)
A 2x3 should give a difference of 20
10 x (2-1) x (3-1) = 20
A 2x5 should give a difference of 40
10 x (2-1) x (5-1) = 40
A 3x5 should give a difference of 80
10 x (3-1) x (5-1) = 80
These are correct. This proves that 10 x (χ - 1) x (γ - 1) is the formula for rectangles.
This can be simplified to: 10(χ-1)(γ-1)
The next step is to see if this formula works for squares as well. It should work, because the square formula (10(χ-1) ²) just assumes that χ is equal to γ, whereas 10(χ-1)(γ-1) specifies both the χ and γ separately.
I will test this formula with the same three squares I used to prove my square formula.
A 2x2 should give a difference of 10
10 x (2-1) x (2-1) = 10
A 3x3 should give a difference of 40
10 x (3-1) x (3-1) = 40
A 4x4 should give a difference of 90
10 x (4-1) x (4-1) = 90
These are all correct, proving that my formula works for squares as well.
So I now know that the formula 10(χ-1)(γ-1) works for every rectangle and every square on any rotation on a 10x10 grid.
The last thing that needs to be investigated is how a change of grid size or grid shape affects the formula.
I am going to try a few shapes out on an 8x8 grid. From my investigations on the 10x10 grid I know that it really doesn’t matter where the shape is on the grid, and I know that because of this I only need to do one test of each shape on the grid.
I will use 4 shapes. This gives a good average and allows for any anomalous results to be seen. It also allows the use of 2 rectangles and 2 squares for the investigation. I will use a 2x2, a 3x3, a 2x3 and a 4x5.
1 x 10 = 10
2 x 9 = 18
18 – 10 = 8
4 x 22 = 88
6 x 20 = 120
120 – 88 = 32
33 x 43 = 1419
35 x 41 = 1435
1435 – 1419 = 16
36 x 64 = 2304
40 x 60 = 2400
2400 – 2304 = 96
For the 10x10 grid the formula was 10(χ-1)(γ-1). I think that the 10 at the start of the formula relates to the size of the grid in some way.
So for an 8x8, the formula should be 8(χ-1)(γ-1)
I will test this by using this formula to calculate the differences for the manual tests I have just done, and if the answers match then I know this is the formula for an 8x8 grid.
For a 2x2 shape the difference should be 8
8 x (2 – 1) x (2 – 1) = 8
This is correct
For a 3x3 shape the difference should be 32
8 x (3 – 1) x (3 – 1) = 32
This is correct
For a 3x2 shape the difference should be 16
8 x (3 – 1) x (2 – 1) = 16
This is correct
For a 5x4 shape the difference should be 96
8 x (5 – 1) x (4 – 1) = 96
This is correct
This proves that, for a square grid, you just have to change the number at the start of the equation to the size of the grid.
But what happens on a rectangular grid? I expect this to need a different formula, because the X and Y-axes are different, as they were when I changed to square shapes to rectangle.
I will now try the formula out on 2 different sized rectangular grids, a 4x5 and a 9x7. For each of these I will test with 3 different shapes to prove my argument and to show up any anomalous results.
1 x 6 = 6
2 x 5 = 10
10 – 6 = 4
7 x 16 = 112
8 x 15 = 120
120 – 112 = 8
13 x 20 = 260
16 x 17 = 272
272 – 260 = 12
The formula for this should be 4(χ-1)(γ-1)
The difference for the 2x2 should be 4
4(2-1)(2-1) = 4
This is correct
The difference for the 3x2 should be 8
4(3-1)(2-1) = 8
This is correct
The difference for the 2x4 should be 12
4(2-1)(4-1) = 12
This is correct
This has disproved my first theory about needing to change the formula for a rectangle because it has a different sized X-axis than Y-axis.
It has told me that on a 4x5 grid that you simply replace the number at the beginning of the formula with the number of squares going across the top of the grid (the x-axis of the grid).
I will now test this on a 7x9 grid to prove this, and to make sure that it is definitely the X-axis that needs inserting into the formula.
1 x 9 = 9
2 x 8 = 16
16 – 9 = 7
17 x 28 = 476
21 x 24 = 504
504 – 476 = 28
50 x 63 = 3150
56 x 57 = 3192
3192 – 3150 = 42
If my theory is correct the formula for these should be 7(χ-1)(γ-1).
The difference for the 2x2 should be 7
7(2-1)(2-1) = 7
This is correct
The difference for the 2x5 should be 28
7(2-1)(5-1) = 28
This is correct
The difference for the 2x7 should be 42
7(2-1)(7-1) = 42
This is correct
This has proved my theory.
The final formula, for finding the difference between the products of the numbers in the opposite corners of any rectangle, on any size grid is:
α(χ - 1) (γ - 1)
Where: α = The horizontal width (χ-axis) of the grid
χ = The horizontal width (χ-axis) of the shape
γ = The vertical depth (γ-axis) of the shape