- 1 x 13 = 13
2 x 12 = 24
∴ 24 – 13 = 11
- 19 x 31 = 589
20 x 30 = 600
∴600 – 589 = 11
- 48 x 60 = 2880
49x 59 = 2891
∴2891 – 2800 = 11
- 85 x 97 = 8245
86 x 96 = 8256
∴ 8256 – 8245 = 11
- 92 x 104 = 9568
93 x 103 = 9579
∴ 9579 – 9568 = 11
Looking at these calculations above, it is clear that the prediction is correct and that this can be used for any 2 x 2 box in an 11 x 11 grid.
I will now use the algebraic formula, using 11 in place of ‘g’ to prove that this is correct.
n(n + g + 1) = n2 + ng + n
n + 1(n + g) = n2 + ng + n + g
∴ [n2 + ng + n +g] – [n2 + ng + n ] = g
I will then replace g by 11 for this exercise:
n(n + 11 + 1) = n2 + 11n + n
n + 1(n + 11) = n2 + 11n + n + 11
∴ [n2 + 11n + n +11] – [n2 + 11n + n] = 11
I will use no. 5 (above) and apply this to the formula
92(92 + 11 + 1) = 922 + 1012 + 92
92 + 1( 92 + 11) = 922 + 1012 + 92 + 11
= 11
This proves that my equation and original prediction are both correct and this can be used for any sized grid.
However, I would like to repeat this with another grid size so I will use a 19 x 19 grid.
As before, the highlighted boxes are the ones I am going to calculate, to see if the difference between the product of multiplying the opposite corners, equals the size of the grid. In this case, I am looking for that number to be 19.
- 1 x 21 = 21
2 x 20 = 40
∴ 40 – 21 = 19
- 32 x 52 = 1664
33 x 51 = 1683
∴1683 – 1664 = 19
- 177 x 197 = 34869
178 x 196 = 34888
∴ 34888 – 34869 = 19
- 302 x 322 = 97244
303 x 321 = 97263
∴ 97263 – 97244 = 19
- 305 x 325 = 99125
306x 324 = 99144
∴ 91444 – 99125 = 19
Once again, I can conclude that the difference between the cross multiplied products is the size of the grid; 19.
To confirm this, the number 19 has been inserted into the formula to prove that this is correct.
n(n + g + 1) = n2 + ng + n
n + 1(n + g) = n2 + ng + n + g
∴ [n2 + ng + n +g] – [n2 + ng + n ] = g
where g = 19;
n(n + 19 + 1) = n2 + 19n + n
n + 1(n + 19) = n2 + 19n + n + 19
∴ [n2 + 19n + n +19] – [n2 + 19n + n] = 19
My prediction of what the difference would be was correct. So for any 2x2 square taken from any size grid, the difference will be the number of the grid size.
I can now work out the difference of any 2x2 square as long as I know the grid size.
5 x 5 grid = Difference of 5
19 x 19 grid = Difference of 19
100 x 100 grid = Difference of 100
31 x 31 grid = Difference of 31
58 x 58 grid = Difference of 58
1000x1000 grid = Difference of 1000
3 X 3 Squares
Having studied 2 x 2 boxes within different sized grids, I would like to study 3 x3 boxes within grids of varying sizes to see if there is an emerging pattern.
I have decided to look at an 8 x 8 grid first to see if there is a pattern.
I am going to leave out all of the numbers in the middle of the 3 x 3 box and will concentrate on the four corners of this box.
Firstly, I need to work out the difference of the products for the highlighted 3 x 3 box.
26 x 44 = 1144
28 x 42 =1176
∴ 1176 – 1144 =32
From this, I can see that the difference is not the size of the grid (g). However, I will formulate an expression to predict the size of the difference for any 3 x 3 box within an 8 x8 grid,
The first top left hand box is shown as ‘n’ and the size of the grid is still ‘g’. The top right hand corner of the box is 2 more than the top left hand corner, so this is n+ 2. The bottom left hand corner of the box is two rows down exactly, so this n + 2g and finally, the bottom right hand corner of the box is n + 2g + 2.
Therefore;
After multiplying the diagonal corners of the box, the algebraic formula is as follows:
(n )( n + 2g + 2) = n2 + 2ng + 2n
(n + 2)(n + 2g) = n2 + 2ng +2n + 4g
Then subtract either sides of the equation from each other to confirm what the difference is.
(n2 + 2ng +2n + 4g) – (n2 + 2ng + 2n) = 4g
This is accurate because the 8 x8 grid had a difference between the products of 32, which is equal to 4 x 8. Therefore, I can confirm that, according to my calculations, the difference between the products of the diagonal corners is 4g.
I will now use a different 3 x 3 box within the same 8 x8 grid to see if this rule remains the same;
Here, I will also multiply the corners only so:
6 x 24 = 144
8 x 22 = 176
∴ 176 – 144 = 32.
This proves that my prediction is correct.
I will now investigate if this formula for 3 x 3 boxes works in a different sized grid.
The box I have highlighted will be used to work out the product of the corners and the difference between them.
1 x 17 = 17
3 x 15 = 45
∴45 – 17 = 28.
This fits the pattern, as 4 x 7 is equal to 28, where 7 is the grid size.
4 X 4 Square
- 4 x 31 = 124
1 x 34 = 34
∴124 – 34 = 90
- 24 x 57 = 1368
27 x 54 = 1458
∴ 1458 – 1368 = 90
- 61 x 94 = 5734
64 x 91 = 5824
∴ 5824 – 5734 = 90
- 67 x 100 = 6700
70 x 97 = 6790
∴ 6790 – 6700 = 90
In a 10 x 10 grid, the difference between the diagonal corners of a 4 x 4 square is always 90.
2 x 3 Rectangle
As yet, I am unable to determine a general formula to predict the difference between multiplying the diagonals of a square or a rectangle in any sized grid. I will therefore investigate the difference between the products of the diagonal corners of a rectangle.
I am going to begin with a 2 x 3 rectangle in a 10 x 10 grid.
- 13 = 13
3 x 11 = 33
∴ 33 – 13 = 20
- 46 x 54 = 2484
44 x 56 = 2464
∴ 2464 – 2484 = 20
- 30 x 38 = 1140
28 x 40 = 1120
∴ 1140 – 1120 = 20
Rectangles can be aligned in two ways, so I will now determine if the difference is always 20 if arranged the other way.
- 1 X 22 = 22
2 X 21 = 42
∴ Difference is 42 – 22 = 20
- 35 x 56 = 1960
36 x 55 = 1980
∴ Difference is 20
- 69 x 90 = 6210
70 x 89 = 6230
∴ Difference is 20!
I can confirm that there is not a change in the difference between the products of diagonal corners in a rectangle if the alignment is changed.
For the following sized rectangles, I am going to give just one example of each and I am going to use a 10 x 10 grid.
I will now investigate rectangles that are all 2 x Y, where Y is a differing length each time:
2 X 4 Rectangle
2 x 15 = 30
5 x 12 = 60
∴ Difference = 30
2 x 5 Rectangle
1 x 15 = 15
5 x 11 = 55
∴ Difference = 40
It appears that every time I increase the width of the square by 1, the difference increases by 10. Therefore, I predict that the 2 x 6 rectangle will have a difference between the products of the corners as 50.
2 x 6 Rectangle
1 x 16 = 16
6 x 11 = 66
∴ Difference = 50
My prediction appears to be correct.
I also predict that the 2 x 7 rectangle will have a difference of 60:
1 x 17 = 17
7 x 11 = 77
∴ Difference = 77 -17 = 60
Calculating a general algebraic formula
My theory on the difference increasing by 10 for every increase of 1 in width of the squares in a 10 x 10 grid is correct. This is correct for a 2 x 2 square also.
Having obtained a widespread range of results, I have drawn up the table below that represents the differences for squares in a 10 x 10 grid of varying heights:
Observations:
The area of the rectangles and squares in a 10 x 10 grid always increase by the value of 2 each time the height is increased by 1.
The difference increases by 10 each time the height is increased by 1 in a 10 x 10 grid. This is due to the fact that each time a number is picked from a 10 x 10 grid, the next number vertically below it is 10 more.
Predicting formulae:
L-1 x 10 The length of the square subtracted by one then multiplied by 10.
However, this formula does not take into account the height of the square and so must be tested with rectangles with height of greater than 2. I am not confident that this will be the correct formula.
L-1 x 5H The length minus one and then 5 multiplied by the height. Again, this needs to be tested to see if it is correct.
Using predicted formulae to see if they work with rectangles with differing heights:
In order to see if the same patterns are in use for rectangles of differing heights, I will firstly look at a height of 3 rows.
3 X 4 Rectangle
1 x 24 = 24
4 x 21 = 84
∴ Difference is 60
3 x 5 Rectangle
1 x 25 = 25
5 x 21 = 105
∴ Difference = 105 – 25 = 80
The difference is increasing by multiples of 20 so the 3 x 6 rectangle should have a difference of 100, if the formula is L-1 and 20H
3 x 6 Rectangle
1 X 26 = 26
6 x 21 = 126
∴ Difference is 100
From this information I have created a second table that highlights the difference for squares and rectangles of varying lengths and heights. Once I have studied this table, I should be able to predict the difference for squares and rectangles of varying heights and lengths.
Observations:
My earlier predicted formula of L-1 X 5H does not fit the pattern, however, L-1 X 20 appears to fit this pattern. However, I am aware that the “20” will not fit all squares and rectangle sizes.
For 2 x χ --- L-1 x 10
For 3 x χ --- L-1 x 20
Perhaps the rule is:
Difference = Length - 1 x 10 x (Height - 1)
= L – 1 x 10(H - 1)
Testing the formula:
In order to test the formula I predict works, I am going to look at 4 x X squares in a 10 x 10 grid.
4x5 rectangle
1 x 35 = 35
5 x 31 =155
∴ Difference is 120
Looking at the predicted formula, I am going to replace the symbols with numbers to see if it correct:
D = L – 1 x 10(H - 1)
= 5-1 x 10(4 – 1)
= 4 x 30
= 120
This is correct, however, I wish to further investigate this using rectangles of other lengths.
4x6 rectangle
1 x 36 = 36
6 x 31 = 186
∴ Difference is 150
4x7 rectangle
1 x 37 = 37
7 x 31 = 217
∴ Difference is 180
Again, I have tabulated the results for the 4 x X rectangles to see if the formula fits the pattern;
←
D = L-1 x 10(H -1)
I will use the row that has been arrowed to see if this is correct.
D = 4-1 x 10(7-1)
= 3 x 10 (6)
= 180
I can confirm that the rule does work and that D = L-1 x 10(H-1)
Further questions:
I believe that the “x 10” in the formula directly correlates with the fact that this is a 10 x 10 grid. Therefore what would happen if I changed the size of the grid? To test this idea I will try my rule on a different sized grid.
Changing the size of the grid
Using the same rule but substituting 10 for 8, which is the width of the new grid, I have calculated the difference in the highlighted 3 x 5 square to be:
D = L-1 x 8(H-1)
D = 3-1 x 8 x (5-1)
= 2 x 8 x 4
D = 64
I must test this with the highlighted box so,
19 x 53 = 1007
21 x 51 = 1071
∴ Difference is 1071 – 1007 = 64
This is correct. Therefore, I need to replace the number that the H is multiplied by ( once is has been subtracted by 1) by the letter ‘g’. This represents the width of the grid. So for a generalised formula that predicts the difference between the diagonal corners of any rectangle or square grid;
L – 1 x G (H -1)
Utilising algebraic formulae to calculating the rectangle values in relation to ‘n’
If the number in the top left hand corner is n then the following squares will be…
For a 2 x 3 rectangle when the first square is n the difference is:
(n+2) (n+G) = n2 + nG + 2n + 2G
n (n+G+2) = n2 + nG + 2n
∴ (n+2) (n+G) - n (n+G+2) = 2G
The algebraic expression for the difference of a 2 x 3 rectangle is 2G, the width of the grid multiplied by two.
For a 2x4 rectangle when the first square is n the difference is:
(n+3) (n+G) = n2 + nG + 3n + 3G
n (n+G+3) = n2 + nG + 3n
∴ (n+3) (n+G) - n (n+G+3) = 3G
The algebraic expression for the difference for this size of rectangle is 3G, the width of the grid multiplied by three.
From this information, I have been able to tabulate my results:
From the table above, it is apparent that between 2 x 3 and 2 x 4 rectangles, n increases by 1. This could be due to the height increasing by 1. I think that the height is probably related in some way to the difference value.
Evaluation:
I have formulated the algebraic formula that can predict the difference for any sized rectangle or square for any sized grid. Having investigated this observation, I can only ascertain that this is correct and that;
L – 1 x G (H -1)
Perhaps further investigations on different grid sizes would be suitable. Also, given more time, I would have liked to look at different shapes in different sized grids and to determine a general formula for them. For example, I would like to know if the same formula fits a rhombus on a 10 x 10 grid.
