(N+4)(N+40)-N(N+44)
= N²+40N+4N+160-N²–44N
= 44N-44N+160
= 160
I am now going to put my results into a table and then predict the following product’s, i.e. 6x6, 7x7, 8x8 and 9x9.
Orange / Grey – see results clearer
Green – predicted results
I have noticed that there is a formula that you can work out all the results from. If I divide each individual result by 10 then you come out with a square number.
The formula is:
10(N-1)²
To make sure this formula and my table is true I will test 6x6, 7x7, and 8x8 grid sizes.
6x6
(36x81)-(31x86) = 2916-2666
= 250
7x7
(38x92)-(32x98) = 3496-3136
= 360
8x8
(20x83)-(13x90)
= 1660-1170
=490
I am now going to test my formula:
10(N-1)²
N=2 = (2-1) ²x10
= 1² x10 = 1x10
= 10 = 2x2
N=3 = (3-1) ²x10
=2²x10 = 4x10
= 40 = 3x3
N=8 = (8-1) ²x10
=8²x10 = 64x10
= 640 = 8x8
My formula has successfully worked and my table has the correct results.
I am now going to investigate further by changing the MAIN GRID SIZE, i.e. 9x9, 11x11, 12x12 e.c.t. So I will start with a 9x9 grid.
9x9
(9x81)-(1x90) = 729-90
= 639
10x10
(10x91)-(1x100) = 910-100
= 810
11x11
(11x111)-(1x121) = 1221-121
= 1100
12x12
(12x133)-(1x144) = 1596-144
=1452
13x13
(13x157)-(1x169) = 2041-169
= 1872
14x14
(14x183)-(1x196) = 2562-196
= 2366
I am finally going to get the result for the 15x15 grid and compare all the results for the main grid size feature.
15x15
(15x211)-(1x225) = 3315-225
= 3090
By looking at all the diagrams I have come up with a formula that works. Although this is presented on a 3x3 grid it works for any main grid size feature, i.e. 10x10, 11x11, 5x5, 15x15 e.c.t.
For example, I have looked at the results of the previous grid squares and I have transferred the information into a formula grid square. :
CPD (Cross Product Difference):
[N+(g-1)] [N+M(g-1)] –N [N+M(g-1)+(g-1)]
= N² + M(g-1)N + N(g-1) + M(g-1)²- N²- M(g-1)N - N(g-1)
= N² + M(g-1)N + N(g-1) + M(g-1)²- N²- M(g-1)N - N(g-1)
= M(g-1)²
Therefore my formula works. I have retested this to see if this works on a 16x16 grid and it has proven that it does. In the formula the letters stand for:
N = the original number
g = grid size on side of actual grid
M= Main grid size.
16x16
(16x241)-(1x256) = 3856-256
= 3600
If you place these numbers into the formula the answer should be correct to the formula. For example:
[1+(16-1)] [1+16(16-1)] –1[1+16(16-1)+(16+1)]
= 1²+16(16-1)1 + 1(16-1) +16(16-1)² - 1² - 16(16-1)1 – 1(16-1)
= 1²+16(16-1)1 + 1(16-1) +16(16-1)² - 1² - 16(16-1)1 – 1(16-1)
= 16(16-1)1
As you can see the formula and the equation above work in the same way, by cancelling out and making a final formula/result.
Now I am going to investigate STEP SIZE. When changing the step size it does matter what the main grid size is to now what the maximum number is to determine what the next row will start on which will determine the rest of the sequence in the grid.
I am going to investigate on a 10x10 main grid size with a step size of +2 each time starting on 1, therefore being odd numbers. I will start with 2x2 grids moving onto 3x3, 4x4 …
2x2
(3x21)-(1x23) =63-23
=40
(111x129)-(109x131) = 14319-14279
=40
(173x191)-(171x193) = 33043-33003
=40
3x3
(53x89)-(49x93) = 4717-4557
=160
(159x195)-(155x199) = 31005-30845
=160
(15x51)-(11x55) = 765-605
=160
4x4
(107x161)-(101x167) = 17227-16867
=360
(77x131)-(71x137) = 10087-9727
=360
(9x63)-(3x69) =567-207
=360
I have spotted a similarity with the step size of the normal 10x10 grid, ranging from 1-100 and the changed 10x10 grid, ranging from 1-199. As all the numbers are odd the sequence follows the odd pattern of 10x10 grid. So 3x3, 5x5, 7x7…
Bold – predicted occurrence
Therefore the same would happen for all changed step sizes. For example:
Another example is a main grid size of 11x11 and a step size of +3.
This shows that there is a logical pattern to all the step sizes. Once again this can be summarised into a general formula:
[N+s(g-1)][N+Ms(g-1)]–n[N+Ms(g-1)+s(g-1)
=N²+MsN(g-1)+ Ns(g-1)+ Ms²(g-1)²-N²-MsN(g-1)-Ns(g-1)
=N²+MsN(g-1)+ Ns(g-1)+ Ms²(g-1)²-N²-MsN(g-1)-Ns(g-1)
=Ms²(g-1)²
I am going to examine the change in shape, so 2x2 to 2x3, 2x4, 2x5 e.c.t. I am exploring the similar patterns of squares in rectangles.
2x2 grid has already been done before, so I will start with 2x3.
2x3
(16x24)-(14x26) = 384-364
= 20
(44x52)-(42x54) = 2288-2268
= 20
(79x87)-(77x89) = 6873-6853
=20
2x4
(29x36)-(26x39) = 1044-1014
= 30
(37x44)-(34x47) = 1628-1598
= 30
(84x91)-(81x94) = 7644-7614
= 30
2x5
(70x76)-(66x80) = 5320-5280
= 40
(35x41)-(31x45) = 1435-1395
= 40
(N+4)(N+10)-N(N+14)
=N²+10N+4N+40-N²–14N
= 14N-14N+40
= 40
Orange / Grey – see results clearer.
Turquoise – predicted results
Now I will draw up a table of results for the 2x2, 2x3, 2x4, 2x5 and then predict what the 2x6, 2x7 and 2x8’s results should be.
I have once again noticed another formula can be formed, to find the result of each grid size result.
The formula is:
10(N-1)
I am now going to show if the formula works and test the predicted grid size results. Where N is the grid size.
2x6
(6x11)-(1x16) = 66-16
= 50
2x7
(10x14)-(4x20) = 140-80
= 60
The results are a success, so now I will test the formula:
10(N-1)
N=3
(10x3)+(10x-1) = 20= 2x3
N=5
(10x5)+(10x-1) = 40= 2x5
N=9
(10x9)+(10x-1) = 80= 2x9
This proves that the formula is correct. The next step is to change the grid sizes again this time, 3x3, 3x4, 3x5, 3x6 e.c.t.
3x4
(46x63)-(43x66) = 2898-2838
= 60
(8x25)-(5x28) = 200 – 140
= 60
(74x91)-(71x94) = 6734-6674
=60
3x5
(30x46)-(36x50) = 1380-1300
= 80
(77x93)-(73x97) = 7161-7081
= 80
(19x35)-(15x39) = 665-585
= 80
3x6
(69x84)-(64x89) = 5796-5696
= 100
(17x32)-(12x37) = 544-444
= 100
(46x61)-(41x66) = 2806-2706
= 100
Orange / Grey – see results clearer.
Turquoise – predicted results
Now I will draw another table of results, and then predict the further 3x7, 3x8 and 3x9 results.
I have once again noticed another formula can be formed, to find the result of each grid size result.
The formula is:
20(N-1)
I will now test to see if my formula is correct.
N=7
(20x7)+(20x-1) = 140-20
= 120
N=8
(20x8)+(20x-1) = 160-20
=140
N=9
(20x9)+(20x-1) = 180-20
= 160
I have now found out a formula for the 2x3 – 2x10 shape sizes and 3x4 – 3x10 shape size so now I will change the shape size to 4x5, 4x6 and 4x7.
4x5
(9x35)-(5x39) = 315-195
= 120
4x6
(27x52)-(22x57) = 1404-1254
=150
4x7
(36x60)-(30x66) = 2160-1980
= 180
I notice that every time you increase the shape size of the 4x4 grid the cross product difference increases each time by 30, therefore the formula would be 30(N-1)
N=7
(30X7)-(30x1) = 210-30 = 180
N=9
(30x9)-(30x1)=270-30= 240
Now I have got some results for the feature shape size I have found out a overall formula that would work.
Now I have drawn up a formula table I will now find out how to find out the cross product difference of all the changed shapes through the means of a formula.
[N+s(L-1)][N+Ms(w-1)] – N[N+Ms(w-1)+s(L-1)]
=N²+NMs(w-1) + Ns(L-1) + Ms (L-1)(w-1) – N² –NMs(w-1) -Ns(L-1)
=N² +NMs(w-1) + Ns(L-1) + Ms (L-1)(w-1) – N² –NMs(w-1) -Ns(L-1)
= Ms²(L-1)(w-1).
Know I am going to test to see if my formula is right to find the cross product difference.
The grid will have the following specifications:
Main grid size: 10X10
Step Size: 1
Length of shape: 2
Width of shape: 2
[17+1(2-1)] [17+10x1(2-1)] -17 [17+10x1(2-1)+1(2-1]
= 289 + 170 + 17 +10 – 289 – 170 – 17
= 10
If I input the numbers of the tested grid into the formula the answer should be the same as the answer 10.
Ms²(L-1)(w-1)
= 10x1²(2-1)(2-1)
=10x1(1)(1)
=10
Therefore the formula must be correct as the answer of the tested grid is the same as the answer from the formula.
I have found out many different formulas during the course of the coursework and through each feature changed the feature before is linked in someway to the next, whether it was the Main grid size or like the one above all features linked together.
M=Main grid size
s= Step size
L=Length of shape
w=width of shape.
In conclusion I have successfully completed 3 different features that could be changed. The Main grid size(M), the step size(s) and the change in shape(L,w). I feel that I productively carried out my task finding out the cross product difference not only in number work and grids but the formulas as well.
