The product of the numbers in the first grid is
12 x 23= 276
13 x 22= 286
This means that the products have a difference of 10.
The products of the second grid are
36 x 47= 1692
46 x 37= 1702
Another product difference of 10
I will try looking at some other grids
The products for the first number grid are
32 x 43= 1376
33 x 42= 1386
Again a product difference of 10.
Stephanie Evans – 20002303
The second grid is
54 x 65= 3510
64 x 55= 3520
This again shows a difference of 10.
In order to check my theory further I need to change the size of the grid within a grid that I look at. My prediction says that as long as the smaller grids are the same size, no matter where they are placed in the larger grid the difference between products of the opposing numbers will be the same. Ill next look at a 3x3 number grid.
Now let’s have a look at these grids
12 x 34=408
14 x 32=448
A product difference of 40
46 x 68=3128
48 x 66=3168
Again a product difference of 40
Thus indicating that my initial theory is correct, but I need to make sure so Ill try another series of 3x3 grids.
Stephanie Evans – 20002303
18 x 40=720
20 x 38=760
The product difference shows again as 40.
43 x 65=2795
45 x 63=2835
Again the difference is 40.
So far my theory is being proved correct. In order to eliminate any doubt in my theory I will try some other variations using 4 smaller grids within the main grid that are of equal size and see if I have similar results.
A 4x4 grid
11 x 44=484
14 x 41=574
A product difference of 90
Stephanie Evans – 20002303
16 x 49=784
19 x 46=874
Again product difference of 90
61 x 94=5734
64 x 91=5824
Difference=90
66 x 99=6534
69 x 96=6624
Product difference = 90
1 x 45=45
5 x 41=205
Product difference =160
6 x 50=300
10 x 46=460
Product difference= 160
51 x 95=4845
55 x 91= 5005
Product difference = 160
56 x 100=5600
60 x 96=5760
Product difference= 160
Stephanie Evans – 20002303
Looking at square grids my theory is proved. Now can we find a rule? A simple algebra equation which would allow us to find out what the product difference is providing we know the smaller grid sizes without working it all out.
The first thing I notice is that the area of the previous grid multiplied by 10 is equal to the difference of the following grid. I.e. the area of the 2x2 grid =4 x 10 = 40 and the difference of the product difference of the 3x3 grid is 40. This is the case with the other grids I tested.
So a possible rule could be the length/ height of the grid minus one squared multiplied by ten. The rule looks like this:
(L-1) x (H-1) squared x 10.
So in the first grid 2-1=1 and 1 squared is 1 x 10 =10,
3-1 = 2 squared is 4 x 10 = 40,
4-1= 3 squared =9 x 10 = 90,
5-1=4 squared = 16 x 10 = 160.
The rule works so far. Taking what we have found out into consideration, can we then predict that a 6x6 grid would have product differences of 250, and a 7x7 grid would be equal to 360?
Stephanie Evans – 20002303
1 x 56=56
6 x 51=306
The product difference is 250 as predicted.
12 x 67= 804
17 x 62= 1054
Again showing the rule works giving a difference of 250.
Stephanie Evans – 20002303
Looking now at the 7x7 grid:
4 x 70=280
10 x 64=640
Difference=360
22 x 88=1936
28 x 82=2296
Difference = 360
Again we see here that the rule is proved right. Could the same rule apply to rectangles within the grid? If my rule is correct a 2x3 rectangle in the grid will have product differences of 20 because the width of 2-1 x height of 3-1 = 2 x 10 = 20.
So if a 2x3 or 3x2 rectangle should have a difference of 20, then a 3x4 or 4x3 rectangle will have a difference of 60.
Stephanie Evans – 20002303
Lets again test the theory. A 2x3 or 3x2 rectangle should have a product difference of if my (L-1) x (H-1) x10 rule is correct the product difference should be 20.
11 x 23=253
13 x 21=273
Difference = 20
15 x 27=405
17 x 25=425
Difference = 20
42 x 63=2646
43 x 62=2666
Difference = 20
46 x 67=3082
47 x 66=3102
Difference = 20
The rule is working so far.
For a 3x4 or 4x3 rectangle the difference should be 60, because 3-1=2 and 4-1=3. 2x3=6 6 x 10=60.
1 x 24 = 24
4 x 21 = 84
Difference = 60
7 x 30 = 210
10 x 27 = 270
Difference = 60
41 x 73 = 2993
43 x 71 = 3053
Difference = 60
46 x 78=3588
48 x 76=3648
Difference = 60
Stephanie Evans – 20002303
My Rule is proved the length of the grid (L) minus 1 multiplied by the height of the grid (H) minus 1 multiplied by ten will give me the product difference for any square or rectangle size in a 10 x 10 number grid.
My next question is what if I change the size of the larger number grid. Will my rule still work? If I try a 5x10 grid does my (L-1) x (H-1) x 10 rule still apply?
In my initial test a 2x2 grid had a product difference of 10.
1 x 7= 7
2 x 6= 12
The difference is 5 my rule doesn’t work in this case. Could it be that my rule is therefore relative to the large grid width. In the first grid the width was 10 and my rule was (L-1) x (H-1) x 10.
By looking at my first small grid in this 5x10 grid the difference is 5.
Could my rule then be (L-1) x (H-1) x grid width (G)?
The length of the grid is irrelevant as the order in which the
Numbers appear would not change. If the new rule works a 3x3 square in this grid will have a difference of 20 because 3-1 = 2 and 2x2x5 = 20, and a 4x4 square will have a difference of 45.
1 x 13= 13
3 x 11= 33
Difference = 20
21 x 39=819
24 x 36=864
Difference = 45
Stephanie Evans – 20002303
A 3x2 or 2x3 rectangle should have a difference of 10, and a 3x4 or 4x3 a difference of 45.
1 x 8= 8
3 x 6= 18
Difference = 10
31 x 49= 1519
34 x 46= 1564
Difference= 45
Summary
In conclusion my rule is correct and will work for and sized square or rectangle within any grid.
H = Height of the square/ rectangle
L= Length of square/ rectangle
G= Width of the number grid
In algebra terms the equation can then be written as (L-1) x (H-1) x G.
My investigation has progressed in such a way that I have found a rule which I am convinced will apply to any square or rectangle in any given grid width. I am happy that I have extended the investigation in such a way that has proved my theory, and that my rule works.