Number Grid Investigation.
Number Grid Investigation
I was first given A 10x10 grid, counting from 1-100. Inside the grid was A 2x2 box surrounding the numbers, 12, 13, 22 and 23;
2
3
4
5
6
7
8
9
0
1
2
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4
5
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7
8
9
20
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I was asked to;
* Find the product of the top left number and bottom right number in the box.
* Do the same with the top right and bottom left numbers n the box.
* Calculate the difference between these numbers.
I did this,
2 x 23 = 276
3 x 22 = 286 286 - 276 = 10
The difference between them was 10
I decided to try it with A 3x3 box surrounding the numbers;
2
3
4
22
23
24
2 x 34 = 408
4 x 32 = 448 448 - 408 = 40
The difference between them was 40
I also tried this with A 4x4 box, and A 5x5 box:
(4x4) 12 x 45 = 540
15 x 42 = 630 630 - 540 = 90
(5x5) 12 x 56 = 672
16 x 52 = 832 832 - 672 = 160
...
This is a preview of the whole essay
I decided to try it with A 3x3 box surrounding the numbers;
2
3
4
22
23
24
2 x 34 = 408
4 x 32 = 448 448 - 408 = 40
The difference between them was 40
I also tried this with A 4x4 box, and A 5x5 box:
(4x4) 12 x 45 = 540
15 x 42 = 630 630 - 540 = 90
(5x5) 12 x 56 = 672
16 x 52 = 832 832 - 672 = 160
I decided to try and find A pattern between these numbers, I thought that since the boxes grows in an even number, so should the totals.
Put into order they looked like this:
Side of box length 2 3 4 5
Difference in numbers 10 40 90 160
Difference 30 50 70
20 20
There was A definite pattern between the numbers, this gave me the idea that I might be able to find A formula for the difference.
I thought that if I were to start working out a formula, I should name some of the variables:
Side of box = M
Top of box = N
Side of grid = X
Bottom of grid = Y
Difference = L
I now had to start by trying to work out the sequence of the numbers:
M = 2 3 4 5
L = 10 40 90 160
30 50 70
20 20
To work out A sequence, we first see how many times we have to look for an equal difference, in this case we have to go look twice, this means that M will be squared;
M2
We next look at the difference, in this case 20, we halve it, this will be multiplied by M;
M2 x 10
This should be the formula, I will test it;
42 x 10 = 160
This is clearly wrong, as this is the answer for the next part, but I can see what's wrong, so I can fix it. The problem is that I am going to have to take 1 away from M before it is multiplied by 10 to get the correct formula;
(M - 1)2 x 10, or factorised, 10(M - 1)2
This will now work, (4 - 1)2 x 10 = 90
I have found the formula for any square box, but the box must be square, as in the formula I only have one variable for size of box, and if the two sides are different, the formula will take the side put in to be both sides.
This is A problem, I have decided to work on rectangular boxes.
Rectangular Boxes
When working out the rectangular boxes, I must keep one side the same, otherwise I will never find A pattern in totals, so I will work out the following boxes, keeping one side at 2 squares;
2
3
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9
0
1
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9
20
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I will not write here all of my working, as it would take up too much space, so I will just write the results;
N x M = L
2x2 = 10
3x2 = 20
4x2 = 30
5x2 = 40
6x2 = 50
There is A certain pattern here, as each time we add 1 to N, the difference goes up by 10.
Instead of taking M to be the variable in this formula, I will use N as it is the one changing;
N --> 2 3 4 5 6
L --> 10 20 30 40 50
10 10 10 10
As I only had to look once for A common difference, N will not have to be squared.
The formula turned out like this:
0 (N - 1)
I tested this for the 7th part of the 'sequence',
0 (7 - 1) = 60, which, as you can see from the numbers above, is the next part of the sequence, I have found A formula for rectangles where M = 2.
