Pattern:
x 10 x 10 x 10 x 10 x 10
As the differences for the square box was a pattern of square numbers as well as multiples of ten, I was curious to find what would happen if I changed the shape of the box. I chose to draw a rectangular box around the selected numbers. To do this I used the same process as with the square box.
10 x 10 Grid
Box size: 3 x 2
I therefore chose to repeat this exercise again with a larger rectangle to try to prove my prediction. Following the same action as previously I drew a 4 x 3 rectangle around the numbers and again omitting the numbers in the middle.
Box size: 4 x 3
To find the difference for box 7 x 6 use box 6 x 5
6 x 5 = 30
30 x 10 = 300
Pattern:
x 10 x 10 x 10 x 10
On closer examination, the differences for both box shapes display another pattern. For the square box the difference is: add 30 to the 10 to make the difference 40, then add 20 to the 30 to find the next difference and so on.
+30 +50 +70 +90 +110
+20 +20 +20 +20
The process to find the difference for the rectangular box is, again, almost the same as for the square box:
+40 +60 +80 +100
+20 +20 +20
I believe that the patterns are so similar because the grid used was the same for both shaped boxes, I also believe that the number sequences of the differences are multiples of ten because the grid has ten numbers in each row.
Because of this belief I chose to try another number grid, this time I would be using a grid that was 8 x 8.
I will be using the same process as before of finding the product of the top left number and bottom right number, then the same with the top right number and bottom left number. I will then find the difference by subtracting the smaller of the two products from the larger.
8 x 8 Grid
Box size 2 x 2
12 x 30 = 360 392 – 360 = 32
14 x 28 = 392
Box size 3 x 3
2 x 47 = 94 294 – 94 = 200
7 x 42 = 294
Pattern:
x 8 x 8 x 8 x 8
I believe that this proves that the differences for the square box on the 8 x 8 grid will definitely be a multiple of eight, but, I have decided to try a rectangular box on the 8 x 8 grid to see what the changes with the differences will be if any.
8 x 8 grid
Box size 3 x 2
Pattern
x 8 x 8 x 8 x 8
Although I believe that this evidence proves all of my predictions, I have decided to try another number grid to be sure that any results I gain are accurate.
I have decided to use a 12 x 12 grid for the next example.
12 x 12 Grid
Instead of drawing tables for all of the square boxes that I drew around the numbers, I chose to draw just one table including the box sizes and the differences, I used the same process four times for each box, 2 x 2, 3 x 3, 4 x 4 and 5 x 5.
Pattern for square box
x 12 x 12 x 12
Pattern for rectangular box
x 12 x 12 x 12
Conclusion
For this project, I drew a box around a selected amount of numbers,
I then found the product of the top left number and bottom right number, then the top right number and bottom left number. I then found the difference by subtracting the smaller product from the larger one. To ensure that I ended with accurate results I changed the variables, I started with a 10 x 10 grid and drew both square and rectangular boxes, I then repeated the exercise for an 8 x 8 grid and a 12 x 12 grid.
I have recorded my results in tables and drawings to show any patterns between the differences.
I believe that through my work during this project I have, proven every one of the predictions that I have made. I have proven that the reason for the difference being a multiple of 10 for the 10 x 10 grid is because there are ten numbers in each row. I believe this as the 8 x 8 grid and the 12 x 12 grid follow the same example.
On closer inspection of the patterns, I was able to see an easier way to explain how to work out the difference using algebra. By using the tables showing the number patterns, I could draw a simpler version that also enabled me to find the nth term:
Number pattern for square box, 10 x 10 grid
Therefore n = box number and d = difference
d = 10n2
Example: d = 10 x 22
= 10 x 4
= 40
Therefore the next term of the sequence would be:
d = 10 x 32
= 10 x 9
= 90
This way to find the way the number pattern works also works for the 8 x 8 grid:
Number pattern for square box, 8 x 8 grid
d = 8n2
Therefore: d = 8 x 22
= 8 x 4
= 32
And the next term would be:
d = 8 x 32
= 8 x 9
= 72
As before this same, process applies to the 12 x 12 grid:
Number pattern for square box, 12 x 12 grid
d = 12n2
Therefore: d = 12 x 22
= 12 x 4
= 48
And the next term in the sequence:
d = 12 x 32
= 12 x 9
= 108
The same rule for finding the nth term applies for all number patterns with regards to the square boxes, and, although the patterns for the rectangular boxes are very similar to that of the square boxes, I was unable to find the nth term. I believe that this is because the patterns created by the rectangular boxes are not linear sequences.
