I have been given a 10 by 10 grid, with a 2 by 2 box going around numbers 12, 13, 22 and 23, as shown below.
I have been asked to find the product of the top right number and the bottom left number in the box (2 by 2 box). Then I was told to do the same with the top left number and the bottom right number. After that I had to calculate the difference between these products.
13 x 22 = 286 _
12 x 23 = 276
10
By doing this I have decided to investigate further and see what happens with more 2 by 2 squares.
3 by 3 boxes
4 by 4 boxes.
5 by 5 boxes
Results:
Term (n)…… 1 2 3 4 5
Differences in
Opposite corners…… 0 10 40 90 160
10 30 50 70 20 20 20
We know that the nth term will have to be squared. We know this by the second difference is all the same.
I have found the rule. nth term squared x ten = n+1. to make it simpler I’ll put it in algebraic form, 10(n-1)²
I would now like to find out if this will be the same for rectangles. I’m going to start by a 2 by 3 rectangle.
2 by 3 rectangle
2 by 4 rectangle
2 by 5 rectangle
I predict that every time I increase the width by one, the difference increases by 10.
2 by 6 rectangle
2 by 7 rectangle
Is the difference different when the rectangle is aligned so that its shortest sides are at the top and bottom?
We also know this rule works for a 2 by 2 rectangle as it was done before.
Results:
Term (n)…… 1 2 3 4 5 6
Differences in
Opposite corners 0 10 20 30 40 50
10 10 10 10 10
The nth term is 10(n-1)
Does this rule work with other rectangles?
I believe I shall require only one example of each and will proceed on this principle. I will also be changing the layout to make it easier.
3 x X
3 x 3
3 x 4
3 x 5
Results:
Term (n)…… 1 2 3 4 5
Differences in
Opposite corners 0 20 40 60 80
20 20 20 20
The nth term is 20(n-1)
