Opposite Corners

We are investigating the difference between the products of the numbers in the opposite corners of any rectangles that can be put on a 100 square.

2 x  3 Rectangles

                         

To keep things simple I have started with rectangles with a width of 2 squares. I kept the width to two squares and increased the length by one square.  (see results table above).  I discovered that the width increases by 10 every time the length increases by 1.

The difference can be worked out for all rectangles with a width of 2 squares by using several formulas:

1. (Length – 1 x 10 = Z)

3 – 1x 10 = 20 = Z

Then

(Width x  Z ) – Z = difference of opposite corners

2 x  20 – 20 = 20

OR

2. L = Length,  W = Width

(L – 1) (10 (W-1)) = difference of opposite corners

Example:

(3 – 1) x (10 (2 – 1)) = 20

OR

3.

Using algebra and going on the theory that the width increases by 10 when the length is increased by 1, I have calculated the value of the corners.  This formula can also work out the difference.

(y+10) (y + 2) = y+ 20 +10y+ 2y

 = y + 20 +12y

y ( y + 12)

= y + 12y

(y + 20 +12y) – (y + 12y) = difference between product.

Extending the problem

The difference between the opposite corners will still be the same even if you make a billion square grid because the length will still increase by 1 and the width will increase by 10.

6 x 3 Rectangles

I have changed the size of the rectangle to see if my formulas will work for it.  (The results are in the table at the top of the first page)

1. 6 – 1 x 10 = 50

Then

3 x 50 – 50 = 100

2. (6 – 1) x (10 (3 – 1) = 100

3.

(y+20) (y + 5) = y+ 100 +20y+ 5y

 = y + 100 +25y

y ( y + 25)

= y + 25y

(y + 100 +25y) – (y + 25y) = difference between product.

= 100

Conclusion

I have come to the conclusion that my three formulas work for all types of rectangles and squares.  There are several ways to achieve the end result for the difference of the opposite corners.