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Opposite Corners

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Introduction

Louis Franks 10PC  10X2                                                                                                         10/12/01

Opposite Corners

Wx L

Difference

Increase

Wx L

Difference

Increase

2 x  3

20

10

6 x  3

100

50

2 x  4

30

10

6 x  4

150

50

2 x  5

40

10

6 x  5

200

50

2 x  6

50

10

6 x  6

250

50

2 x  7

60

10

...read more.

Middle

10

6 x  9

400

50

2 x  10

90

10

6 x  10

450

50

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

1

2

3

11

12

13

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.

1

2

3

11

12

13

...read more.

Conclusion

  1. 6 – 1 x 10 = 50

Then

3 x 50 – 50 = 100

2. (6 – 1) x (10 (3 – 1) = 100

3.

1
2
3
4
5
6
11
12
13
14
15
16
21
22
23
24
25
26

(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.  

...read more.

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