Number Grid

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NUMBER GRID

Look at this number grid:

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

20

21

22

23

24

25

26

27

28

29

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32

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89

90

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98

99

00

0 x 10 grid

* A box is drawn round four numbers.

* Find the product of the top left number and the bottom right number in this box.

* Do the same with the top right and bottom left numbers.

* Calculate the difference between these products.

Investigate further.

The purpose of this investigation is to prove or disprove that there is a correlation between the products of corner numbers of any size box within any size grid. I shall calculate the diagonal difference (d) for a box. There are two ways to calculate the difference between the products:

V

W

Y

Z

i. W x Y - V x Z

ii. V x Z - W x Y

I shall begin with the 10 x 10 grid, as shown above, and a 2 x 2 box.

2 x 2 box #1

I have substituted numbers for the letters in my two formulae.

2

1

2

i. d = 2 x 11 - 1 x 12 = 22 - 12 = 10

ii. d =1 x 12 - 2 x 11 = 12 - 22 = -10

d = +/-10

I will use formula i, because formula ii creates negative numbers, which could make my calculations more complex than necessary.

2 x 2 box #2

2

3

22

23

d = 13 x 22 - 12 x 23

d = 286 - 276

d = 10

I think that d = 10 for any 2 x 2 box in a 10 x 10 grid. I will test this theory again, using the highest numbers possible in the grid, to check that it is not a coincidence.

2 x 2 box #3

89

90

99

00

d = 90 x 99 - 89 x 100

d = 8910 - 8900

d = 10

Algebraic proof: formula #1:

+1

+10

a

a + 1

a + 10

a + 10 + 1

d = (a + 1) (a + 10) - a (a + 10 + 1)

= a2 + 10a + a + 10 - a2 - 10a - a

= 10

I have noticed that d = 10, and the grid is 10 columns (G) wide. To test if this is a coincidence, I have amended my formula.

Algebraic proof: formula #2:

+1

+G

a

a + 1

a + G

a + G + 1

d = (a + 1) (a + G) - a (a + G + 1)

= a2 + aG+ a + c - a2 - aG - a

= G, where G = 10, d = 10

I will test formula #2 using a 2 x 2 box on a different size grid. I will use a 5 x 5 grid.

2 x 2 box in a 5 x 5 grid

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

20

21

22

23

24

25

5 x 5 grid

G = 5

2 x 2 box #4

2

6

7

d = 2 x 6 - 1 x 7

d = 12 - 7

d = 5

2 x 2 box #5

7

8

2

3

d = 8 x 12 - 7 x 13

d = 96 - 91

d = 5

2 x 2 box #6

9

20

24

25

d = 20 x 24 - 19 x 25

d = 480 - 475

d = 5
Join now!


The results show that d = G. I will use formula #2 from now on.

I shall go back to using a 10 x 10 grid. I shall test the formula using different sized boxes. I shall draw up a table to record my results. I shall compare the results to see if there is a pattern forming. This would prove correlation.

NB: To save time I shall only indicate the corner numbers in each box. The other numbers in the box are irrelevant for these calculations.

3 x 3 box #1

3

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