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Number Grids Investigation

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*My coursework task is to investigate why, in a number grid square of 1-100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10. *### Preview

Number Grid Investigation

My coursework task is to investigate why, in a number grid square of

1-100, when a section of two by two squares is extracted and the two

opposite squares are multiplied and then subtracted the result is

always 10.

I will also be testing and studying whether it is true for three by

three, four by four, five by five e.t.c number squares. I shall also

be studying what will happen if I change the size of the grid square

upon which I am extracting the numbers from.

E.g.

[IMAGE]

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I shall also be using 4x4, 5x5 6x6 up to 10x10

[IMAGE]

[IMAGE] 2x2 square

[IMAGE]

[IMAGE]

[IMAGE]

3x3 square

2 by 2 Analyses

[IMAGE]

[IMAGE] (2x11) - (1x12) = 10

[IMAGE]

[IMAGE] (35x44) - (34x45) = 10

[IMAGE]

[IMAGE] (3x12) - (2x13) = 10

[IMAGE]

[IMAGE] (99x90) - (89x100) = 10

As we can see the results clearly show that no matter what selection

of 2x2 square we use the result will always be 10.

We can show how and why the result is always 10 by using Algebra,

(representing numbers by using letters).

[IMAGE][IMAGE]

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This is how we can express numbers using letters. As with numbers we

can also put letters into a formula. This is how it would look:

(P+10)(P+1)- P (P+11) = 10

When we multiply out this formula it looks like this:

P²+11P+10 -P² - 11P = 10 As it shows the result is 10 once again

P²+11P+10 -P² - 11P = 10

Testing the Formula

Now we can put the formula to the test by using it with numbers:

(23x32)- (22x33) =10

[IMAGE][IMAGE]

=

As we can see ...

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