Number Grids

(5x5) every time gave a difference of 160

(6x6) every time gave a difference of 250

(7x7) every time gave a difference of 360

(8x8) every time gave a difference of 490

(9x9) every time gave a difference of 640

(10x10) every time gave a difference of 810

Next I tried to find a pattern; I saw that the differences had a difference between them going up in twenty's starting with 30 then 50 etc. but realized that it was not helping or going anywhere.

I then looked at the similarity that all these answers were factors of 10 and thought of the possibility this could be involved in an algebraic equation.

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I came up with the equation 10(n - 1) and tested it. My answers were the same, but not proved.

Proof

Firstly I figured out how to find the number in each corner from just the first in a square.

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I came up with:

X=X

Y=X + (n-1)

Z=X + ((n-1) x 10)

Q=X + ((n-1) x 10) + (n-1)

I used this to formulate the equation:

X x X+11(n-1)) - ((X+ (n-1)X+10(n-1)))

This cancels down to:

(X2+11X(n-1))-(X2+11X(n-1) +10(n-1)2)

To work out my overall formula I will subtract the first part of the Formula from the other:

X2+11X(n-1) - X2+11X(n-1)) +10(n-1)2

0 + 0 + 0 +10(n-1)2

=10(n-1)2

Rectangles

Solving the puzzle of rectangles in a grid came quite quickly to me as I realized that n was just for use in a square because the dimensions were the same. I labeled the vertical length now as m, but keeping the horizontal length as n. I adapted the equation from 10(n-1)2 into 10(n-1)(m-1). I did this because I saw if you write out the full equation for a square it is 10(n-1)(n-1) the second part (in red) is just the second dimension.

Testing

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Firstly I worked out the answer before putting it into my equation.

Now to test the formula, 10(4-1)(2-1)= 30! 10(2-1)(3-1)= 20!

Different Sized Grids

When I came to the extension on different sized grids I noticed that the size of the grid I had been using was 10 x 10 and in the equation I had formulated before, 10(n-1)2, that the first number was 10 and decided to change this number according to the size of my grid. E.g. 9x9 grid = 9(n-1)2, 8x8 grid = 8(n-1)2 etc. In my equations the vertical dimension of the grid will be equal to g. This leaves us with the following formula.

(n-1) (m-1) g

N.B. n represents the width of the inner grid, m represents the length of the inner grid, g represents the width of the outer grid.

Now I tested my new formula, which should work, for any sized square or rectangle in any sized grid.

Firstly, I tried a 3x3 (red) inner grid in a 7x7 (black) outer grid.

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(3-1)(3-1)7

=2x2x7

=28

x 17= 17, 3 x 15=45, 45-17=28 so my formula works. However, it should work with a rectangular grid as well. I tried a 3x4 inner grid in a 6x7 outer grid.

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(3-1)(4-1)7

=2x3x7

=42

The formula still works.

4x23 = 92, 2x25 = 50, 92-50 = 42 so I have found a formula which will work whatever of the size, inner or outer grid.

Conclusion

I have now found three equations, from these I can find any size of rectangle or square in any size of number grid. They are:

* 10(n-1)2, for any square inside a 10x10 grid

* 10(n-1)(m-1), for any rectangle inside a 10x10 grid

* g(n-1)(m-1), for any rectangle or square inside any grid

Josef Jeffrey Math's G.C.S.E coursework