Maths Coursework - Number Grid

Kayleigh McCormack

Maths Coursework – Number Grid

For this task I will first be looking at a number grid from 1 to 100, like the one below :

I will start my investigation by looking at 2 by 2 squares. I will draw a square around 4 numbers, find the product of the bottom left and top right numbers and the product of the top left and bottom right numbers, then calculate the difference between the 2 products. I will see if there are any patterns and if so I will try to work it out algebraically.

I will then look at changing the size of the squares to see if there are any patterns. I will try looking at 3 by 3 squares, 4 by 4 squares and 5 by 5 squares; I will do the same with these squares as I have with the 2 by 2 squares, I will find the products of the top left and bottom right and the bottom left and top right numbers then calculate the difference between them.

Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles. I will start by looking at a 3 by 2 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern. I will then try changing the size of the rectangles and looking for patterns there. I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles.

I will also look at changing the size of the number grids to see if this has an affect on the patterns. I will look at a 9 by 9 grid, an 11 by 11 grid and a 5 by 5 grid. I will be looking for patterns in 2 by 2 squares within the different size grids and trying to find an algebraic formula to explain my findings.

First of all I am looking at 2 by 2 squares within a 10 by 10 grid. The squares I am looking at are highlighted in the grid below. I chose these squares randomly; there is no particular reason why I chose them.

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I will now calculate the products of the diagonals and the difference of these products for each square in the grid above. I will display my findings in a table to help me identify a pattern if there is one.

I have noticed that the difference of the products in each square is always ten.

The formula for this pattern is:

(N+1)(N+10) - N(N+11) = D

N2 + 10N + N + 10 - N2 - 11N = D

10 = D

Difference = 10

This shows that the difference between the two products in ...

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I have noticed that the difference of the products in each square is always ten.

The formula for this pattern is:

(N+1)(N+10) - N(N+11) = D

N2 + 10N + N + 10 - N2 - 11N = D

10 = D

Difference = 10

This shows that the difference between the two products in each square is always 10.

I will next look at 3 by 3 squares within a 10 by 10 grid. The squares I am looking at are highlighted in the grid below. I chose these squares randomly; there is no particular reason why I have chosen them.

I am now going to calculate the products of the diagonals and the difference of these products for each square in the grid above. I will display my findings in a table to help me identify a pattern.

I have noticed that the difference of the products in each square is always forty.

I have worked out the formula for this pattern below:

(N+2)(N+20) - N(N+22) = D

N2 + 20N + 2N + 40 - N2 - 22N = D

40 = D

Difference = 10

This shows that the difference between the two products in each square is always -40; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 40 rather than -40.

I will next look at 4 by 4 squares within a 10 by 10 grid. The squares I am looking at are highlighted in the grid below. I chose these squares randomly; there is no particular reason why I have chosen them.

I am now going to calculate the products of the diagonals and the difference of these products for each square in the grid above. I will display my findings in a table to help me identify a pattern.

I have noticed that the difference of the products in each square is always ninety.

I have worked out the formula for this pattern below:

N(N+33) - (N+3)(N+30) = D

N2 + 33N - N2 - 30N - 3N - 90 = D

-90 = D

Difference = 90

This shows that the difference between the two products in each square is always -90; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 90 rather than -90.

Now that I have investigated squares within a 10 by 10 grid and found algebraic formulas to explain the patterns I saw, I will move on to investigate rectangles within a 10 by 10 grid. I will first be looking at 3 by 2 rectangles. The rectangles I am looking at are highlighted in the grid below.

I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above. I will display my findings in a table to help me identify a pattern.

I have noticed that the difference of the products in each square is always twenty.

I have worked out the formula for this pattern below:

N(N+12) - (N+2)(N+10) = D

N2 + 12N - N2 - 10N - 2N - 20 = D

-20 = D

Difference = 20

This shows that the difference between the two products in each rectangle is always -20; I have shown the difference as 20 rather than -20 as I am only interested in the number and not the sign in front of it (+/-).

I will next look at 2 by 4 rectangles within the same grid. The rectangles I am looking at are highlighted in the grid below. I chose these rectangles randomly.

I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above. I will display my findings in a table so it will be easier to identify the pattern.

I have noticed that the difference of the products in each square is always thirty.

I have worked out the formula for this pattern below:

N(N+31) - (N+1)(N+30) = D

N2 + 31N - N2 - 30N - N - 30 = D

-30 = D

Difference = 30

This shows that the difference between the two products in each rectangle is always -30; I have shown the difference as 30 rather than -30 as I am only interested in the number and not the sign in front of it (+/-).

I am now going to look at 5 by 3 rectangles within the same 10 by 10 grid. The rectangles I am looking at are highlighted in the grid below. I chose these rectangles randomly.

I have noticed that the difference of the products in each square is always eighty.

I have worked out the formula for this pattern below:

N(N+24) - (N+4)(N+20) = D

N2 + 24N - N2 - 20N - 4N - 80 = D

-80 = D

Difference = 80

This shows that the difference between the two products in each rectangle is always -80; I have shown the difference as 80 rather than -80 as I am only interested in the number and not the sign in front of it (+/-).

I am now going to look at 4 by 5 rectangles within the same 10 by 10 grid. The rectangles I am looking at are highlighted in the grid below. I chose these rectangles randomly.

I have noticed that the difference of the products in each square is always one hundred twenty.

I have worked out the formula for this pattern below:

N(N+43) - (N+3)(N+40) = D

N2 + 43N - N2 - 40N - 3N - 120 = D

-120 = D

Difference = 120

This shows that the difference between the two products in each rectangle is always -120; I have shown the difference as 120 rather than -120 as I am only interested in the number and not the sign in front of it (+/-).