Number Grid Investigation

2

2

0

0(r-1)²

3

3

40

0(r-1)²

4

4

90

0(r-1)²

5

5

60

0(r-1)²

Firstly, I noticed that if you took the zeros off the difference, then you would get square numbers for 1, 2, 3 and 4. So I tried to find a way where you get the square numbers:

2-1=1 x10= 10

The formula would be 10(r-1)

So I tried using this formula for a 3 by 3 square:

0(3-1) = 20

This was wrong as I had to get 40 not 20, so I would have to multiply the 10 by 4 somehow. So I squared the (3-1) to get 4.

0(3-1)²

This worked and so I substituted the number of rows and columns for letters and so the formula became:

0(r-1)² where r is the number of rows

I want to see how this formula will be altered for a 9 by 9 grid and a 5 by 5 grid. I predict that the formula for these will be 9(r-1)² and 5(r-1)² respectively.

This is a table to show the differences in squares in a 9 by 9 grid.

No. of Rows (r)

No. of Columns (c)

Difference

Formula

2

2

9

9(r-1)²

3

3

36

9(r-1)²

4

4

81

9(r-1)²

5

5

44

9(r-1)²

This is a table to show the differences in squares in a 5 by 5 grid.

No. of Rows (r)

No. of Columns (c)

Difference

Formula

2

2

5

5(r-1)²

3

3

20

5(r-1)²

4

4

45

5(r-1)²

5

5

80

5(r-1)²

I was right about the formulas and so an overall formula for any sized square would be: x(r-1) ² where x is a square root of the total numbers in the grid

I am now going to prove algebraically that in a 10 by 10 grid, with a 2 by 2 square the difference will be 10.

x

x+1

x+10

x+11

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

(x² + x + 10x + 10) - (x² + 11x)

(x² + 11x + 10) - (x² + 11x)

x² + 11x + 10 - x² - 11x

Difference =10

This proves algebraically that the difference in a 2 by 2 square is 10.

I am now going to prove algebraically that in a 10 by 10 grid, with a 3 by 3 square the difference will be 40.

x

x+2

x+20

x+22

((x + 2)(x + 20)) - x(x + 22)

(x² + 2x + 20x + 40) - (x² + 22x)

(x² + 22x + 40) - (x² + 22x)

x² + 22x + 40 - x² - 22x

Difference =40

This proves algebraically that the difference in a 3 by 3 square is 40.

I am now going to prove algebraically that in a 10 by 10 grid, with a 4 by 4 square the difference will be 90.

x

x+3

x+30

x+33

((x + 3)(x + 30)) - x(x + 33)

(x² + 3x + 30x + 90) - (x² + 33x)

(x² + 33x + 90) - (x² + 33x)

x² + 33x + 90 - x² - 33x

Difference =90

This proves algebraically that the difference in a 4 by 4 square is 90.

I am now going to prove algebraically that in a 10 by 10 grid, with a 5 by 5 square the difference will be 160.

x

x+4

x+40

x+44

((x +4 )(x + 40)) - x(x + 44)

(x² + 4x + 40x + 160) - (x² + 44x)

(x² + 44x + 160) - (x² + 44x)

x² + 44x + 160 - x² - 44x

Difference =160

This proves algebraically that the difference in a 5 by 5 square is 160.

I am now going to prove algebraically that in a 9 by 9 grid, with a 2 by 2 square the difference will be 9.

x

x+1

x+9

x+10

((x + 1)(x + 9)) - x(x + 10)

(x² + x + 9x + 9) - (x² + 10x)

(x² + 10x + 9) - (x² + 10x)

x² + 10x + 9 - x² - 10x

Difference =9

This proves algebraically that the difference in a 2 by 2 square is 9.

I am now going to prove algebraically that in a 9 by 9 grid, with a 3 by 3 square the difference will be 36.

x

x+2

x+18

x+20

((x + 2)(x + 18)) - x(x + 20)

(x² + 2x + 18x + 36) - (x² + 20x)

(x² + 20x + 36) - (x² + 20x)

x² + 20x + 36 - x² - 20x

Difference =36

This proves algebraically that the difference in a 3 by 3 square is 36.

I am now going to prove algebraically that in a 9 by 9 grid, with a 4 by 4 square the difference will be 81.

x

x+3

x+27

x+30

((x + 3)(x + 27)) - x(x + 30)

(x² + 3x + 27x + 81) - (x² + 30x)

(x² + 30x + 81) - (x² + 30x)

x² + 30x + 81 - x² - 30x

Difference =81

This proves algebraically that the difference in a 4 by 4 square is 81.

I am now going to prove algebraically that in a 9 by 9 grid, with a 5 by 5 square the difference will be 144.

x

x+4

x+36

x+40

((x + 4)(x + 36)) - x(x + 40)

(x² + 4x + 36x + 144) - (x² + 40x)

(x² + 40x + 144) - (x² + 40x)

x² + 40x + 144 - x² - 40x

Difference =144

This proves algebraically that the difference in a 5 by 5 square is 144.

I am now going to prove algebraically that in a 5 by 5 grid, with a 2 by 2 square the difference will be 5.

x

x+1

x+5

x+6

((x + 1)(x + 5)) - x(x + 6)

(x² + x + 5x + 5) - (x² + 6x)

(x² + 6x + 5) - (x² + 6x)

x² + 6x + 5 - x² - 6x

Difference =5

This proves algebraically that the difference in a 2 by 2 square is 5.

I am now going to prove algebraically that in a 5 by 5 grid, with a 3 by 3 square the difference will be 20.

x

x+2

x+10

x+12

((x + 2)(x + 10)) - x(x + 12)

(x² + 2x + 10x + 20) - (x² + 12x)

(x² + 12x + 20) - (x² + 12x)

x² + 12x + 20 - x² - 12

Difference =20

This proves algebraically that the difference in a 3 by 3 square is 20

I am now going to prove algebraically that in a 5 by 5 grid, with a 4 by 4 square the difference will be 45.

x

x+4

x+15

x+18

((x + 3)(x + 15)) - x(x + 18)

(x² + 3x + 15x + 45) - (x² + 18x)

(x² + 18x + 45) - (x² + 18x)

x² + 18x + 45 - x² - 18x

Difference =45

This proves algebraically that the difference in a 4 by 4 square is 45.

I am now going to prove algebraically that in a 5 by 5 grid, with a 5 by 5 square the difference will be 80.

x

x+4

x+20

x+24

((x + 4)(x + 20)) - x(x + 24)

(x² + 4x + 20x + 80 ) - (x² + 24x)

(x² + 24x + 80) - (x² + 24x)

x² + 24x + 80 - x² - 24x

Difference =80

This proves algebraically that the difference in a 5 by 5 square is 80.

Now I'm furthering my investigation by looking at rectangles.

This is a table to show the differences in squares in a 10 by 10 grid.

No. of Rows (r)

No. of Columns (c)

Difference

Formula

2

3

20

(10(r-1))x(c-1)

2

4

30

(10(r-1))x(c-1)

2

5

40

(10(r-1))x(c-1)

3

3

40

(10(r-1))x(c-1)

3

4

60

(10(r-1))x(c-1)

3

5

80

(10(r-1))x(c-1)

4

3

60

(10(r-1))x(c-1)

4

4

90

(10(r-1))x(c-1)

4

5

20

(10(r-1))x(c-1)

5

3

80

(10(r-1))x(c-1)

5

4

20

(10(r-1))x(c-1)

5

5

60

(10(r-1))x(c-1)

In this formula, r is the No. of Rows and c is the No. of columns.

Basically, you multiply (r-1) by 10 and then times your answer by (c-1).

Sandeep Kutty