Maths Grids Totals

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Module 4 Coursework

I am going to investigate squares of different sizes and on different grids.

I am going to draw a square around numbers on a grid, and find the product of the top-left and bottom-right numbers, and the top-right and bottom-left numbers. I will then calculate the difference.

10 x 10 grids

2 x 2 squares

12 x 23 = 276

13 x 22 = 286

286 – 276 = 10.

55 x 66 = 3630

56 x 65 = 3640

3640 – 3630 = 10.

25 x 36 = 900

26 x 35 = 910

910 – 900 = 10

I have found that that there is a difference of 10 on any 2 x 2 square, on a 10 x 10 grid.

3 x 3 squares

36 x 58 = 2088

38 x 56 = 2128

2128 – 2088 = 40.

77 x 99 = 7623

79 x 97 = 7663

7663 – 7623 = 40.

4 x 4 squares

25 x 58 = 1450

28 x 55 = 1540

1540 – 1450 = 90.

5 x 5 squares

32 x 76 = 2432

36 x 72 = 3592

3592 – 2432 = 160.

I have now found out the differences for 2x2, 3x3, 4x4 and 5x5 squares on a 10x10, which will be shown in the following table:

All the differences are square numbers of the previous number multiplied by 10 (e.g. 3 x 3 = 10 x 22 = 10 x 4 = 40). This gives me the formula 10(n-1)2.

Using this formula, I predict that the difference for an 8 x 8 square will be 10(8-1)2 = 10 x 72 = 10 x 49 = 490.

I will take a random 8 x 8 square from the 10 x 10 grid:

12 x 89 = 1068

19 x 82 = 1558

1558 – 1068 = 490.

My prediction was correct; therefore this proves that the formula 10(n-1)2 is correct.

I am going to prove 10(n-1)2 works, algebraically. First, I am going to show an n x n square on a 10 x 10 grid (algebraically).

“x” is the number in the top-left corner. The number to the right of it is “x+ (n-1)” because it is “x” added to the length of the square (take away one). Because the square is on a 10 x 10 grid, the formula for the number underneath the square is “x” added to (10 multiplied by “the squares length”-1). The bottom-right number is “x+ 11(n-1)” because it is “x+ 10(n-1) added to another (n-1), so therefore this becomes “x+ 11(n-1).

 

I am now going to multiply the top-left and bottom-right numbers, and the top-right and bottom-left numbers:

 [x + (n-1)][x + 10(n-1)]= x2 + 10x(n-1) + x(n-1) + 10(n-1)2.

x[x+11(n-1)] = x2 + 11x(n-1).

When multiplied out, this becomes: x2 + 10x(n-1) + x(n-1) + 10(n-1)2 x2 – 11x(n-1).

x2 = x × x. 10x(n-1)= x × 10(n-1). x(n-1) = x × (n-1). 10(n-1)2 = 10(n-1) × (n-1).

– x2 = x × x. 11x(n-1) = x × 11(n-1).

The x2 and –x2 cancel each other out, while the 10x(n-1) and x(n-1) add together to cancel out the -11x(n-1). This leaves 10(n-1)2, which is the overall formula for a 10 x 10 grid.

9 x 9 Grids

2 x 2 squares:

  33 x 41 = 1353

  32 x 42 = 1344

1353 – 1344 = 9.

36 x 44 = 1584

35 x 45 = 1575

1584 – 1575 = 9

11 x 19 = 209

10 x 20 = 200

209 – 200 = 9

3 x 3 squares:

3 x 19 = 57

1 x 21 = 21

57 – 21 = 36

61 x 77 = 4697

59 x 79 = 4661

4697 – 4661 = 36

4 x 4 squares:

24 x 48 = 1152

21 x 51 = 1071

1152 – 1071 = 81

8 x 32 = 256

5 x 35 = 175

256 – 175 = 81                

5 x 5 squares:

42 x 74 = 3108

38 x 78 = 2964

3108 – 2964 = 144

24 x 56 = 1344

20 x 60 = 1200

1344 – 1200 = 144.

All of the differences are multiples of 9. The formula for the 10 x 10 grid was 10(n-1)2. Is it possible that the formula for a 9 x 9 grid is 9(n-1)2?

Join now!

A 2 x 2 square would be 9(n-1)2 = 9(2-1)2 = 9 x 1 = 9.

A 3 x 3 square would be 9(n-1)2 = 9(3-1)2 = 9 x 4 = 36.

A 4 x 4 square would be 9(n-1)2 = 9(4-1)2 = 9 x 9 = 81.

A 5 x 5 square would be 9(n-1)2 = 9(5-1)2 = 9 x 16 = 144.

Now that the formula has been established, I am going to use it to predict the difference for a 7 x 7 square. 9(n-1)2 = 9(7-1)2 = 9 x 62 = 9 x 36 = 324.

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