Number Grid Investigation

37*46=1702

702-1692=10

I did this a few times and kept on getting ten as the difference. I then worked out if the difference was always ten by working out what the value of each corner in the square is relative to the top left square.

X

X+1

X+10

X+11

((X+1)(X+10))-(X(X+11))=Difference

(X²+10X+X+10)-(X²+11X)=Difference

(X²+11X+10)-(X²+11X)=Difference

Difference = 10

This shows that the difference with a 2 by 2 square is always ten when it is in a grid with ten squares in each row.

I then tried to find out what the difference between the product of the top left and bottom right numbers and product the top right and the bottom left numbers would be in a 3 by 3 square using the same grid as before with ten numbers along the top:

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9

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9

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64*86=5504

66*84=5544

5544-5504=40

I tried this again with this square.

48

49

50

58

59

60

68

69

70

48*70=3360

50*68=3400

3400-3360=40

I then worked out what the difference was by using the method I used before:

X

X+2

X+20

X+22

((X+2)(X+20))-(X(X+22))=Difference

(X²+20X+2X+40)-(X²+X22)=Difference

(X²+22X+40)-(X²+X22)=Difference

Difference=40

This shows that the difference of a 3 by 3 square is always 40 on a grid with ten squares in each row.

I used this method to find out what the difference would be for a square with any width and height. I did this by using the same method as before only finding out what the corner squares would be relative to the height and width:

X

X+(W-1)

X+10(H-1)

X+10(H-1) +(W-1)

Difference=(X+(W-1))*(X+10(H-1)) - (X(X+10(H-1)+(W-1))

Difference=X²+X(10(H-1))+X(W-1)+(W-1)10(H-1) - X²+X(10(H-1))+X(W-1)

Difference=10(H-1)(W-1)

This is how I tested the formula:

53

54

55

56

57

58

59

63

64

65

66

67

68

69

73

74

75

76

77

78

79

Difference=10(H-1)(W-1)

Difference=10(2*6)

Difference=120

53*79=4187

59*73=4307

4307-4187=120

Difference=120

Another thing that could be changed is the length of the grid. If you use this formula on a grid with a different length along the top it doesn't work:

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48

Difference=10(H-1)(W-1)

Difference=10(2*3)

Difference=60

0*29=290

3*26=338

339-290=49

X

X+(W-1)

X+10(H-1)

X+10(H-1) +(W-1)

If the grid is 8 squares long you would have to go 8 squares along from a square to get to the one beneath it. This means the number would be greater by 8. This shows that if the grid was 8 squares along the top, the formula for the bottom left square in the rectangle would be X+8(H-1). I used this bit of information to try working out the formula the same way as before only replacing the 10's with L's to represent the amount of squares the grid length is.

X

X+(W-1)

X+L(H-1)

X+L(H-1) +(W-1)

Difference=(X+(W-1))*(X+L(H-1)) - (X(X+L(H-1)+(W-1))

Difference=X²+X(L(H-1))+X(W-1)+(W-1)L(H-1) - X²+X(L(H-1))+X(W-1)

Difference=L(H-1)(W-1)

This is how I tested the formula:

2

3

4

5

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7

8

9

0

1

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3

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9

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36

Difference=L(H-1)(W-1)

Difference=6(2*4)

Difference=48

3*29=377

7*25=425

425-377=48

Difference=48

The final formula I got was Difference=L(H-1)(W-1)