• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Magic E Coursework

Extracts from this document...

Introduction

Magic E Coursework

For the E-Total coursework, I am investigating different algebraic formula for the letter “E”, when it is placed on a numbered grid. In each E, there are 11 squares.

I will first map out an “E” on a 10 X 8 Grid

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

From this diagram you can see that the total of the numbers in the E is:

1+2+3+9+17+18+19+25+33+34+35 = 196

We can move the “E” along the grid to find out the totals of other E’s.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

From this diagram you can see that the total of the numbers in this E is:

2+3+4+10+18+19+20+26+34+35+36 = 207

Instead of drawing out the next E, we can make a table of E’s and there e-totals. If we call the top left number the e-number (shown as “e”) then we can see if there is a pattern between the e-number and e-totals. We use the number in the top left hand corner as the e-number because it’s the smallest and therefore easiest to work with

E-number

Numbers in E

E-total

1

1,2,3,9,17,18,19,25,33,34,35

196

2

2,3,4,10,18,19,26,34,35,36

207

3

3,4,5,11,19,20,21,27,35,36,37

218

4

4,5,6,12,20,21,22,28,36,37,38

229

5

5,6,7,13,21,22,23,29,37,38,39

240

From the table you can see that when the e number goes up by 1, the e-total goes up by 11. So each time we move the E one space to the right, the total goes up by 11. This is because each square in the E has + 1 added to it when moved to the right.

...read more.

Middle

11

 17

25

26

27

33

41

42

43

11e + 185 = e-total

(11x9) + 185 = e-total

99 + 185 = 284

9 + 10 + 11 + 17 + 25 + 26 +27 + 33 + 41 + 42 + 43 = 284

This proves that the formula is correct and we can now work out the e-total of any E on a 10 x 8 grid

I will now plot out an E on a 10 x 9 grid to see if I can find a formula for the e-total of and E on any grid size. The formula I already have (11e +185) is only correct when used on a 10 x 8 grid.

If we follow the example of the 10 x 8 grid, then we can work out the formula for this grid fairly quickly by converting the E into algebra and then checking it with a random number.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

When converted to algebra the E is:

e

e+1

e+2

e+8

e+16

e+17

e+18

e+24

e+32

e+33

e+34

Therefore the formula should be:

e-total = e + (e + 1) + (e + 2) + (e + 9) + (e + 18) + (e + 19) + (e + 20) + (e + 27) + (e + 36) + (e + 37) + (e + 38)

e-total = 11e + 207


Now lets check with a random number, 10.

10

11

12

19

28

29

30

37

46

47

48

11e + 207 = e-total

(11x10) + 207 = e-total

110 + 207 = 317

10 + 11 + 12 + 19 + 28 + 29 + 30 + 37 + 46 + 47 + 48 = 317

This proves that the formula is correct and we can now work out the e-total of any E on a 10 x 9 grid


...read more.

Conclusion

Therefore the formula for the middle row is R = x(e + 2g) + ½ x(x – 1)

To work out the bottom row we do exactly the same except there is an extra 4g added to the e for each square.

Therefore the formula for the bottom row is R = x(e + 4g) + ½ x(x – 1)

The final E will look like this:

e

e+1

e+2

-----

e+(x-1)

e+g

e+2g

e+2g+1

e+2g+2

-----

e+2g+(x-1)

e+3g

e+4g

e+4g+1

e+4g+2

-----

e+4g+(x-1)

To get the final formula for arm length we need to add up the formulae from each of the rows and add up the 2 squares in between. (e +g and e +3g)

e-total = xe + ½ x(x – 1) + x(e + 2g) + ½ x(x – 1) + x(e + 4g) + ½ x (x – 1) + e + g + e + 3g

e-total  = xn + x (n + 2g) + x (n + 4g) + 1.5x (x – 1) + 2n + 4g

e-total = xn + xn + 2gx + xn + 4gx + 2n + 4g + 1.5x (x – 1)

e-total  = (3x + 2) (n + 2g) + 1.5x (x – 1)

Finally we must check that the formula works by choosing random numbers:

E number = 7

Arm length = 5

Grid size = 6

7

8

9

10

11

13

19

20

21

22

23

25

31

32

33

34

35

e-total = 7 + 8 + 9 + 10 + 11 + 13 + 19 + 20 + 21 + 22 +                                                            23 + 25 + 31 + 32 + 33 + 34 + 35

e-total = 353

e-total = (3x + 2)(e + 2g) + 1.5x(x – 1)

e-total =  (3 x 5 + 2)(7 + 2 + 6) + 1.5 x 5 x 4

e-total = 17 x 19 x 30

= 353

This proves the formula correct so finally we have a formula, which can find the e-total of any e on any grid size with any arm length.

...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE T-Total essays

  1. Marked by a teacher

    T-total coursework

    5 star(s)

    then y = 1, while the horizontal translation is also 1 (the T-shape is moved right by 1) then x = 1. The diagram below left shows what n-2w-1 n-2w n-2w+1 10 n-w 12 19 n 21 (n+x+wy-2w)-1 n+x+wy-2w (n+x+wy-2w)+1 10 n+x+ wy-w 12 19 n+x+wy 21 happens to the

  2. T-Total Maths coursework

    N = 13 T = (5 x 13)-7 = 65-7 = 56 Rotation of 900 7 by 7 T-number and T-total table T-number T-total 8 47 9 52 10 57 11 62 Here I have added a prediction of mine when I realized the pattern of the sequence, which goes up by 5 each time.

  1. T-Shapes Coursework

    40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 5th T-Shape: T-Total 5 + 6 + 7

  2. T-Total Course Work

    68 69 70 71 72 73 74 75 76 77 78 79 80 21 T = 20 G = 9 Y = 6 5T-7G+((5xG)xY) = (5x20) - (7x9) + ((5x9)x6) =100 - 63 + 270 = 307 Which is the same as the T-Total of the translated shape 55+56+57+65+74=307 This

  1. Maths Coursework- Borders

    + 5 = 18 - 18 + 5 = 5 This shows that the formulae I have used are correct as the equations give the correct results for cross-shape pattern 3. Now I will test the formulae on pattern 9 - a cross-shape I don't know the answer to.

  2. T-Shapes Coursework

    Therefore, the following logic can be used to create a formula where: n = the Middle Number 1) Using Pattern 1 above, we can say that the Sum of the Wing = 3n 2) Using Pattern 2 above, we can say that the Sum of the Tail = n + 8 3)

  1. T totals. In this investigation I aim to find out relationships between grid sizes ...

    11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Center of Rotation (v) Rotation (degrees) Direction T-Total (t) Difference compared to original T-Total 8 0 N/a 30 0 8 90 Clockwise 42 +12 8 180 Clockwise 50 +20 8

  2. mathematics coursework

    4 46 60 12.5 6 75 66 13.5 19 256.5 85 14.5 15 217.5 100 TOTAL 100 1069 100 To work out the range subtract the highest mid point from the lowest mid point. To work out the mean average of IC Fuel battery life I can divided the total

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work