# Magic E Coursework

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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.

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

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.

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

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