# To look for the general rule I am going to draw out the smallest possible grid to find the smallest E Total.

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Introduction

To look for the general rule I am going to draw out the smallest possible grid to find the smallest E Total.

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 |

I can also writ this in terms of E as follows.

E | E+1 | E+2 |

E+3 | E+4 | E+5 |

E+6 | E+7 | E+8 |

E+9 | E+10 | E+11 |

E+12 | E+13 | E+14 |

E+15 | E+16 | E+17 |

E+18 | E+19 | E+20 |

E+21 | E+22 | E+23 |

E+24 | E+25 | E+26 |

E+27 | E+28 | E+29 |

Instead of a number in the top left hand corner I will cal it E. Then go across the rows to add 1 to the E each time. E.g. E+1.

## Facts

- There are 11 E’s in the shape.
- There are 11 E’s in total in the shape.
- The numbers in the E add p to 75.

I know that for the smallest possible E the total =86 and the E start number is 1 as it starts in the top left hand corner.

: . I can say that E total = 11 x 1 (E start) + 75 = 86

Now if I move the E down I square the new start number would be 4 (E +3). So I can test the rule above.

Et = E total

Et = 11 x 4 + 75 =119

Now I will add the numbers up to see if it works. 4+5+6+&+10+11+12+13+16+17+18 = 119.

IT WORKS!!!!!!

Test | Difference from first E total | |||

There are 11 E's inside the shape | So Et = | 11 x Es + 75 | 86 | 0 |

& the numbers inside add up to 75 | ||||

Moving shape down 1 row | Et = | 11 x Es + 75 | 119 | 33 |

Moving shape down 1 row | Et = | 11 x Es + 75 | 152 | 33 |

Moving shape down 1 row | Et = | 11 x Es + 75 | 185 | 33 |

Moving shape down 1 row | Et = | 11 x Es + 75 | 218 | 33 |

Moving shape down 1 row | Et = | 11 x Es + 75 | 251 | 33 |

Es = E start

So far my rule works for any 3-column grid.

Now I will try to improve it so it works for any number of columns.

I will now try a 4-column grid.

E | E+1 | E+2 | E+3 |

E+4 | E+5 | E+6 | E+7 |

E+8 | E+9 | E+10 | E+11 |

E+12 | E+13 | E+14 | E+15 |

E+16 | E+17 | E+18 | E+19 |

E+20 | E+21 | E+22 | E+23 |

E+24 | E+25 | E+26 | E+27 |

E+28 | E+29 | E+30 | E+31 |

E+32 | E+33 | E+34 | E+35 |

E+36 | E+37 | E+38 | E+39 |

Middle

There are 11 E's inside the shape

So Et =

11 x Es +97

108

108

first E total

& the numbers inside add up to 97

0

Moving shape down 1 row

Et =

11 x Es + 97

152

152

44

Moving shape down 1 row

Et =

11 x Es + 97

196

196

44

Moving shape down 1 row

Et =

11 x Es + 97

240

240

44

Moving shape down 1 row

Et =

11 x Es + 97

284

284

44

Moving shape down 1 row

Et =

11 x Es + 97

328

328

44

Moving shape across 1 column

Et =

11x Es + 97

119

119

0

Moving shape across/down 1 column

Et =

11x Es + 97

163

163

44

Moving shape across/down 2 columns

Et =

11x Es + 97

207

207

44

Moving shape across/down 3 columns

Et =

11x Es + 97

251

251

44

Moving shape across/down 4 columns

Et =

11x Es + 97

295

295

44

Moving shape across/down 5 columns

Et =

11x Es + 97

339

339

44

Es = E start

This proves I need to alter my rule. I noticed that the difference between the first E total on the 4 column grid went up by 44 this time, as it went up by 33 in the 3 column grid.

Also the smallest E total is now increased from 86 in the 3-column grid to 108, which is a difference of 22.

I will now try the next size grid, which is with 5 columns.

E | E+1 | E+2 | E+3 | E+4 |

E+5 | E+6 | E+7 | E+8 | E+9 |

E+10 | E+11 | E+12 | E+13 | E+14 |

E+15 | E+16 | E+17 | E+18 | E+19 |

E+20 | E+21 | E+22 | E+23 | E+24 |

E+25 | E+26 | E+27 | E+28 | E+29 |

E+30 | E+31 | E+32 |

Conclusion

=163

Check: 6+7+8+10+14+15+16+18+22+23+24 =163!!

IT WORKS!!!!!

Now I will try a 5 x 6 grid, with an E start of 8.

Et = 11 x 8 + 75 + 22 x 2 = 207

Check: 8+9+10+13+18+19+20+23+28+28+30 = 207

IT WORKS!!!!!!

Therefore my general rule works.

### Et = 11 x Es + 75 + 22 x Nc

Et = E total

Es = E start

Nc= Number of extra columns.

I am now going to move the E around in the following ways to see if the rule works and if it doesn’t work out a rule: -

- Backwards
- 90 degrees
- 180 degrees.

These are the results I have found.

Minimum Et = | 11Es+75 | 86 | |

Any grid Et = | 11Es+75+22*Nc | 691 | 8x10 grid |

Backwards any grid Et = | 11Es+75+22*Nc+4 | 695 | 8x10 grid |

Min Et at 90 deg Et = | 11Es+67 | 78 | 5x10 grid |

Min at 180 deg Et = | 11Es+87 | 98 | 5x10 grid |

At 90 deg any grid Et = | 11Es+67+9*Nc | 754 | 8x10 grid |

At 180 deg any grid Et = | 11Es+87+9*Nc+4 | 584 | 6x10 grid |

From these results I can make a table showing the final formulae for each position E is displayed on a grid.

Final formulae | |||

Any grid Et = | 11Es+75+22*Nc | * | |

@ 90 deg any grid => 5 columnsEt = | 11Es+67+9*Nc | ||

backwards any grid Et = | 11Es+75+4+22*Nc | Always increases by 4 from formula marked * | |

@ 180 deg any grid => 5 columnsEt = | 11Es+87+4+9*Nc |

Et = E total

Es = E start

Nc = Number of extra columns

I have noticed that there is always an increase in the E total by 4 when the E is backwards from the formula that works from any grid.

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

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