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

Extracts from this document...

Introduction

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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6

7

8

9

10

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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

...read more.

Middle

Difference from

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

...read more.

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

image00.png

*

 @ 90 deg any grid => 5 columnsEt =

 11Es+67+9*Nc

image01.png

 backwards any grid Et =

 11Es+75+4+22*Nc

image02.png

Always increases by 4 from formula marked *

  @ 180 deg any grid => 5 columnsEt =

 11Es+87+4+9*Nc

image03.png

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.

...read more.

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