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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
...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 * @ 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.

...read more.

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