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

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.

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.

Now the difference has gone up to 55, and the smallest E total has increased by 22 again.

Fact

For every column added on from the smallest grid the E total increases by 22.

Et = 86 (3 column grid)

Et = 108 (4 column grid)

Et = 130 (5 column grid)

Since I know the Et for a 3-column grid Is 86 and that is the smallest value for this size E, then I could add 22 for every extra column.

So Et = 11 x Es + 75 + 22 x Nc ( number of extra columns)

I will now test this for a 4 column grid with an E start at 1.

: . Et = 11 x 1 + 75 + 22 x 1 (extra column)

= 108

IT WORKS!!!!

Now I will try the rule for a 5 column grid.

: . Et = 11 x 1 + 75 + 22 x 2 (extra columns)

= 130

IT WORKS!!!!!!

I am now going to see if it works for any E start on any grid.

First I will try a 4x6 grid with an E start of 6.

Et = 11 x 6 + 75 + 22 x 1

=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.

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

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.