# LEVEL connections

How many ways are there to form the word 'LEVEL' by following the lines?

(You can't go back on yourself but apart from that you can move in any direction and can end up on the same 'L' as you started with)

• Does it count only once if we start from either left or right (or up down), or twice, one for each starting L? Commented Apr 30, 2016 at 17:49
• Are you allowed to reuse a vertex without going back on a line, e.g. starting and ending on the same 'L'? Commented Apr 30, 2016 at 18:00
• @KeyboardWielder yes you are Commented Apr 30, 2016 at 18:01
• And do palindromes count? i.e. the exact same path but going in the opposite direction. For example, could a diagonal be 2 paths? Commented Apr 30, 2016 at 18:02
• @KeyboardWielder that counts as going back on yourself Commented Apr 30, 2016 at 18:05

Without allowing backtracking along edges

$60$

Since

"L" only appears at the four corners, and staring at any corner there are fifteen paths to form the word. $15\times 4=60$

The paths

From a corner there is $1$ way to reach the opposite corner, $5$ ways to reach each of the $2$ adjacent corners, and $4$ ways to come back without back tracking along edges ($2$ directions around the $2$ small triangles). $1+2\times 5+4=15$

• could you somehow show the paths? Commented Apr 30, 2016 at 17:57

There are

60 ways.

Proof.

From the top left to the top right L there are 5 ways.

1
'L E V E L'

2
'L E V L'
' E '

3
'L V E L'
' E '

4
'L V L'
' E E '

5
'L L'
' E E '
' V '

And another 5 ways similarly to the bottom left corner.

There are also 4 ways of returning to the same position:

Two paths (one clockwise and one anticlockwise)
'L E V'
' E '

Two paths (one clockwise and one anticlockwise)
'L '
'E E'
'V '

Finally, there is one way to go diagonally to the bottom right corner, which makes for a total of 15. Times 4 for all the corners and we get 60.

• You can do code blocks in quotes. It'll look better than separate pre blocks.
– anon
Commented May 1, 2016 at 5:39
• @QPaysTaxes Cheers, that's a good idea I hadn't thought of. Commented May 1, 2016 at 12:03