What's the shortest path from A to R in the array below, if your moves are to be in the repeated length sequence 3, 2, 1, 3, 2, 1... (such as, say ADJMC...), and no two successive moves are in the same direction or in the reverse direction? Moves may be orthogonal or diagonal. No move may end on a hyphen (-) but may jump over them and the R must be reached on exact count.
ABCD-
EFGH-
--IJK
-LM-N
OP-QR
a) Beginning thoughts
The beginning is forced. It must start ADJ
b) Middle Thoughts
There appear to be four letters that cannot be reached using a 3 hop: GIJK. That means that they also cannot be reached via a 1 hop, because there is then no way to leave it within the rules. If they are involved in the solution, we can only reach G or I or J or K with a 2 hop and leave it with a 1 hop.
c) Ending thoughts
There are apparently 6 ways to reach R:
With a 3 hop from F or P.
With a 2 hop from I or K
With a 1 hop from N or Q.
But three of these fail quickly.
Per my middle thought, we cannot exit I or K with a 2 hop.
A 1 hop from Q fails, because the preceding move must be a 2 hop from J, which is not possible per my middle thought.
So the ending must be one of the following:
FR
PR
NR. If NR, it must be CMNR.
And that's as far as I got.
Is it possible to draw a network which will solve this, or is a computer the way to go?
More thoughts later, maybe.
Some Thoughts
