We have a virtual ant which uses a chemical scent trail to navigate along a path. As the ant explores in new directions it lays down chemical blobs at regular intervals. The presence and pattern of these blobs helps the ant navigate back to the nest and other ants navigate to where the pioneer ant is located.
Since the blobs are placed regularly, it seems more than one chemical marker is necessary, because a string of "A A A A A..." doesn't help the ant determine which direction is toward the nest and which is away from it.
So if you had two chemical markers A and B and had to lay them in a regular pattern, so that an ant traversing the pattern could tell WHICH WAY they were going, how small would the smallest such pattern be?
For example, "A A B" doesn't work, because the reverse, "B A A" is a cyclic permutation of "A A B" and thus the ant would not be able to tell which way it was headed. So you're looking for the smallest string of A's and B's whose reversal is not a cyclic permutation of itself.