  Using a new approach based on automatic sequences, logic, and a decision
procedure, we reprove some old theorems about circularly squarefree words and
unbordered conjugates in a new and simpler way. Furthermore, we prove three new
results about unbordered conjugates: we complete the classification, due to
Harju and Nowotka, of binary words with the maximum number of unbordered
conjugates; we prove that for every possible number, up to the maximum, there
exists a word having that number of unbordered conjugates, and finally, we
determine the expected number of unbordered conjugates in a random word.
