Label each edge that Tamara walks as *(e, d, t) *where *d*
is the direction along the edge that Tamara walks, *e* is the edge
and* t *is the turn the toad made to reach the edge (either left
or right).

Observe that if Tamara executes *(e, d, t)* on step *k* >
1 then the triple (*e', d'.t'*) that she executes on step *k*
-1 is determined. *t'* is the opposite of *t*. There are two
edges incident with start of edge *e* under direction d and only
one is consistent with *t'*. And then *d'* is also now determined.
Since there are only a finite number of such triples, Tamara must eventually
repeat one of the triples, assuming that she doesn’t stop unless
she reaches home. This must be the first triple, since otherwise the prior
one will also have been repeated. Acknowledgement Mike Schuresko and Brian
Thompson provided correct solutions.