  In this paper we study 1/k-geodesics, those closed geodesics that minimize on
any subinterval of length $L/k$, where $L$ is the length of the geodesic. We
investigate the existence and behavior of these curves on doubled polygons and
show that every doubled regular $n$-gon admits a $1/2n$-geodesic. For the
doubled regular $p$-gons, with $p$ an odd prime, we conjecture that $k=2p$ is
the minimum value for $k$ such that the space admits a $1/k$-geodesic.
