  We study the bipartite composition of elementary toy systems with state
spaces described by regular polygons. We provide a systematic method to
characterize the entangled states in the maximal tensor product composition of
such systems. Applying this method, we show that while a bipartite pentagon
system allows two and exactly two different classes of entangled states, in the
hexagon case, there are exactly six different classes of entangled states. We
then prove a generic no-go result that the maximally entangled state for any
bipartite odd gon system does not depict Hardy's nonlocality behaviour.
However, such a state for even gons exhibits Hardy's nonlocality, and in that
case, the optimal success probability decreases with the increasing number of
extreme states in the elementary systems. Optimal Hardy's success probability
for the non-maximally entangled states is also studied that establishes the
presence of beyond quantum correlation in those systems, although the resulting
correlation lies in the almost quantum set. Furthermore, it has been shown that
mixed states of these systems, unlike the two-qubit case, can depict Hardy's
nonlocality behaviour which arises due to a particular topological feature of
these systems not present in the two-qubit system.
