The dimer problem on the hexagonal lattice is solved for various boundary shapes. The number of arrangements of dimers in the asymptotic limit of large lattices proves to be very sensitive to the precise form of the lattice boundary. If N is the number of lattice points and (for large N) WN/2 the number of dimer configurations, then the molecular freedom W reduces to 1 in the case of standard boundaries (parallelogram, rectangle), and there is, in the thermodynamic limit, no freedom in placing a dimer at all. But in the case of a hexagonal lattice which has been given the shape of a macroscopic honeycomb and which thus exhibits the symmetry properties of the unit cell, the molecular freedom is found to tend to W=(3/4)√3 =1.299 for N→∞. Investigating various other boundary forms, the regular hexagon proves to be a convenient surface shape with a maximum molecular freedom which differs markedly from the known value W=1.381 for the hexagonal lattice wrapped on a torus.