We present a detailed calculation of the recently published exact solution of a
random tiling model possessing an eightfold-symmetric phase. The solution is
obtained using the Bethe Ansatz and provides closed expressions for the entropy
and phason elastic constants. Qualitatively, this model has the same features as
the square-triangle random tiling model. We use the method of P. Kalugin, who
solved the Bethe Ansatz equations for the square-triangle tiling which were
found by M. Widom.