  It is shown that the square-triangle random tiling model is equivalent to an
asymmetric limit of the three colouring model on the honeycomb lattice. The
latter model is known to be the O(n) model at T=0 and corresponds to the
integrable model connected to the affine $A_2^{(1)}$ Lie algebra. Thus it is
shown that the weights of the square-triangle random tiling satisfy the
Yang-Baxter equation, albeit in a singular limit of a more general model. The
three colouring model for general vertex weights is solved by algebraic Bethe
Ansatz.
