The ice model on the three-dimensional Hanoi graph

We study the ice model (with Boltzmann factors equal to one) on the 3D Hanoi graph H(n) at stage n. We derive recursive equations for functions that provide the number of configurations for each type of restricted arrow diagram. The total number of configurations I(n) can be computed exactly using these functions. The entropy per site in the thermodynamic limit is S H = 2 3 ln 2 = 0.462 098 120 373 … for H(n). This is comparable to the entropy per site S s q = 3 2 ln 4 3 = 0.431 523 108 677 … for the square lattice (Lieb 1967 Phys. Rev. Lett. 18 692-4 and Lieb E H 1967 Phys. Rev. 162 162-72).