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Hamiltonian walks on lattices are model systems for compact polymers such as proteins. Here we enumerate exactly the number of Hamiltonian walks on the 4 × 4 × 4 cube and give estimates up to the 7 × 7 × 7 cube through Monte Carlo methods. We find that the number of configurations grows faster with chain length than previously anticipated. Finally, we discuss uniqueness of ground states in the HP model for protein folding.doi:10.1088/1751-8113/46/48/485001 fatcat:z3l7g7jmunf6xktoocj2xiubbu