Prove that Hamming cube has a Hamiltonian cycle

Question

How would one prove that all Hamming cubes with 2 or greater dimensions have a Hamiltonian cycle.

Answer 1

Hint: Induction.$ $

Answer 2

A Hamming cube of dimension $n+1$ is two copies of an $n$ dimensional Hamming cubes, one with $0$ appended and one with $1$ appended. Take an $n$ dimensional cycle of the one with $0$ appended, break one segment, change the last coordinate, and go backwards around, then switch the last bit back to $0$. You have constructed a path.