Decomposing the cube into paths

We consider the question of when the n -dimensional hypercube can be decomposed into paths of length k . For odd n it is necessary that k divides n 2 n − 1 and that k ≤ n . Anick and Ramras (2013) conjectured that these two conditions are also sufficient for all odd n and prove that this is true for odd n ≤ 2 32 . In this note we prove the conjecture.