Can someone please help me solve this problem. I came up with the following observations:

1) the first bit of the binary number should be 1.

2) we can think of each binary prefix(modulo n) as a node of the graph. So finally we need to find a hamiltonian cycle starting at 0 and ending at 0 where each node has an outgoing edge to the nodes (2*node)%n and (2*node+1)%n.

I am unable to progress further. Any help/hint will be really helpful.