A bug is placed at one corner of a wire frame in the shape of a cube. At the diagonally opposite corner is a piece of sugar.

The bug crawls along the 12 wires of the frame searching for the sugar. At each of the 8 corners the bug randomly chooses one of the 3 wires to follow next (including the one it just traveled).

What is the expected number of edges the bug will traverse until it reaches the sugar? What if the bug never doubles back on the wire it just crossed?