I can't quite figure this problem yet. There is an ant at one vertex of a cube. The ant goes from one vertex to another by choosing one of the neighboring vertices uniformly at random. What is the average minimum time it takes to visit all vertices?



What is the cover time of a random walk on a cube?
An additional reference:


Chapter 12 in *Problems and Snapshots from the World of Probability* by 
Blom, Holst, and Sandell is devoted to an elementary exposition of such 
cover problems. 


A related problem: 


The solution to Problem 6556 in the *American Mathematical Monthly* 
(Vol. 96, No. 9, Nov. 1989, pages 847-849) 
looks at the average number of steps for a random walk to visit all the 
**edges** on the cube in dimensions $d=2$, $3$, and $4$. 


For $d=2$ the answer is easily computed to be 10.


For $d=3$ a system with 387 equations in 387 unknowns 
is solved to give an answer of about 48.5.


For $d=4$ the problem is declared hopeless. 


