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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.

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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.