After one turn the spider will be on a diagonal corner of a common face as the ant. So the mean number of turns from the x position is one more than the mean number from the y position:
Once at a y position there is a 2/3 chance it will then move to a z position, and a 1/3 chance back to an x position:
If the spider arrives at a z position there is a 1/3 chance it will move to the ant, and a 2/3 chance it will move back to a y position:
With these three equations and three unknowns it is not difficult to solve for E(x), E(y), and E(z).
Michael Shackleford, A.S.A.