Let y=number of turns to reach ant from diagonal corner on same face as ant.

Let z=number of turns to reach ant from an adjacent corner to ant.

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:

E(x)=1+E(y).

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:

E(y)=(2/3)*(1+E(z))+(1/3)*(1+E(x)).

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:

E(z)=(1/3)*1+(2/3)*(1+E(y)).

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.