Across the cube

Idea. By symmetry, only the distance to the target matters. That reduces the walk on $8$ corners to a chain on $4$ states.

Step 1: transitions. Measure the distance to the target corner.

• From distance $3$ (the start), all $3$ edges lead to distance $2$.
• From distance $2$, $2$ edges lead to distance $1$ and $1$ edge leads back to distance $3$.
• From distance $1$, $1$ edge reaches the target and $2$ edges lead back to distance $2$.

Step 2: equations. Let $e_d$ be the expected number of steps from distance $d$, with $e_0=0$: $$\begin{array}{rl}{e}_3& =1+e_2,\\ e_2& =1+\frac{2}{3}{e}_1+\frac{1}{3}{e}_3,\\ e_1& =1+\frac{2}{3}{e}_2.\end{array}$$

Step 3: solve. Substitute $e_3=1+e_2$ and $e_1=1+\frac{2}{3}{e}_2$ in the middle equation: $$e_2=1+\frac{2}{3}+\frac{4}{9}{e}_2+\frac{1}{3}+\frac{1}{3}{e}_2=2+\frac{7}{9}{e}_2,$$ so $e_2=9$, $e_3=10$ and $e_1=7$.

The ant needs $10$ steps on average, although the shortest path has only $3$.