Ants on a cube

The problem can be solved using a Markov Chain . We assume that both ants can meet either on an edge or on a vertex. Let's consider a cube which vertices are numbered from $1$ to $8$ . Let be $S=\{\text{E}, \, \text{C}, \, \text{O}, \, \text{0} \}$ the set of all possible states the ants can be in.

• $\text{E}$ is the state in which at the next step the two ants could meet on an edge.
• $\text{C}$ is the state in which at the next step the two ants could meet on a corner.
• $\text{O}$ is the state in which the two ant are on opposite vertices.
• $0$ is the state in which the ants are on the same edge/corner.

Let's indicate with $|s|$ , $s \in S$ , the number of couples of each state indicating where the ants are at. For example, $0=\{(1,1), \, (2,2), \, ... \,, (8,8) \}$ , so $|0|=8$ . You can figure out from the picture that $|E|=24$ , $|C|=24$ , $|O|=8$ . Now, we can write the transition matrix $\textbf{P}$

\displaystyle \textbf{P}= \begin{array}{[cccc]c} \\ E & C & O & 0 \\ \frac{6}{9} & 0 & \frac{2}{9} & \frac{1}{9} & E \\ 0 & \frac{7}{9} & 0 & \frac{2}{9} & C \\ \frac{6}{9} & 0 & \frac{3}{9} & 0 & O \\ 0 & 0 & 0 & 1 & 0 \end{array}

Following the standard theory of Markov chains, we evaluate $\textbf{N}=(\textbf{I}-\textbf{Q})^{-1}$

$\displaystyle \textbf{N}=(\textbf{I}-\textbf{Q})^{-1} = \left( \begin{array}{c}\\ 9 & 0 & 3 \\ 0 & \frac{9}{2} & 0 \\ 9 & 0 & \frac{9}{2} \end{array} \right)$

We define $\text{U}=u_i$ , for $1 \leq i \leq 4$ ,as the stochastic vector indicating the probabilities of each state being the starting state. We have

$\displaystyle u_1=\mathbb{P}(E)=\frac{|E|}{8^2-8}=\frac{24}{56} \\ \displaystyle u_2=\mathbb{P}(C)=\frac{|C|}{8^2-8}=\frac{24}{56} \\ \displaystyle u_3=\mathbb{P}(O)=\frac{|O|}{8^2-8}=\frac{8}{56} \\ \displaystyle u_4=\mathbb{P}(0)=0$

Eventually, the expected value of arriving at the absorbing state is

$\displaystyle E[0]=\sum_{i=1}^3 \sum_{i=1}^3 n_{ij}u_i=\frac{24}{56} \cdot 12+\frac{9}{2} \cdot \frac{24}{56} + \frac{27}{2} \cdot \frac{8}{56} = \boxed{9}$