Mickey mouse travels in a maze with nine $3 × 3$ cells. The cells are numbered as $0, 1, ..., 8$ from left to right and top down. Each step Mickey travels from where it is to one of the surrounding connected cells with equal chance.
Mickey mouse and travels blindly independently in the maze and Donald duck follows Mickey's footstep. Let $M_n$ be the cell number of Mickey after $n$ step transition and and $D_n$ be the cell number of Donald after $n$ step transition Then, $D_n = M_{n-1}$ for $n \ge 1$, with $M_0 = D_1 = 4$(The center cell). Counting $n = 1, 2, ...$,
- $D^2_n$ is a Markov chain.
- $2M_n + D_n$ is a Markov chain.
- $M_nD_n$ is a Markov chain.
- $M_n - D_n$ is a Markov chain.
- None of the above is correct.
Is it option 1 is the right, since $D_n$ is equal to $(0,1,...,9)$, and its square is equal to $(0,1,4,9,16,...)$, if we take the square root, only 1 value is correct, which is mean one-to-one, so that option 1 is correct?
Yes, as you say, option $1$ is right, since squaring the non-negative values $D_n$ is a bijective transformation, so it merely changes the labels of the Markov states.