**Motivation.** With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, and move the clock hands to the right (i.e. from $n$ to $n+1 \text{ mod } 12$) if the coin toss was tails, and move the clock hands in the other way otherwise.


The procedure ends, when the clock hands have been at every of the $12$ positions of the clock.


**Question.** In terms of $n$, what is the expected number of moves until this game ends?



Tossing a coin around $\mathbb{Z}/n\mathbb{Z}$
Suppose you are at position $a$, and have already visited $\{a-i,\dotsc,a,\dotsc,a+j\}$, where $i,j\geq 0$ and $i+j<n$. The expected time $t\_{ij}$ until you have covered the whole clock must satisfy $t\_{i,n-1-i}=0$ and
$$ t\_{ij} = \frac{1}{2}t\_{(i-1)\_+,\;j+1}+\frac{1}{2}t\_{i+1,\;(j-1)\_+}+1 $$
(where $k\_+$ means $\max(0,k)$). The solution is
$$ t\_{ij} = \frac{1}{2}\left(n(n-1)-i(i+1)-j(j+1)\right)$$
In particular, the expected time from the obvious starting position is $t\_{00}=n(n-1)/2$.


