Visiting every hour
You stand on one number of a 12-hour clock. At each step a fair coin moves you one hour forward or one hour back.
Find the expected number of steps until you have stood on all 12 numbers. The starting number counts as visited.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The visited numbers always form an arc of consecutive numbers, and you are at one end of it whenever a new number has just been visited.
A fair random walk starting at distance from one end of an interval and from the other leaves it after steps on average.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The visited numbers form a growing arc. When the arc has numbers and you have just reached one of its ends, the next new number is either 1 step ahead or steps behind you, the other end of the arc plus one. By the gambler's ruin duration formula, that takes steps on average.
Step 1: the arc. You move one hour at a time, so the visited numbers are always consecutive. Each time a new number is visited, you stand at one end of the arc.
Step 2: one stage. With numbers visited and you at an end, the unvisited numbers next to the arc are at distance 1 on one side and on the other. A fair walk started at distance 1 from one exit and from the other leaves after steps on average.
Step 3: add the stages. The arc grows from 1 to 12 numbers, through :
In general. On a cycle with points, the expected cover time is .