# How long will it take to hand out the shuffled papers?

A teacher has $$n$$ students sit in a circle in her classroom. She holds in her hands a perfectly shuffled stack of the students' graded homework, with Juan's on top. She is currently standing in front of Juan, and intends to hand out the homework by repeatedly (1) seeing whose paper is on the top of the stack, and then (2) walking clockwise to that student to hand over the homework. She will finish by continuing to walk clockwise back to Juan.

For example, if the stack were perfectly in order with the students' clockwise positions, then she would do exactly one rotation in total. In the worst case scenario, if the stack were perfectly in order with the students' counterclockwise positions, then she would do exactly $$n-1$$ rotations in total.

How many rotations will she be performing on average?

• Hint: rot13(Rhyrevna Ahzoref). Feb 1 at 21:01