I came across this question from a trading interview. There are 4 chairs in a circle. You roll a die. If it comes up 4, you move one chair counterclockwise. If it is any other face, you move clockwise that number of chairs. For example, if I got a 6 I would move 6 chairs clockwise, which is equivalent to 6 mod 4 = 2 chairs clockwise. The question is what is the probability of ending up in the same chair you started in after 6 rolls?
I tried tackling this with the law of total probability, conditioning outcomes on X, the number of 4's we get, so then the other 6-X rolls have to sum to a number that satisfies some modulo operation, but I found that approach to devolve into a tedious counting exercise. Is there any more intuitive, straightforward approach that relies on some key insight I am missing?