You are one of ten friends who are going to play a few rounds of a simple two-player game of chance. Each player has a 50% chance of winning the game each round.
There are 5 playing squares on the floor, arranged in a circle.
In the beginning, two people go to each square. Each round the people at a square play the game and determine the winner. A round ends when the winner walks clockwise to the next square. The loser stays where they are.
After 10 rounds, what is the probability you are on the square you started on?
If d(x) is the number of positive divisors of x, and n(x) is the number of distinct prime factors of x, show that d(A)=Σ(2^n(A/i²)) for all positive i such that A/i² is an integer.