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?
On a recent trip to the nation of Imalas, you violated an obscure custom and were subsequently arrested, convicted, and thrown in jail. You have no set sentence, because the people of Imalas believe that anyone should have the opportunity to get out of jail at any time.
You are placed in a cell with five feet thick steel walls, a heavily bolted door with a small opening at the bottom where food is given to you, and no windows. It also has all the typical jail cell furniture. All of the pipes are too small to fit through. You have been told that you may request and keep any kind of plant or small animal in your cell.
You are free to go if you can break out. How do you do it?