I learned a game today. A group of people sit in a circle, looking down. The game leader counts to three, and on three everyone lifts their head and looks at one other person. If the other person is looking back at you, you’re both out. The game continues until there are one or two people left (depending on if your group has an even or odd number). The wonderful part is that no one feels bad about getting knocked out, because it’s impossible not to laugh when you look up to see the person you chose staring right back at you – the power of human eye contact.
The question is: for a given number of players, , and assuming each person chooses who to look at randomly, how many rounds do you expect the game to take before ending?