You have a chain with N links numbered 1 through N. Every minute, you draw a random link from a bag, and connect it to any other consecutively-numbered link that you drew before. For example, if you drew 4,1,5,7,3, you would end up with three subchains: 1,(3,4,5),7. You keep on drawing until you have drawn all N links and connected them into a single chain of length N. Let M be the maximum number of subchains in this process.
What is the mean of the distribution of M if N=8?
What is the standard deviation of the distribution of M if N=8?
What is the mean of the distribution of M if N=16?
What is the standard deviation of the distribution of M if N=16?
What is the mean of the distribution of M if N=32?
What is the standard deviation of the distribution of M if N=32?
A fun coding question.