Consider the following classic problem:
Four people on the west side of a river wish to use their single boat to get to the east side of a river. Each boat ride can hold at most two people, and the time it takes to get across will be the time preferred by the slower occupant. The time preferences are: $1, 2, 5$ and $10$ minutes. What is the minimal amount of time in which you can get all four people across the river, where an eastbound trip must have two occupants and a westbound trip must have one occupant?
Answer: $17$ minutes. (Though many mistakenly believe it is $19$ minutes.)
Question 1: If you look at all possible ways of ferrying these four people across the river, subject to the above constraints, what would the average (mean and median) times be?
Question 2: What if you replace the time preferences with $x_1, x_2, x_3$ and $x_4$ minutes?
Question 3: What if you have time preferences $x_1, \ldots, x_n$ minutes for $n$ people, respectively?
Probably the easiest way to broach Question 1 would be to write a quick program to compute the answer, and perhaps doing this for several different time preferences would give some insight into the mean and median in the general four person case. I'm not quite sure how I would start thinking about the general $n$ person case; perhaps by solving it for $n = 1, 2, 3, 4$.
Answers (even partial ones) to any or all of my questions would be greatly appreciated!