Note that there exists a solution symmetrical to reddishbrownbeard's, wherein 1-minute man returns after the first trip and 2-minute man returns after the second trip.
An analytical approach to the problem goes something like this:
- I need to move four men across the bridge.
- By the constraints of the problem (men move in pairs, one man must return after each trip) I must make a total of two return trips.
- A return trip is wasted time, because it does not contribute to the solution.
- It follows that the fastest movers must make the return trips, in order to minimize wasted time.
- There is no way that the two slower movers can make better time by travelling separately than by travelling together (10+n+5+n < 10+5+n).
- Therefore, the slower movers must travel together, leaving the faster movers to travel together as well.
- Having deduced the proper pairing for the four men, and knowing that faster pair must make a return trip, we have enough information to solve the problem.
Although it's perfectly possible to solve problems like this by trial and error or by simple intuition, it is more valuable in the long run if you can explain to yourself why and how you came up with your solution. This will give you new tricks for solving similar problems in the future, as well as improving your problem-solving skills on the whole.