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"Four men crossing a bridge" is a logic puzzle that I came across a few years ago. I particularly enjoy this puzzle because it appears simple at first glance but becomes deviously complex as you attempt to solve it. I did not make it up, and take no credit for it's invention. Unfortunately, I am unaware of it's origin.

Four men are travelling through a forest when they come upon a rickety old bridge spanning a gaping chasm. The bridge can only hold the weight of two men at once, and it is so dark that a flashlight is necessary to successfully navigate. Since the men only have one flashlight among them, they will keep sending two men across the bridge and have one bring the light back until all four are across. The four men are of different physical ability, and they can cross the bridge in one minute, two minutes, five minutes and ten minutes respectively. If two men cross the bridge, they must both travel the speed of the slower man. For example, if the one minute man and the ten minute man travel across together, it will take them both ten minutes to get across. What is the fastest time all four can successfully cross the bridge?



Although lateral thinking is a good thing, it is not used to solve this puzzle (i.e. the men do not use the flashlight to signal a helicopter to take all four across in one minute.) There are no word tricks here, just try to figure out the most efficient use of each man's ability to span the bridge. Feel free to add your own solutions to this node, and we'll see who can come up with the fastest time.
I claim it can be done in 17 minutes.
First, 2 minute man and 1 minute man cross. Time = 2 minutes
Now send 2 minute man back. Time = 4 minutes.
10 minute man and 5 minute man hobble across. Time = 14 minutes.
1 Minute man grabs the flashlight and sprints back to the first side. Time = 15 minutes.
2 minute man and 1 minute man hurry to the far side just as the battery dies. Now they've all safetly crossed in 17 minutes, but they're stuck in the woods without a flashlight. Hopefully the moon will come out.

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:

  1. I need to move four men across the bridge.
  2. 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.
  3. A return trip is wasted time, because it does not contribute to the solution.
  4. It follows that the fastest movers must make the return trips, in order to minimize wasted time.
  5. 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).
  6. Therefore, the slower movers must travel together, leaving the faster movers to travel together as well.
  7. 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.

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