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Bridge and Torch Puzzle

Four people, one torch, a bridge that holds two. Can you get everyone across in 17 minutes?

Type
Logic
Takes
about 2-4 min
Result in
minutes

Bridge and Torch

~2-4 min

  1. Everyone must cross a narrow bridge at night, with one torch.
  2. Two at most cross at once, carrying the torch, at the slower one's pace. Someone brings it back.
  3. Use as few minutes as you can. The fewest here is 17.

Tap one or two people, then Cross.

Your result: — min

How it's measured

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What is the bridge and torch puzzle?

Four people must cross a narrow bridge at night. They have one torch, and the bridge holds at most two at a time. Crossing alone takes them 1, 2, 5 and 10 minutes; two crossing together walk at the slower one's pace, and someone always has to bring the torch back. What is the least time to get everyone over? The answer is 17 minutes. It is also known as the bridge riddle, and TED-Ed's 2015 video of it uses the same four times.

How to play it here

Tap one or two people on the side where the torch is, then press Cross. The minutes add up at the top. Undo takes back a crossing and Start over resets the puzzle. When everyone is across, your total is rated against the fewest possible.

Why isn't the answer 19 minutes?

The first idea most people have is to let the fastest person walk everyone over and come back each time. That takes 19 minutes. The trick is to send the two slowest across together, so the 10-minute walk also covers the 5-minute one, and to have the two fastest do the running back and forth:

  1. The 1- and 2-minute walkers cross (2 min).
  2. The 1-minute walker brings the torch back (1 min).
  3. The 5- and 10-minute walkers cross (10 min).
  4. The 2-minute walker brings the torch back (2 min).
  5. The 1- and 2-minute walkers cross (2 min).

That adds up to 17 minutes, and a computer search over every possible plan finds nothing shorter.

The five-person version

With five people taking 1, 3, 6, 8 and 12 minutes, the fewest is 29 minutes. The same two moves do the work: either the two fastest ferry the two slowest across together, or the fastest escorts them one at a time, whichever is quicker for that pair. Günter Rote proved in 2002 that choosing between those two moves, slowest pair first, always gives the best answer, however many people there are. One best plan:

  1. The 1- and 3-minute walkers cross (3 min).
  2. The 1-minute walker brings the torch back (1 min).
  3. The 8- and 12-minute walkers cross (12 min).
  4. The 3-minute walker brings the torch back (3 min).
  5. The 1- and 3-minute walkers cross (3 min).
  6. The 1-minute walker brings the torch back (1 min).
  7. The 1- and 6-minute walkers cross (6 min).

What does my rating mean?

Your minutes are compared with the fewest possible: exactly the fewest is Perfect, and every extra minute lowers the rating.

Rating (minutes)4 people5 people
Perfect1729
Excellent18–1930–33
Good20–2234–39
Fair23–3040–52
Keep trying31+53+

Like crossing puzzles? Try the River Crossing Puzzle or the Water Jug Puzzle.

Questions

What is the answer to the bridge riddle?
17 minutes: the two fastest cross, the fastest comes back, the two slowest cross together, the second fastest comes back, and the two fastest cross again.
Can three people cross at once?
No. The bridge holds at most two, and whoever crosses must carry the torch.
Can someone cross without the torch?
No. It is dark, so every crossing needs the torch, which is why someone always has to bring it back.
Is there a time limit?
Not here: you can take as long as you like to think, and your total minutes are rated against the fewest possible, 17 for four people.
Who invented the bridge and torch puzzle?
Nobody knows. The oldest known printed version is in a 1981 puzzle book, and in 2002 the mathematician Günter Rote worked out the best plan for any number of people.