Four Travellers and One Torch
The problem
Four hikers must cross a rickety bridge at night. They have one torch, the bridge holds at most two people at a time, and anyone crossing must carry the torch. A pair walks at the speed of its slower member. Individually they need , , and minutes. What is the shortest total time in which all four can be across?
- At most two people on the bridge at any moment.
- The torch must be carried on every crossing; a torch left on the far side cannot be used by someone still on the near side.
- Walking together is limited by the slower walker.
Take a minute
Reveal hint 1
The naive plan — fastest person shuttles everyone — costs minutes. Can the two slowest travellers be made to cross together?
Reveal hint 2
To pair the slow walkers you need the torch back on the near side after they cross, and that return trip costs at least minutes.
Solution
Answer minutes.
Derivation
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Send the two fastest across.
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Bring the torch back cheaply.
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Send the two slowest across together.
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Bring the torch back again.
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Finish.
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Why nothing shorter works.