Bridge and torch night crossing

Question

Four people need to cross a rickety bridge at night. They have one torch, and the bridge holds at most two people at a time. Anyone crossing must carry the torch (there is no throwing it back, someone must walk it back).

Their individual crossing times are 1, 2, 5, and 10 minutes. When two people cross together, they move at the slower person's pace.

What is the minimum total time for all four to get across, and what schedule achieves it?

Accepted Answer

The answer is 17 minutes. Label the people by their times: 1, 2, 5, 10. The naive plan sends the fastest person (1) back and forth as the escort: - 1 and 2 cross (2 min), 1 returns (1), total 3 - 1 and 5 cross (5), 1 returns (1), total 9 - 1 and 10 cross (10), total 19 minutes. We can do better by making the two slowest people cross together, so their long times overlap instead of adding. The optimal schedule (17 minutes): 1. 1 and 2 cross together → 2 min (elapsed 2). Now 1,2 on far side. 2. 1 returns with the torch → 1 min (elapsed 3). 1 on near side; 2 far. 3. 5 and 10 cross together → 10 …

Bridge and torch night crossing

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