Bridge and Torch: Racing Against the Dark
This classic puzzle tests our ability to optimize under constraints. Four people—A, B, C, and D—need to cross a bridge at night. They have only one torch, and at most two can cross at a time. The crossing speed of any pair is determined by the slower person.
Crossing Times:
- A: 10 minutes
- B: 5 minutes
- C: 2 minutes
- D: 1 minute
The goal: get everyone across as quickly as possible.
Naive Approaches Don’t Work
It might seem intuitive to always try to send the fastest people back and forth, or to send the slowest people together—but that doesn’t always minimize total time.
Key Insight
To minimize time, we minimize the number of slow crossings, and use the two fastest people (C and D) as returners whenever possible.
Optimal Strategy
Let’s use the fastest-first return strategy:
- D and C cross → 2 minutes
- D returns → 1 minute
- A and B cross → 10 minutes
- C returns → 2 minutes
- D and C cross again → 2 minutes
Total Time = 2 + 1 + 10 + 2 + 2 = 17 minutes
Why It’s Optimal
- We only send the slowest pair (A and B) together once.
- The fastest (C and D) handle all the return trips.
- Any alternative involves multiple crossings with A or B, pushing the total time above 17 minutes.
Final Answer
Minimum total time: 17 minutes