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2. It is a dark and stormy night. Four people must evacuate from an island to the mainland. The only link is a narrow bridge which allows passage of two people at a time. Moreover, the bridge must be illuminated, and the four people have only one lantern among them. After each passage to the mainland, if there are still people on the island, someone must bring the lantern back. Crossing the bridge individually, the four people take 2, 4, 8 and 16 minutes respectively. Crossing the bridge in pairs, the slower speed is used. What is the minimum time for the whole evacuation?
Solution Exactly five passages are required,
three pairs to the mainland and two individuals back to the island.
Let the fastest two people cross first.
One of them brings back the lantern.
Then the slowest two people cross,
and the fastest people on the mainland brings back the lantern,
The final passage is the same as the first.
The total time is 4+2+16+4+4=30 minutes.
To show that this is minimum,
note that the three passages in pairs take at least 16+4+4=24 minutes,
and the two passages individually take at least 4+2=6 minutes.