Four people must cross a bridge at night. They only have one flashlight, and no one can cross without it. A maximum of two people can cross at a time. Each person takes a different time to cross: - 1 minute 2 minutes 5 minutes 10 minutes When two cross together, they take as long as the slower one takes. What is the shortest total time in which the four can cross?

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The bridge and the lantern

The quickest way is not to always send the fastest one back. The key is to decide when it is best for the two slowest ones to cross together.

Four people must cross a bridge at night. They only have one flashlight, and no one can cross without it. A maximum of two people can cross at a time. Each person takes a different time to cross: - 1 minute

2 minutes
5 minutes
10 minutes When two cross together, they take as long as the slower one takes. What is the shortest total time in which the four can cross?

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Hints

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No basta with that the more fast haga all the regresos.
Sometimes it is better to send the two slowest ones together.
Compare the cost of bringing back the flashlight with 1 and 2.

Solution

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Answer: the shortest possible time is 17 minutes. Let's call people by their times: 1, 2, 5 and 10. The optimal strategy is: 1. they cross 1 and 2: it takes 2 minutes;

returns 1: takes 1 minute;
they cross 5 and 10: it takes 10 minutes;
returns 2: takes 2 minutes;
They cross 1 and 2: it takes 2 minutes. The total time is: $$

2+1+10+2+2=17.
$$ The idea is to use the fastest two to carry the lantern, but allow the slower two to cross together only once. If the fastest one made all the returns, more time would be lost.

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