There are four people who want to cross a rickety bridge; they all begin on the same side. You have 17 minutes to get them all across to the other side. It is night, and they have one flashlight. A maximum of two people can cross the bridge at one time. Any party that crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth; it cannot be thrown, for example. Person 1 takes 1 minute to cross the bridge, person 2 takes 2 minutes, person 3 takes 5 minutes, and person 4 takes 10 minutes. A pair must walk together at the rate of the slower person’s pace. Required:
What is the shortest time needed for all four of them to cross the bridge?

The shortest time needed for all four of them to cross the bridge = 17 minutes

Explanation:

As given ,  to cross the bridge

Person 1 takes 1 minute,

Person 2 takes 2 minutes,

Person 3 takes 5 minutes, and

Person 4 takes 10 minutes.

For the first time ,

Person 1 and Person 2 will go to cross the bridge

As Person 2 takes 2 minutes

So, total time taken by Person 1 and person 2 together will be 2 minutes

Now,

Person 1 will come back with flashlight.

He takes 1 minutes.

So, Total time becomes 2 + 1 = 3 minutes

Then,

Person 3 and Person 4 will cross the bridge

As Person 4 takes 10 minutes

So, total time taken by Person 3 and person 4 together will be 10 minutes

So, Total time becomes 3 + 10 = 13 minutes

Now,

Person 2 will come back with flashlight.

He takes 2 minutes.

So, Total time becomes 13 + 2 = 15 minutes

Then,

Person 1 and Person 2 will cross the bridge

As Person 2 takes 2 minutes

So, total time taken by Person 1 and person 2 together will be 2 minutes

So, Total time becomes 15 + 2 = 17 minutes

∴ we get

The shortest time needed for all four of them to cross the bridge = 17 minutes