Consider the generalization of the bridge crossing puzzle in which we have n $>1$ people

whose bridge crossing times are t$1,$ t$2,...,$ tn$.$ 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 and it cannot be thrown. For example, suppose there are four people who

want to cross a bridge and they all begin on the same side. 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. If person $1$ and person $4$ walk across first, $10$ minutes have elapsed when

they get to the other side of the bridge. If person $4$ returns the flashlight, a total of $20$

minutes have passed.

Design a greedy algorithm for this problem and find how long it will take to cross the

bridge by using this algorithm. Does your algorithm yields a minimum crossing time for

every instance of the problem? If it does, prove it; if it does not, find an instance with

the smallest number of people for which this happens.