Crossing the Bridge at Night

We solve the general case of the bridge-crossing puzzle. 1 The Puzzle Four people begin on the same side of a bridge. You must help them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at a time. Any party who crosses, either one or two people, must have the flashlight to see. The flashlight must be walked back and forth, it cannot be thrown, etc. Each person walks at a different speed. A pair must walk together at the rate of the slower person’s pace, based on this information: Person 1 takes t1 = 1 minutes to cross, and the other persons take t2 = 2 minutes, t3 = 5 minutes, and t4 = 10 minutes to cross, respectively. The most obvious solution is to let the fastest person (person 1) accompany each other person over the bridge and return alone with the flashlight. We write this schedule as + {1, 2}− 1 + {1, 3}− 1 + {1, 4}, denoting forward and backward movement by + and −, respectively. The total duration of this solution is t2 + t1 + t3 + t1 + t4 = 2t1 + t2 + t3 + t4 = 19 minutes. The interesting twist of the puzzle is that the obvious solution is not optimal. A second thought reveals that it might pay off to let the two slow persons (3 and 4) cross the bridge together, to avoid having both terms t3 and t4 in the total time. However, starting with + {3, 4}− 3 + · · · or + {3, 4}− 4 + · · · incurs the penalty of having person 3 or person 4 cross at least three times in total. The correct solution in this case is to let persons 3 and 4 cross in the middle: + {1, 2}− 1 + {3, 4}− 2 + {1, 2}, with a total time of t2 + t1 + t4 + t2 + t2 = t1 + 3t2 + t4 = 17. 1