logic puzzles

Question

Alex, Bob, Carol and Daniel must cross a bridge at night before next morning; they all begin on the same side. It is very likely that the bridge will be exploded any time after 56 minutes. 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 by a person until all of them cross to the other end. Alex takes 4 minutes to cross the bridge, Bob takes 7 minutes, carol takes 29mins and Daniel takes 31 mins. A pair must walk at a slower person's pace.

Find two alternative ways to get all of them to cross the bridge. Show how they cross the bridge using the most suitable notations.

Answer 1

This is variation of a classic logic puzzle, all of which pose the same general quandary; that is, getting people across a bridge, a river, etc, in the least amount of time, while holding a flashlight, a torch, etc.  Here is how this one plays out.

Alex and Bob cross first.  Elapsed time: 7 minutes

Bob stays on the other side and Alex comes back across alone. Elapsed time: 7 + 4 = 11 minutes

Now Carol and Daniel cross together.  Elapsed time: 11 + 31 = 42 minutes

Carol and Daniel stay on the other side and Bob comes back across.  Elapsed time: 42 + 7 = 49 minutes

Finally Alex and Bob cross together. Elapsed time: 49 + 7 = 56 minutes

The alternative way to also get this done is for Alex to stay behind after the first crossing, Bob to come back across, Carol and Daniel to cross, Alex comes back across, and then Alex and Bob cross together again at last.

Answer 2

The key to solving this problem is to realize that the slower person sets the pace for the pair. So, if two people with different crossing times cross together, the time taken will be equal to the slower person's crossing time.

Step 1: Alex and Bob cross the bridge together, taking 7 minutes (because Bob is the slower one).
AB ----> (7 minutes)
Step 2: Alex goes back alone with the flashlight, taking 4 minutes.
A <---- (4 minutes)
Step 3: Carol and Daniel cross the bridge together, taking 31 minutes (because Daniel is the slower one).
CD ----> (31 minutes)
Step 4: Bob goes back alone with the flashlight, taking 7 minutes.
B <---- (7 minutes)
Step 5: Alex and Bob cross the bridge together, taking 7 minutes (because Bob is the slower one).
AB ----> (7 minutes)

Total time taken = 7 + 4 + 31 + 7 + 7 = 56 minutes.

Therefore, they cross the bridge in 56 minutes.

Notations: A, B, C, and D represent Alex, Bob, Carol, and Daniel, respectively. The arrow (---->) indicates crossing the bridge from one side to the other, and the less than symbol (<) indicates coming back with the flashlight.