Four safety engineers set out to inspect a newly cut tunnel through Mt. Popocaterpillar in the Andes. Each person walks at a different constant integer speed measured in meters per minute. In the tunnel there is a mine car which travels along a fixed track, automatically going from end to end at a fixed integer speed. When people board the car they may reverse its direction, but cannot change its speed.
At noon on Monday all four engineers start at the south end, while the mine car starts at the north end. The first (fastest) engineer meets the car, and takes it some distance north. The engineer gets out and continues going north, while the car resumes heading south. Then the second engineer meets the car and also takes it some distance north. Likewise for the third and fourth engineers. All the people, and the mine car, travel continuously with no pauses. The inspectors always go north. Each person enters and exits the car at an integral number of minutes.
All four engineers reach the end of the tunnel simultaneously. What is the earliest time this could happen?
I have established to my satisfaction that 20 minutes does not work as a solution. Further, all solutions must be even integers, assuming that the mine car finishes at the north end. Therefore, I claim (without proof) that a new lower limit on the solution is 22 minutes.
By the way, the 3 person solution of 17 minutes given by Steven Lord should actually be 16 minutes. I was suspicious of the odd number of minutes. I confirm that a 16 minute trip is possible with Steven's car travel times of 1,2 and 5 minutes. However, I have an initial walking time of 5 minutes, walking speeds of 154, 110 and 70/min, car speed of 385/min and a tunnel length of 2,695. All Engineers are picked up at the 770 mark, 2/7 of the way through the tunnel.
I think I will check the 34 minute solution.
New lower limit and new 3 person solution
