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?
"man" here is short for male or female engineer.
1) The car must go faster than the 1st man; otherwise it will be unable to drop-off the 2nd man ahead of the first (a requirement).
2) The tunnel length is an integer number of meters, since the car and 1st man, traveling at integer rates from either end, meet on an integer minute.
3) The car time of the 2nd man is fixed by the following need: he must be dropped at a position ahead of the 1st such that the difference in speeds (1st-2nd) yields the same time needed to close their gap as for each to reach the finish line.
At the moment of the 2nd drop-off, their separation is integral, as are their distances to the finish.
4) For a simulation, a reasonable range of integer possibilities may be tried to represent walking speeds, car speeds, and tunnel lengths (avoiding physical improbabilities).
5) For a simulation, specifying 7 integers will allow a determination if a possible solution has been given, which may be compared in total time with all instantiations of another 7 integers: car speed, 1st man foot speed, 1st man foot time, 1st man car time, 2nd, 3rd, and 4th men's foot speeds.
5a) The 1st three numbers will give the tunnel length.
5b) The 2nd man's car time is set by the car pick-up time and the car speed such that the distance separating the 1st and 2nd man on drop-off will close their gap at the same time they reach the finish. (Note that the car speed must be sufficient to get to the 2nd man before the 1st has crossed the finish and likewise for the others).
5c) Similarly, the car time from the 2nd man drop-off to North and return will determine the 3rd man's foot time and car time, and likewise in turn the 4th man's foot time and car time.
Edited on April 22, 2019, 12:40 pm
preliminary thoughts
