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 should explain my method a little more, starting with
1) Engineer 1 (faster) gets into car at 2 minutes
2) Engineer 1 exits car -- 1 minute later
3) Engineer 2 (slower) gets into car -- 1 minute later
4) Engineer 2 exits car -- 2 minutes later
Then:
a) Total time is 6 minutes
b) After exiting the car, Engineer 1 walks in 3 minutes the same distance that the car did in 1. So the car is 3 times as fast as engineer 1.
c) Engineer 1 walks for 2 minutes, then rides in 1 minute the same distance he could have walked in 3, then walks for 3 minutes. 2 + 3 + 3 = 8. So in his first 2 minutes he walks 2/8 of the way. Let's arbitrarily say that the distance = 8 units. Then his speed is 1 unit/minute, and the car goes 3 units/minute.
d) At the 4 minute mark, the car meets engineer 2 right where he met engineer 1. So engineer 2 walks half as fast as Engineer 1, namely 0.5 units/minute.
e) But this is not integral, so scale everything up by multiplying by 2. Then Engineer 1 walks 2 units/minute, Engineer 2 walks 1 unit/minute, the car goes 6 units/minute, and the distance is 16 units.
The two person case (supporting calculations)
