Rescuing the Paratroopers

Another week, another Riddler. The question:

During their descent, three paratroopers were blown off course and crash-landed in the desert — an infinite, uniform, two-dimensional plane. They come to at a random time after the crash and must find a way to meet up. The only tool they each have is a device that displays a snapshot of the positions (but not the velocities) of all three of them in the desert. They can each use this tool only once. To make things more annoying, they’ve all been nearly blinded by sandstorms and won’t be able to physically see the others until they’ve all arrived at the same point.

Can you devise a strategy that they can agree upon beforehand and that will guarantee they will meet up? (Note that the snapshot does not indicate the specific identities of the other paratroopers, so a strategy like “let’s agree that A stands still and B and C walk to him” will not work.)

Extra credit: Is there a strategy that would work for any number of paratroopers?

Here is the strategy I devised: Everyone gets to decide their unique walking velocity — the only thing that matters is that everyone’s speed is different than the others. Then we will, when we wake up, draw a circle that is constrained by the three points of our buddies (including ourselves). This represents the path that we will walk at our set velocity in a set direction (say, clockwise). Then as we walk if another person were to wake up they would inevitably draw the same circle and embark along the same path, eventually running into one another. If we meet up we will stay exactly where we met. The third person will either be on the same path or wake up and see that the other two have met and make a bee line for the group of two.

Example of path determined by a snapshot

Extrapolating this to more people, we just make a closed shape that has as many variables as there are people. For four we can use an ellipse with horizontally constrained foci, for five we can use an ellipse with unconstrained foci, etc.

Example of a path made with four people

In the case with more than three people if two meet they continue to move along their path. Anyone who wakes up after two have met will again just make a bee line for where ever the two person group is at that time, knowing they will be collected at the next revolution. This should work for all cases.