A troop of N immortal point mass soldiers (with N >= 3) are attempting to infiltrate Fogland (an infinite 2-dimensional plane covered in fog).
They will jump out of an airplane and, after being buffeted by the winds, will each land in an independent random location in Fogland.
Unfortunately, being point mass soldiers in a foggy country, they will not be able to see each other, and will not even know if they are in the precise same location as each other.
The landing will knock them unconscious for a random amount of time. Each soldier carries two devices, both of which are only good enough for a single use.
The first device is a GPS device, which will tell them the instantaneous distances and directions of the other point mass soldiers at the time of its use.
The second device is a detonator.
The only way that the infiltration can succeed is if all N soldiers activate their detonators at the same location (but not necessarily at the same time).
For each value of N, come up with a strategy that the soldiers can use for the infiltration to succeed with probability 1. (You can specify probability 0 conditions which will cause the infiltration to fail)
* The soldiers have a finite speed limit beyond which they cannot travel.
* Given any region on the plane with non-zero area, the probability that any given soldier lands in that region is non-zero.
* Given any non-zero period of time after the landing, the probability that any gien soldier awakes during that period is non-zero.
* The only means of communication the point mass soldiers have with each other is seeing each others locations on the GPS device.
NB: infinite 2d planes don't have magnetic poles. Therefore there is no such concept as "North"
I got this puzzle from a colleague who used to send the team a brain teaser every week. This was one of them and came with the following note:
"WARNING: this week's brainteaser is ridiculously difficult. I know few people who have independently worked out a solution for any N, and nobody who has independently worked out a solution for all N."
I found an answer for N=3 and he accepted it (different from the answer that was provided here), but I never knew if it was what he expected and if it can be generalized. Therefore, I am posting my answer and will give a bounty if someone manages to do it. Good luck!