Suppose you are supervising three students, each armed with an inclinometer. You wish to place them in the plane so that they can determine the maximum height of a small rocket that you will launch after they are placed. When the rocket reaches its apex it will emit a flash, and at that time each student will measure the angle from the rocket to the horizontal through his or her position. So a student at point A will measure angle RAX where R is the rocket's position and X is the point on the ground underneath R.
Is it possible to place the students so that, with the three angles that they provide to you, you can determine the height of the rocket (no matter where it is)?
Note: The observers do not have compasses and get absolutely no information about the bearing (azimuth) to the rocket. All you get is three angles, and of course the positions of the students in the plane. The rocket veers in flight so that we have no information about its coordinates except that z > 0.
Source: This problem was suggested to me by Clifford Stoll, who teaches high-school physics in Berkeley (and owns the Acme Klein Bottle company, which makes beautiful Klein bottles). Cliff was unsure whether this could be done with three observers, and offered a Klein bottle for a good solution. Dan Flath (Macalester) and I solved it, and now have a fairly complete theory about what is going on here (and a nice Klein bottle).