In the spirit of wizards and circular prisons:
You have been imprisoned by an evil wizard in a perfectly circular prison cell of unknown size. You're shackled to the wall, unable to move about the cell.
Fortunately, the wizard isn't so evil that he won't give you at least one opportunity to escape. He tells you that if you can compute the area of the prison cell exactly (that is, deterministically, and to arbitrarily high precision), he'll let you go free.
To help you accomplish this task, he's given you three flying monkey drones, Arthur, Bob, and Dale, and a set of teleportation spells you can use as often as you like:
- Teleport any drone to your location (you can pick which drone).
- Teleport yourself to Arthur or Bob's location (you can pick which drone).
- Teleport yourself to a random location on the prison wall. (You remain unable to move even after teleporting. Also, you never teleport to exactly the same location twice.)
Conveniently, if you pull on Arthur's tail, he will tell you exactly what the shortest angle $\theta$ between himself and Dale is, as shown below.
If you pull on Dale's tail, he will tell you exactly what the distance $d$ between himself and Arthur is, as shown below.
There are three restrictions on measurement, however:
- You must be exactly at a drone's location to pull his tail.
- After you have teleported a drone to your location, you cannot pull his tail until you have teleported at least once.
- You cannot measure both $d$ and $\theta$ across the same pair of points (even in reverse).
All three drones start at your location on the wall, which is unknown. You may assume that you have a perfect memory, and that you can compute any mathematical function to arbitrarily high precision.
Are you able to escape? And how?
Note: Generally speaking, I'm looking for a solution that
- has 100.0% probability of being correct (not e.g. a probability of correctness that approaches 1 asymptotically)
- has finite $E\left[ n\right]$, where $n$ is the RV describing the number of steps needed to reach a solution