Droning On in Circles

Question

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:

1. Teleport any drone to your location (you can pick which drone).
2. Teleport yourself to Arthur or Bob's location (you can pick which drone).
3. 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:

1. You must be exactly at a drone's location to pull his tail.
2. After you have teleported a drone to your location, you cannot pull his tail until you have teleported at least once.
3. 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

1. has 100.0% probability of being correct (not e.g. a probability of correctness that approaches 1 asymptotically)
2. has finite $E\left[ n\right]$, where $n$ is the RV describing the number of steps needed to reach a solution

Answer 1

1. Start at A
2. Go to random B
3. Call Arthur to B
4. Go to Bob at A
5. Get $\overline{AB}$
6. Go to Arthur at B
7. Call Bob to B
8. Go to random C
9. Call Arthur to C
10. Go anywhere
11. Go to Arthur at C
12. Get $\angle{AC}$
13. Go to Bob at B
14. Call Dale to B
15. Go anywhere
16. Go to Bob at B
17. Get $\overline{BC}$
18. Go to Arthur at C
19. Call Bob to C
20. Go to random D
21. Call Arthur to D
22. Go anywhere
23. Go to Arthur at D
24. Get $\angle BD$
25. If $\angle{BD}$ sucks, forget point D and goto step 20
26. Go to Bob at C
27. Call Dale to C
28. Go anywhere
29. Go to Bob at C
30. Get $\overline{CD}$

$\overline{AC}^2=\overline{AB}^2+\overline{BC}^2\pm2\times\overline{AB}\times\overline{BC}\cos\frac{\angle{AC}}{2}$, giving two possibilities for radius $r=\frac{\overline{AC}}{2\sin\frac{\angle{AC}}{2}}$, one of which can be confirmed by solving the triangle $BCD$ in the same way.

At step 25, $\angle{BD}$ sucks with probability 0, but we can still guarantee a solution in a finite number of steps against an adversarial teleporter because only four sucky locations exist. The geometry of sucking is left as an exercise for the reader.

Answer 2

Remark: This is not a complete solution. See the edit below.

Yes, it is possible to escape, using the following sequence:

1. Teleport Dale to your location
2. Teleport to a random location
3. Teleport Arthur to your location
4. Teleport to a random location
5. Telport to Arthur
6. Pull Arthur's tail, receiving a measurement $A$.
7. Teleport Bob to your location
8. Teleport to a random location
9. Teleport Arthur to your location
10. Teleport to Bob's location
11. Teleport to Arthur's location
12. Pull Arthur's tail, receiving a measurement $B$
13. Teleport to Bob's location
14. Teleport Dale to your location
15. Teleport to Arthur's location
16. Teleport to Bob's location
17. Pull Dale's tail, receiving a measurement of $d$

This is depicted in the image above. Your initial location is marked with $0$, the first random place you teleport to is marked with $1$, and the third random place you teleport to is marked $3$. The angle subtended by the chord of length $d$ is $2\pi-A-B$, the radius of the room is $$ \frac{d}{2\sin\left(\frac{2\pi-A-B}{2}\right)}, $$ and the area of the room is

$$\pi\left(\frac{d}{2\sin\left(\frac{2\pi-A-B}{2}\right)}\right)^2.$$

Edit: As pointed out in the comments, the situation could be different depending on the relative locations of teleportation. The picture might also look like the following:

In this case the angle subtended by the chord of length $d$ is $B-A$, and the area of the room is

$$\pi\left(\frac{d}{2\sin\left(\frac{B-A}{2}\right)}\right)^2.$$

The difficulty is, without some knowledge of the relative positions of teleportation locations, we cannot know which formula to use.

Answer 3

Based on Julian's solution, I came up with the following. Mostly I just added in the repeat-as-necessary at the end to determine which calculated radius is the correct one.

• Teleport to Random Location: Point 1.
• Summon Dale.
• Summon Bob.
• Teleport to Random Location: Point 2.
• Summon Arthur.
• Teleport to Bob's Location (which is also Dale's location).
• Pull Dale's tail. This gives you the distance between points 1 and 2.
• Teleport to Arthur's location.
• Summon Bob.
• Teleport to Ransom Location: Point 3.
• Summon Arthur.
• Teleport to Bob's location.
• Teleport to Arthur's location.
• Pull Arthur's tail. This gives you $\angle 13$.
• Teleport to Bob's location.
• Summon Dale.
• Teleport to Arthur's location.
• Pull Arthur's tail. This gives you $\angle 23$.

As indicated in Julian's answer, there is ambiguity as to whether $$\angle 12 = 2\pi - (\angle 13 + \angle 23)$$ or $$\angle 12 = (\angle 13 - \angle 23)$$

Calculate the radius of the circle for both cases. One is correct, the other is not.

Repeat the entire process. One calculated radius from each measurement should match one from the other. That is the true radius. There is a vanishingly small chance that the incorrect radii would also match. Even if they do, repeat the whole process again.

Once you have the radius, calculating the area is straightforward.

Answer 4

This gets a bit odd, and assumes something about the rules.

My algorithm is:

1. Identify starting point as point A
2. Teleport at random (to point B).
3. Summon Dale.
4. Pull Dale's tail, revealing distance AB (call it a).
5. Summon Arthur.
6. Teleport at random (to point C).
7. Summon Dale.
8. Pull Dale's tail, revealing distance BC (call it b).
9. Summon Arthur.
10. Teleport back to Bob.
11. Summon Dale.
12. Pull Dale's tail, revealing distance CA (call it c).

13. Calculate the diameter of the circumscribed circle (the circle that touches all three points on the triangle, which were all on the wall) according to the formula:

    diameter = 2abc / sqrt((a+b+c)(-a+b+c)(a-b+c)*(a+b-c))

14. Multiply the diameter by Pi for the area of the circle.

15. Escape.
16. Profit.

The formula for the diameter is explained in great detail at https://en.wikipedia.org/wiki/Circumscribed_circle

Answer 5

My answer assumes a few things about the situation that I must put forth:

1. Although I may not move throughout the prison I am able to move my arms as I can pull the tails of the drones but have no "pull" spell.
2. The drones are flying and therefore their mass does not matter when moving them.
3. The drones fly low enough for me to grab them, as I can pull their tails.
4. The drones will accept actions I take upon them as they allow me to pull their tails.

With these assumptions this is my answer:

1. push Arthur as far from me as possible in what appears to be the direction of the far side of the wall.
2. Teleport to Arthur.
3. Repeat steps 1 and 2 until we reach a wall.
4. Use two pieces of clothing torn from my garments pulled taught to make a line that is perpendicular to the line connecting two points on the circumference (one piece is on two points, the other placed perpendicular to it)
5. Push Arthur out to tip of perpendicular piece of clothing
6. Teleport to Auther and pull his tail
7. If result is not 180 degrees move clothing and Arthur untill answer is 180 degrees. Teleporting to Arther each time to pull.
8. Push Arther along perpendicular cloth until he is against the wall.
9. Teleport to Bob
10. As bob and Dale are in the same place pull Dale's tail
11. I now have the diameter of the circle and can calculate the area as A = (d/2)*pi^2

Answer 6

ok, let try this for minimal Blinkage:
Blink to somewhere random
summon dale
summon bob
Blink to art
YANK! (i can has angle now)
Blink to bob (quik cause arts gonna bite i'm sure)
YANK! (i can has a chord now, B#!)
Blink away from dale (again avoiding the aforementioned consequence)
now i can do my math,
you can calculate the radius as you now has an isosceles triangle
call the center of the room c
angle AcD and dist AD are known
angles cDA and cAD are equal to (180-AcD)/2
call midpoint AD as m
again use symmetry to get angles Dcm and Adm as 90-AcD
i can has a right triangle and all the angles is now knowed
Ac = Am/sin(Acm)
and the area is (Ac)^2
lemme go!
if i was a masochist i could do it with one less blink
but then maybe i wouldn't wanna leave...