In a plane, there is a robotic arm consisting of $n \ge 2$ segments of length 1, like this:
- The first segment is fastened to a single point ("origin"), but it can rotate freely around that point.
- All other segments are connected to the previous segment by means of a joint, so they can bend with respect to the previous segment. However, the joints have limited allowance: the segment cannot deviate from the direction of the previous segment by more than some angle $\alpha$ (or, in other words, the angle between no two consecutive segments can be smaller than $\pi-\alpha$). All the joints have the same allowance.
Here's a rough picture of such an arm with $n = 4$ segments:
The blue circles are the joints, the blue lines are the segments. The dashed lines show the possible angles of the first joint.
The question: What is the least allowance $\alpha$ that makes it possible for the robotic arm to reach any point in the disk of radius $n$ around the "origin" (the point where the first segment is fastened)?
Source: I made up this puzzle on my own.
P. S.: As you can see, this is my first puzzle here. Please point out anything that I could have done better. (The solution is pretty nice and while it's nothing terribly complicated, you will need to come up with a bit of elegant reasoning. So I hope that makes it a math puzzle and not just a math problem.)