Hungry Lion

A hungry lion runs inside a circus arena which is a circle of radius 10 meters. Running in broken lines (i.e. along a piecewise linear trajectory), the lion covers 30 kilometers. Prove that the sum of all turning angles is at least 2998 radians.

Imagine the lion is static, facing North, and instead, the center of the arena moves around. Then, each time the lion runs X meters in some direction, this translates into the center moving X meters South. Each time the lion makes a turn of Y radians, this translates into the center moving along an arc of Y radians.

Thus, the problem translates to a point inside the arena alternating between traveling straight South and then moving along arcs around the center of the arena. Since the total distance traveled straight South by the point is 30KM and the distance between the starting and the ending points is at most 20M, the total distance traveled North must be at least 30KM – 20M = 29980M. Therefore, the total length of the arcs traversed by the point is at least 29980M, and since the radius of each arc is at most 10M, the total angle of the arcs must be at least 2998 radians. The sum of all turning angles of the lion is the same, so this concludes the proof.

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  1. It has to be at least 3000 radians. The best the lion can do is to run around the circumference of the circle. What method involves less turning than that, and how do you know he ran 30km south?

    1. Hello Kevin. Your intuition is correct that moving along the edge of the arena is an excellent strategy that achieves 3000 radians. However, the lion can improve it a little bit… For example, imagine that he starts by traveling 20M along the diameter of the arena, then makes a 90° turn, then travels 29960M along the circumference of the arena, then makes another 90° turn, and finally travels along the diagonal extra 20M. The total turning of the lion will be less than 3000 radians.
      In any case, the task is to provide a lower bound for the total turning. The solution now is expanded and a picture is added for clarity. If you want to discuss it further, I have created a special thread in the forum.