Hungry Lion

Source: Puzzle Toad, CMU

Problem: 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.

Update(05/02/10):
Solution: Solution posted by me in comments!!

Comments

  1. Pratik Poddar10:41 PM, February 05, 2010

    Posting solution since no one has solved it till now.

    Solution at http://www.cs.cmu.edu/puzzle/Solution16.pdf

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  2. Pratik Poddar9:29 AM, February 06, 2010

    Let us look at the world through the eyes of the hungry lion, that is, the lion is stationary but everything else moves around. Assuming furthermore that the poor animal has a stiff neck and cannot rotate its head and is always facing North. The trajectory of the center of the arena looks in the the following way: It is a combination of linear segments pointing South and circular arcs. The total length of the linear segments is 30 km. Since the center is never more than 10 meters away from the lion, the initial and final positions of the center are at most 20 meters apart. By a form of the triangle inequality, the total length of arcs is at least 30, 000 − 20 = 29, 980 meters. But each arc has radius at most 10 meters, so the sum of all arc angles is at least 2, 998 radians.

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  3. sumit somani4:07 PM, November 29, 2010

    very old puzzle but still I would like to say that assume lion is very smart and wish to reduce the amount of angle it requires to turn, so he moves in infinitesmall segments along the circumference so that in min movement in terms of angle he gets the max distance. So 2*pi radians gives him 20*pi m, i.e. 10 metre per radian, So for 30 km angle min required wud be atleast 3,000 radians.

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  4. Pratik Poddar1:21 PM, September 08, 2026

    The circumference intuition gives the right scale but 3000 is not quite the right constant - here is the rigorous argument from the CMU solution. Put the camera on the lion with a stiff neck always facing North: when the lion runs straight, the arena's center moves due South; when the lion turns through an angle, the center traces a circular arc of radius at most 10 m through the same angle. The straight segments total 30 km and all point South, but the center's start and end points are at most 20 m apart, so the arcs must cover at least 30000 - 20 = 29980 m. With radius at most 10 m, the arc angles sum to at least 29980/10 = 2998 radians. Your 10 m/radian efficiency is right; the slack of 2 radians is exactly the 20 m of freedom in the endpoints. (replied using AI)

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