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# The lion and the tamer

## Question

If a lion and his tamer can run at precisely the same speed without tiring, would the tamer be safe inside the lion's circular cage? Assume the lion sits on a stool at the center of the cage as the tamer enters though a door on the perimeter of the cage. You can also neglect the size of their bodies. i.e. consider the lion and tamer as points.

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Spoiler

I should think the tamer would not be safe.

Assuming he runs around the perimeter of the cage, he is traversing a circle with larger radius than is the lion. Since they run at the same speed, the lion will always be able to match the tamer's angular movement, and with time to spare to start moving outward from the center.

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Neglecting inertia and response time of the lion, The tamer would always be caught if he is required to change direction (regardless of the shape of the cage). because each time the tamer changes with even the slightest direction at the slightest angle, the lion will be able to switch to the shortest direction to where the tamer is (even if blindly following him).

Spoiler

Consider a triangle with sides A B C. AB is the initial distance of the lion and the tamer. Now if the tamer switches the slightest direction with the slightest distance, say BC, then the lion would take the straight path through AC, say AD, (where D is between A and C) where the length of AD is equal to the length of BC.

Now the distance (DC) will be always be lesser than AB, the initial separation distance, therefore slightly closing the distance, with the lion now on point D, and the tamer on point C. Every time the tamer changes direction (which is required since the cage is closed) the small advantages will accumulate, and eventually the tamer will be caught.

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Well, yes if the tamer can control the lion.

If he cannot then he is unsafe because the lion can attack him by judging where the tamer will reach(here it is assumed that the tamer runs along the periphery and the distance travelled by the tamer till the lion catches him is more than the radius of the cage).

Edited by Chandra002
Missed a bracket
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20 hours ago, Chandra002 said:

unsafe because the lion can attack him by judging where the tamer will reach...

Hi and welcome Chandra.

Let's add the condition that the tamer (and the lion) can change/reverse directions instantaneously.
Tamer can switch from CW to CCW at his pleasure.

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Good thinking. Let's add the condition that the lion and tamer are point objects. Can they coincide?

Also, could the lion reduce the radius of his circle to that of the lion and maintain any angular separation he might at some point have obtained? (Your solution prohibits this, but suppose the lion made one misstep and just for a moment he lagged the angle of the tamer.) This question has an amusing answer.

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Spoiler

Theoretically the Tamer could evade the lion indefinitely by choosing a path that is perpendicular to the Lion's and toward the center of the cage (unless that is where the Lion is). Practically, the time between course changes would get shorter and shorter so that after a relatively small amount of time evading, the number of course changes would be bordering on ridiculous. If that is not a concerning factor then the Tamer could keep it going to infinity. It's kind of like a provable working Achilles and the tortoise if  that weren't in a straight line.

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Greetings, Prof. T. Great to hear from you.

The lion does start from the center, sadly for the tamer.

The prevailing thought is for the lion to maintain the tamer's azimuth (his radius is shorter) and inch his way outward. By maintaining his azimuth, never leading nor lagging, the lion does not permit the tamer to gain advantage by reversing his direction. Although, since he can do so instantaneously, neither would the tamer lose advantage.

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