The Lion and the Man Puzzle - Can the Man Escape, Or Is the Proof I Read Incorrect?

This is a delightful problem I came across and took a long time to find a solution. Apparently it was incorrectly unsolved for 25 years.

"A man is stuck in a perfectly circular arena with a lion. The man can move as fast as the lion. Is it possible for the man to survive? (Assume each has infinite strength so they can both continue to move indefinitely if needed)."

I posted a detailed solution from a proof I read in a book. Apparently a lot of people disagree with this proof. I think it's a wonderful problem, was curious what you guys thought.

Writeup of solution

References: The post I wrote up followed a proof presented in this book: Famous Puzzles of Great Mathematicians. The author of that book said he based his proof on 2 papers.

How the Lion Tamer was Saved, by Richard Rado, Mathematical Spectrum Volume 6 (1973/1974).

More About Lions and Other Animals, by Peter Rado and Richard Rado, Volume 7 (1974/1975).

• This doesn't appear to be a question. Jun 28, 2013 at 15:21
• "Apparently a lot of people disagree with this proof. I [...] was curious what you guys thought.". I think, this is a question, in other words: "Do you agree with this proof?" Jun 28, 2013 at 15:27
• Eliding "I think it is a wonderful problem, " before "was curious what you guys thought" obviously clarifies what he wrote. But he didn't write that. @Tomas Jun 28, 2013 at 15:49
• Okay, I guess I just got it the other way. Jun 28, 2013 at 15:56
• It’s perfectly clear what is being asked. What’s the point of this petty nitpicking? Jun 28, 2013 at 20:48

According to the different links you and Presh provided, here is how I understand the solution :

Let d be the distance between the man and the lion, and let t be the time passing. $$\lim\limits_{t \to \infty} d = 0$$

What does it mean ? It means that the man can theoretically escape if we assimilate the man and the lion to points. However, if we consider that they are circles with non null diameter, the lion catches the man.

Thanks to a comment from "Blue," I came across this, and I feel that settles the matter as it confirms the references provided adequate proofs.

Wolfram Mathworld's entry on this problem states the man can survive. http://mathworld.wolfram.com/LionandManProblem.html

A lion and a man in a closed arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal? This problem was stated by Rado in 1925 (Littlewood 1986).

An incorrect "solution" is for the lion to get onto the line joining the man to the center of the arena and then remaining at this radius however the man moves. Besicovitch showed the man had a path of safety, although the lion would come arbitrarily close.

• What does it mean to "remain at this radius"? On its face, it would mean staying at the same distance from the centre, which is pointless... Jun 29, 2013 at 22:04
• @DJohnM It's goofy wording, but the intent is to define the radius to be the line segment (not a length) joining the center of the arena to the outer edge of the arena that passes through the man; this segment moves as the man moves. The idea is that the lion can stay on this segment with movement to spare, so it can always approach the man along this segment. May 9, 2022 at 16:53