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A duck is in the center of a square pond and cannot fly. A fox is in the corner of the pond and cannot swim. If the duck can swim with speed 1, and both play optimally, what is the minimal fox movement speed $v$ such that the fox can prevent the duck from escaping the pond?

I expect there is a closed form solution, and here is my thinking about how to solve it. The combined positions of the duck and the fox form a point in three dimensional space. At the critical speed $v$, there should be a critical set of positions such that the duck can approach arbitrarily close to the set, but if the duck reaches just beyond it, the duck can escape. Analysis of the puzzle, or perhaps numerical simulation of the perfect play, should reveal the geometry of this set, after which the problem may become straightforward.

For the circular pond (of radius $r$), the solution is well-known: Near the critical speed, the duck stays opposite of the fox until the duck gets to distance $r/v$ from the center, and then the duck continues on the straight path (tangent to the radius $r/v$ circle) to the edge of the pond, just barely escaping at $v=4.603...$. For the square pond problem, the lack of rotational symmetry makes the game state three-dimensional and thus harder.

Update: See this question for the generalization to a regular polygon and some good but (at least as of this writing) suboptimal strategies. I am looking either for an exact solution, or an accurate numerical simulation of the perfect play, including a graph showing the perfect play at the critical speed (minus epsilon, thus allowing the duck to escape). Despite the simplicity of the problem, the continuous nature of the play makes efficient accurate simulation tricky.

Dmytro Taranovsky
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  • Your question is a great question, though it isn't new. It has been covered in quite a bit of detail here. Still, the final answer remains elusive. – Jens Aug 16 '17 at 18:35
  • @Jens Thank you. I added a note to my question accordingly. – Dmytro Taranovsky Aug 16 '17 at 21:00
  • I think the previous question also sought an exact solution. It simply is the case that none of the people attempting to answer the question has been able to finish such a solution yet. Thanks for the reminder about this interesting problem, but it does not seem to me that this is really a new question at all, even with the update. – David K Aug 17 '17 at 02:51
  • @DavidK If someone can solve the problem for the square (which is a natural and perhaps easier case), but cannot solve it for an arbitrary regular n-gon, we still want to encourage that person to give an answer. For questions with unknown answers, it sometimes makes sense to keep both the general case and a natural special case open (but linked and perhaps with the special case marked as duplicate once the general case is solved). However, I searched the documentation about duplicates and could not find guidance on this issue. – Dmytro Taranovsky Aug 17 '17 at 04:04
  • We now have a question about n-gon answered by incomplete solutions for a square, and this question specifically about a square but marked as a duplicate. I admit that the current situation does not seem ideal to me. I would have preferred a question about a square (like this one), with answers specifically addressing the square, and a related question trying to generalize to an n-gon. Maybe there's a way to improve the situation after all. – David K Aug 17 '17 at 12:04
  • One thing that is new about this question is that you suggested a solution method that hasn't yet been tried. Perhaps the question can be reworded so that it specifically requests answers that apply that method to case $n=4$ of the other question. That would say why someone should post an answer here rather than under the n-gon question. I think that might be enough to reopen this question. – David K Aug 17 '17 at 12:09

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