A duck is at the center of a circular pond with radius 1. A wolf is at the edge of the pond and is able to move around the circumference. The duck wins if it is able to get to the edge of the pond with no wolf there. Find a strategy that can guarantee the duck can escape to the circumference assuming that the wolf moves 3.5 times as fast as the duck.
For clarity, you can assume the duck is pointlike and the wolf has a small size epsilon (the wolfs size is needed to dodge weird solutions involving the duck touching an irrational point on the circumference and the wolf cannot access him). To qualify as a solution, it must work for some epsilon greater than zero. I will say that the proper solution involves no weird spirals or things, all shapes are simple geometrical objects that you can calculate lengths of easily.
There are a few follow-up questions to this one, I'll post one of them later, but for now:
What is the maximum speed of the wolf for which this strategy still works?
4 comments:
Interesting problem. I found a strategy that appears to work (it should be correct by a pseudo-inductive argument, but I haven't calculated the full number of steps). From calculations for that 'proof' I have a procedure to solve for the max speed, but the algebra is ugly and I'm without mathematica, so I'm interested in seeing if you have a more elegant solution.
-AT
There are funny solutions that need complicated math to solve. Thats why I said in my post the solution uses simple shapes. Probably your solution works also, but I guarantee that if you are needing mathematica there is a simpler way.
Wolf's strategy is head to the point on the circumference of the pond wherever the duck is currently headed? I've got a solution although I doubt it is optimal.
- !Bob
In my strat (or obvious variations for various speeds of wolf), it fails hard at wolf speed = (pi+1)*Duck speed
I think that may clarify what my strategy is.
-!bob
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