Zorba the Hutt (zorbathut) wrote,
Zorba the Hutt

Duck problem continued

So the question I left you with was, "What happens if the fox travels at exactly 1+pi times the duck speed?" jude pointed out that the duck and fox would then arrive at the exact same instant at the shore, and that the fox would probably win. But it's actually worse than that.

See, with the original spiral the duck takes, the duck merely *approaches* the radius. As the duck gets closer and closer, it spends more and more of its effort just keeping the wolf from catching it. Essentially, while the duck *can* arrive at the shore at the exact same instant as the fox, it will take it an infinite amount of time to do so . . . so the duck is *really* doomed.

Now, as for the "better than 1+pi" solution, it now seems possible that the duck can curve away from the fox while it's making a run for shore. It can't beat the fox, but it can, possibly, delay the fox long enough to get more of an advantage. Also working in the duck's favor is the fact that the shoreline is curved towards it - if it goes five inches away from the straight-on path, it won't have to go quite as far straight forward. (However large the pond is.)

But nobody I know has managed to get an optimal solution yet, or prove it's optimal, to my knowledge. So this is still up in the air.

And this is the last I'll say about this, unless something new comes up. Next: another logic puzzle. :)
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