猫捉老鼠

fzy · 发表于 2005-2-27 21:39:45

I have posted a variation of the problem at 灵机一动: Suppose the speed of the cat is k times the speed of the mouse. What is the upper limit of k so that the mouse can escape? (Hint: It is larger then pi + 1.)

fzy · 发表于 2005-3-2 00:43:51

Or you all know know the answer already?

fzy · 发表于 2005-3-2 01:26:11

Yours is 4.6085.... I have 4.6033..., which is the solution of the equation

sqr(k^2 - 1) = pi + acos (1/k).

Not nearly as nice as yours. But how did you get yours?

fzy · 发表于 2005-3-2 17:59:06

TO solve your equations, we get k = pi + 1. This is the best you can get with the strategy you described. (Remember the hint I give?) It is correct to reach the farthest point on the r circle. (r = R/k actually.) But this is only the first step. The real interesting part is what happens next.

This variation is a much more difficult problem than the original. I will post My solution if the problem is put on top, so more people can see it. Otherwise I will post a link to a discussion about the solution.

fzy · 发表于 2005-3-3 20:24:17

That is basically the idea. The actual strategy is a little nore complicated. Because the cat can stop or change direction, the mouse needs to modify her strategy accordingly. Furthermore, since the cat can also change his strategy according to what the mouse does, we need to prove tha the mouse can win no matter what the cat does.

Apparently our monitor does not like the problem. So here is the link to a discussion:

http://www.heidisun.com/qqsh/viewtopic.php?t=222

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猫捉老鼠

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Nobody wants to try?

Very close to what I got

Not very close now

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