## Trivia

Subject:
Re: Spiral solution
Response To:

Larry Barrett
Recall our old familiar formula, D=RT, or T=D/R.
At the R/4 point, the time it takes for the duck to swim around this circle matches the time it takes for the cat to run around the pond.
The time for the duck to go around the R/4 circle is 2π(R/4)/1 = πR/2.
The cat runs at a speed 4 times the duck, so the time it takes for the cat to go around the pond is 2π(R)/4 = πR/2. Ed has already pointed out that inside this R/4 circle the duck is faster than the cat. Outside the R/4 point, the cat is faster.

It has already been pointed out that if the duck can get to the R/4 point, with the cat directly opposite with respect to the center of the circle, the duck can escape. In fact if the duck can get to a point that is just beyond the (R-πR/4) point, with the cat directly opposite with respect to the center of the circle, the duck can escape. At the R-πR/4 point it will be a tie (and the cat wins ties).

So the duck should swim to a point that is just beyond the (R-πR/4) point but less than the R/4 point. At this point the duck starts swimming in a circle at that radius. The duck will gradually pull ahead of the cat with respect to the center of the circle. Once the duck gets to a point where the cat is directly opposite with respect to the center of the circle, the duck will be able to swim to the shore before the cat can reach that point.

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