Friday Puzzler -- The Duck and The Cat

Re: Friday Puzzler -- The Duck and The Cat

Ed in Leaside

I think this is a spiral problem, and the duck can exscape as long as he keeps the cat on his 6.

Re: Friday Puzzler -- The Duck and The Cat

Larry Barrett

If the duck can get within about 3/4Radius of the shore, and directly opposite the cat, the duck can swim directly to shore and escape.

Or, the duck can swim a short way toward the shore directly opposite the cat. When the cat runs half way around the pond, the duck starts swimming back the other way. The cat runs back and the duck turns and swims back when the cat gets near the starting point. The duck does this a few more times and the cat is so tired the duck can easily swim to the shore and escape.

Re: Friday Puzzler -- The Duck and The Cat

Henry Higginbotham

I propose we award Larry extra points for messing with the cat.

Dead Duck

Kent B

Pond Circumference is PI*D

From starting spot, cat can get to any point on the shore by running no more than halfway around, or [PI* D/2], or [PI*R], or approx [3.14156 * R]

Duck has to swim distance = R.

[3.14156 * R] = cat distance

[4 * R] = cat velocity

<0.85844> cat recovery time before dinner

Dead Duck

Christmas is coming

Henry Higginbotham

I hear the duck wants a laser pointer.

Spiral solution

Alex Y

Ed, this may well work, but I can't convince myself.

Convince me.

3/4 R from the edge

Alex Y

Larry, this is correct. But how do you know he can get that close to shore with the cat 108* away?

Not Dead Duck

Alex Y

But he has to swim farther than R

Re: 3/4 R from the edge

Alex Y

meant 180 degrees away

Re: Spiral solution

Ed in Leaside

Within the 1/4 radius inner circle the duck is actually faster than the cat is, relative to the full circumference. So, the duck can start the exscape* run with the cat on his 6.

*I had in my employ a Czechoslovakian pressperson, in 1986, who had "exscaped" from communism ...

Re: Spiral solution

Alex Y

I agree that when the duck is at R/4 from the enter, with the cat opposite, he can sprint to the edge and escape.

But i just can't see the spiral with the duck keeping the cat on his 6.

I think that as long as the duck is within R/4 from the center, he is able to move closer to the edge while keeping the cat directly across the pond from him. The duck's velocity vector at any point has a circular component equal to the cat's angular velocity plus a component away from the center. As the duck reaches R/4 (or gets arbitrarily close to it) just before his dash to shore, the cat will be at his 3 or 9, not his 6.

It seems to me (and I say this with very little confidence) that keeping the cat on his 6 would end up in a stable situation with the duck at R/4 and the cat only 90* behind him, easily catching him when he sprinted to shore.

Re: Spiral solution

Ed in Leaside

I think I would have to draw it to show the ducks progress relative to the cat ...

Re: Spiral solution

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.

Re: Spiral Illustration

Ed in Leaside

Cat is in red, Duck in Green.

Re: Spiral Illustration

Larry Barrett

I’m sure Alex will chime in, but FWIW I think your solution works and is the fastest way for the duck to get to the R/4 circle with the cat on its 6 and once there can escape. I laid it out with compass and dividers and got the same result.

Re: Spiral Illustration

Alex Y

Ed and Larry, I agree that the illustration provided by Ed is the quickest way for the duck to get to the critical point at which he heads straight for shore. I did it with a spreadsheet and came up with the duck reaching critical point after spiraling through 90 degrees, just like Ed's illustration.

My only disagreement is the description of this spiral as one in which the duck keeps the cat on his 6 (and maybe I misread the intent?). I think instead, the duck keeps the cat across the pond from him. Ed's illustration shows the cat being on the duck's 6 as the escape begins, but by the time the duck gets through 60-80 degrees of the spiral, the cat is across the pond, on the ducks ~4 o'clock position. The cat will be on the duck's 3 o'clock when the duck gets ready to turn 90 degrees to the left and swim for the shore.