The Endless Cat-Mouse Chase: Can the Mouse Outrun the Cat?

[Bartholomew]

Now here's a problem I thought up that I actually know the answer to

A cat and a mouse start at opposing points by the inside wall of a cylindrical box 1 meter in radius with sides too high to climb. The cat and mouse each move at a constant speed, and the cat always moves straight towards the mouse wherever the mouse is. When the cat gets within half a meter of the mouse, the cat leaps and catches the mouse. How much faster than the cat does the mouse have to be to perpetually evade capture?

 

[Icebreaker]

I don't understand the question. How exactly are the cat and mouse placed, and the tube?

 

[Bartholomew]

In the view from above, the box (open to the sky) appears as a circle--the cylindrical box is positioned like a grain silo. Pick a diameter of this circle. The cat starts at one end of the diameter, just within the box, and the mouse starts at the other end, also just within the box. The circle's edge is a vertical wall which the mouse can't climb.

 

[Gokul43201]

In other words, the cat and mouse are constrained within a unit circle.

 

[Bartholomew]

Basically, yes. I can't just _say_ they're stuck in a circle; cats and mice get stuck in boxes. Round boxes.

 

[Rogerio]

How much faster than the cat does the mouse have to be to perpetually evade capture?

It's easy :

Vmouse > 2 * Vcat

 

[Bartholomew]

Well, if the cat's smart, the mouse has to be that fast. But in the problem the cat's dumb--he always goes straight towards the mouse.

 

[Rogerio]

You are right!
Then, probably Vmouse > Vcat * 2/sqrt(3) is enough.

 

[Bartholomew]

You got it Rogerio.

 

[Bartholomew]

Problem Two: Two Cats

Explanation for Rogerio's answers (Rogerio you can supply your own if you want and then I'l delete this): If the cat is smart, he will go halfway between the center and the edge of the circle and start moving in a circle with radius 1/2 m. The mouse's best try is moving around the outer edge of the circle, ahead of the cat's revolutions. If the mouse moves inward to reduce the distance he must travel around the circle, the cat moves inward too and reduces his distance even more (proportionally), so the mouse must stay to the edge for his best chance. Since the mouse's circle is twice as big as the cat's circle, the mouse must move twice as fast.

If, as in the problem, the cat is dumb and just chases straight after the mouse, the mouse's best shot is again to circle around the circumference. The cat will end up chasing behind him on a circle. If the mouse's speed is the minimum it has to be, the distance between the cat and the mouse is .5 m. Since the mouse is directly ahead of the cat, draw another circle inside the 1 m circle to represent the cat's path and pick a point on it. Then draw a tangent ray to it in one direction out to the mouse's path; this distance is the forward distance between the cat and mouse, .5 m. This forms a right angle with the radius from the cat's position to the center. The cat's radius, the mouse's radius (length 1 m), and the .5 m distance between them now form a right triangle. Solve for the cat's radius and you get (3^.5)/2. So the mouse's radius is 1/((3^.5)/2) times longer than the cat's radius, so the mouse must move 2/(3^.5) times as fast as the cat.

If the mouse leaves the circumference of the circle, the cat follows him and since the cat was more to the "inside" anyway, you can see that the new path of the mouse simply follows a tighter shape where the cat is proportionally closer to the center than in the original circle, so it is worse for the mouse. In any case, once the situation gets set up, the mouse cannot leave the edge even a tiny bit or he will be instantly caught by the looming shadow of the cat's range (if the mouse is moving at the minimum speed).

(you can think of why the mouse must stay to the edge in those 2 problems in another way: at every moment the mouse does not want to go closer to the cat; he wants to take the path that moves him as far away from the cat as he can. If you visualize this you will see that it means the mouse must move on the circumference)


Two Cats
Now here's a harder one, which I also have an answer to:

What if you have the same round box, the same mouse, but TWO cats, which both move at the same speed and which both use the best tactics they can to catch the mouse--and they can't catch the mouse unless they are exactly on it.

Can you think of a strategy the two cats can use to catch the mouse, if the mouse moves exactly as fast as the cats do? Assume the cats have instant reaction time.

(Reminder: the box constrains the cats and mouse to a circle with radius 1)