# A mathematician explores the cat-and-mouse problem

Originally published at: https://boingboing.net/2019/05/28/a-mathematician-explores-the-c.html

3 Likes

Depends where they start out.

(RTFA? Hah!)

2 Likes

I feel like I’m missing something. The cat never enters the water, so the mouse “getting away” from the cat is a non-event; the cat can never “get” the mouse in the first place.

Mouse is going to drown, which is a kind of getting away

3 Likes

I’d assume the mouse cannot swim forever, whereas the cat could sit and wait for a a time much longer than that which the mouse can swim before it drowns.

I mean, I suppose drowning can be seen as a preferable demise when compared to being torn apart by a (comparatively) giant killing machine filled with sharp teeth and claws, but I would hesitate to call that a win.

4 Likes

Now if they could just get to work on the chicken-and-egg problem.

2 Likes

Assuming the cat’s limbs are shorter that the radius;
and that the cat always runs to get as close to the mouse as possible;
the mouse waits out of paw’s reach until the cat is on the edge of the pond nearest the mouse.
then the mouse swims along the diameter to the opposite edge.
the cat hasn’t been given instructions as to how to decide which way to run so it must remain still.
the mouse swims to the far shore and escapes.

i really must watch this video.

2 Likes

since the mouse can run faster, the question is, can the mouse get to the edge of the pool and get out before the cat running around the perimeter can get there. If he can get out without the cat being there waiting for him, then he can run away. If not, he is trapped in the pool.

So then it turns into a simple algebra problem. Which is a shorter amount of time: Mouse to swim to edge of pool, or cat to run to spot where mouse is trying to exit pool.

Mouse should start from most advantageous position - dead center. His swimming distance is 1/2 diameter. He can swim to the opposite point from cat forcing the cat to run his max distance of 1/2 circumference.

(Iwatched the video now and I was only partial right - there is a strategy to get a head start and beat the cat)

half circumference is pi times the radius. the cat can run four times the rate the mouse can swim. unless you live in a particularly backward state, pi is less than 4. om nom nom.

2 Likes

The tension between Second and Third Laws subsequently cause the cat to appear to be inebriated.

2 Likes

6 Likes

the felonious mouse cannot escape the long arm of the feline forever!

1 Like

The explanation does give a tactic for the mouse to escape but it is not quite optimised.

In the first half, the mouse should swim away for the cat, and go across the centre. The cat will pursue, so the mouse can spiral out to the quarter radius event ‘horizon’, keeping the cat behind it. This is a shorter path than keeping at a constant radius.

Then, a the mouse heads for the edge, it can see which way the cat is going around the pond, and head slightly in the opposite direction. So, the mouse’s path has a slight curve at the beginning, but by the time the cat is about half-way round, it is going straight for the edge.

My guess is that the two curves join smoothly when you have the best solution. There are probably two clear ‘best’ solutions - one where the mouse is furthest from the cat, and one where the mouse does the least swimming.

1 Like

That’s easy: the egg can’t move so the chicken always catches it.

3 Likes

WhatMeRTFA? The answer should be found relatively simply using calculus – you know, derivatives and all that. You do need to start with a generic function for the mouse’s swimming path
Position_in_pool = f(x,y,t).
EDIT: going strictly by Fraunfelder’s text, you need to find a solution that works for any starting location (X0,Y0). Presumably that’s the center, since any other starting position is on a successful path from the center.

Any self-respecting kitty would just wait until the mouse comes within striking distance.

1 Like

I’m with you. I think the optimal strategy in the dash phase would be for the mouse to swim directly away from the tangent point on the circling boundary that is closest to the cat. Which results in a longer dash (1m vs .75m), but a more than proportionately further distance for the cat (1.5pi m vs pi m). If the cat sticks to the short way around the mouse will dash in a straight line, if the cat starts going the long way round the mouse will zag as the tangent point to swim away from switches, but will still go only 1m in the dash while the cat will have to go even further than if it stuck to the optimal strategy.

Edit: I got the distances a bit wrong, it’s more like 1.5pi - .25 m for the cat, and just shy of 1 m for the mouse. But even so this strategy should still work for a SpeedRatio of 4.4 .