bonanova Posted December 21, 2008 Report Share Posted December 21, 2008 Tabby and his intended dinner, the gray mouse, are loose in a circular ballroom. There are no exits, chairs, stairs, curtains, mouse holes ... just walls. They have equal top speeds and equal endurance. So the mouse should be able to stay alive indefinitely, right? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 22, 2008 Report Share Posted December 22, 2008 So here is my guess So I think it doesn't matter. If they both have equal speed and endurance, and they are in a room with no exits, they both run in circles till they are both exhausted and die of thirst or starvation. Or the mouse wins.... I hate cats Quote Link to comment Share on other sites More sharing options...
0 unreality Posted December 22, 2008 Report Share Posted December 22, 2008 Ysan: a logarithmic spiral has infinite length... so if the cat & mouse are "points", the mouse survives, otherwise it dies. Assuming it's possible for the mouse to force the situation into a logarithmic spiral... I'm not sure on that part... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 22, 2008 Report Share Posted December 22, 2008 Ysan: a logarithmic spiral has infinite length... so if the cat & mouse are "points", the mouse survives, otherwise it dies. Assuming it's possible for the mouse to force the situation into a logarithmic spiral... I'm not sure on that part... The length along the curve from some finite distance r to the origin is finite. Not so for a hyperbolic spiral though... Quote Link to comment Share on other sites More sharing options...
0 unreality Posted December 22, 2008 Report Share Posted December 22, 2008 Oh yeah - duh. I'm mistaking "rotations" with "length". It's like 1/2 + 1/4 + 1/8 + ... = 1, a finite distance, but infinite fractions. Nvm, forget I ever said that hehe Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 23, 2008 Report Share Posted December 23, 2008 The mouse cannot live on indefinitely. If the mouse runs around the circumference of the circle the entire time the cat would only need to pause briefly and then cut across the circle. Any distance across the circle is shorter than traveling it's exterior. Chances are the mouse will be darting around and change its direction frequently. Because they have the same speed, same endurance and there is no escape outside the circle for the mouse he will not be able to create any distance between him and the jaws of hunger. Given these factors the cat will continually close off the distance between him and his prey until there is none. There are other factors to consider though: Physical conditions such as hunger, muscular pain and/or failure, age, etc. Environmental conditions such as lack of water or food and the size of the room also come into play. These conditions would insert a fair amount of variables into the equation where many posibilities could occur. Ignoring these factors and considering the highly unlikely scenerio of them both travelling around the outer most limit of the circular room constantly, without changing direction, they would continue until they die making the posibility of the mouse living on indefinitely still not possible. So, I guess I could have answered this in a much simpler fashion. Nothing in our physical materiality goes on indefinitely so the answer would be no. P.S. I have 4 cats and if this scenario where real, in say a 30' DIA. empty ballroom with no escape..that mouse wouldn't have a prayer. I'd give him 30 minutes tops. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 23, 2008 Report Share Posted December 23, 2008 animals might run in circles but not perfect ones i'm sure the cat would cut across the floor, however this mouse in particular could be a super secret but awsome ninja and could use its awesome ninja skills to take the cat out, but I am thinking either way he ends up dead if there are no windows, doors, or holes or any means of escape as he eventually would starve to death himself. Quote Link to comment Share on other sites More sharing options...
0 Jiminy Cricket Posted December 23, 2008 Report Share Posted December 23, 2008 I say the Cat gets the Mouse. As long as the Cat does not trace the Mouse's path it will eventually intersect. I'll leave the math to more capable minds... Tabby and his intended dinner, the gray mouse, are loose in a circular ballroom. There are no exits, chairs, stairs, curtains, mouse holes ... just walls. They have equal top speeds and equal endurance. So the mouse should be able to stay alive indefinitely, right? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 23, 2008 Report Share Posted December 23, 2008 The OP said the room in question is a "ballroom". This would imply reasonable acoustics, I would think. Common acoustic building materials are generally within the capability of a mouse to climb and/or gnaw through easily. The real question is, can the rodent gnaw through, or far enough in to be safe, before stamina runs out? The cat will be patiently waiting for him if he is to fall! So, BN, what acousticly reasonable material is this ballroom made of? Remember that your common house mouse can gnaw through aluminium and such! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 23, 2008 Report Share Posted December 23, 2008 unless we have a very fat, inagile cat with a large turning circle and a mouse that was the complete opposite then I agree with everyone else... the mouse cannot pull away from the cat and every time the mouse turns round the cat can just cut the corner and shorten the distance. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 24, 2008 Report Share Posted December 24, 2008 on the definitions of the cat and mouse. If real life than yes (think race track as Itachi said), If reduced to points than no, There exists a path that the mouse can take so that he cat only gets arbitrarily close, but never catches the mouse. I agree with his point on the dependency on the definition. Also, if the cat continually follows the mouses course doggedly, then it will never catch the mouse. However, based on the current given info., you would have to make alot of assumptions, and you should never assume with math/logic. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 24, 2008 Author Report Share Posted December 24, 2008 Guidance for discussion. Tabby and the mouse are geometrical points. Can the mouse keep the distance between the points strictly positive? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 24, 2008 Report Share Posted December 24, 2008 This situation is very hypothetical with multiple variables. that if there are only walls and no exits, then both the mouse and cat are doomed to die of starvation. In mathematics there are arguments for both sides. If the mouse does not follow a circular path the cat will overtake him by intersection or taking the "inside" as said before. However if we apply the limiting theory behind Zeno's paradox, if the mouse is granted a specific head start the cat will never be able to reach his desired dinner for he will continually get closer but never reach the "limit" being the mouse. Personally this situation reminds me of a dream in which Taby is actually a man who yearns for the grey mouse which symbolizes an unattainable goal which he pursues forever (signified by the room with no exits.) Aren't we all fated to desire what we can't have? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 24, 2008 Report Share Posted December 24, 2008 If the mouse always runs on perimeter, the cat will have to reach the perimeter eventually. Assume that cat is at a point of x, inside the circle and very close to perimeter. Mouse is at perimeter, at point y, very close to x. Mouse runs to another point of z on the perimeter. At the same time cat runs to same z point. The distance beween y and z ( an arc) is smaller than the distance xz, if xy distance is enough small. Thus, when cat reaches z point (perimeter), mouse is a little front of it. Now, if the cat chases mouse on the perimeter, it will never catch, because their speeds are same. If cat wants to leave the perimeter, all this case will repeat, and mouse will stay alive. I didn't work on the limit value of xy distance/radius, but if my opinion is true, it may be formulated? Actually, this is a geometrical result, practically mouse has no chance. Quote Link to comment Share on other sites More sharing options...
0 Prof. Templeton Posted December 24, 2008 Report Share Posted December 24, 2008 Guidance for discussion. Tabby and the mouse are geometrical points. Can the mouse keep the distance between the points strictly positive? The mouse can stay alive indefinitely. See my previous post which has a link to the solution. (5.8MB PDF file, last problem on page 135 of the book). Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 24, 2008 Report Share Posted December 24, 2008 well if they're running at the same speed in the same direction and have the same durability, then eventualy they will tire out at the same time. But the mouse will recover quicker than the cat because its faster heart rate. the mouse will stay alive for some time. Eventually the cat (being the smarter) will change speed or direction and the mouse will die. Quote Link to comment Share on other sites More sharing options...
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bonanova
Tabby and his intended dinner, the gray mouse, are loose in a circular ballroom.
There are no exits, chairs, stairs, curtains, mouse holes ... just walls.
They have equal top speeds and equal endurance.
So the mouse should be able to stay alive indefinitely, right?
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