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 21, 2008 Report Share Posted December 21, 2008 well... if they run in the same direction, yes. if the mouse is, let's say, blind, he could run into tabby's mouth Quote Link to comment Share on other sites More sharing options...
0 Guest 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? The mouse can't run away in a straight line forever, and as soon as he changes direction, the cat has an opportunity to close the gap. But maybe our doomed rodent will get lucky, and the cat just wants to learn the Lindy Hop. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 This might not still be the complete picture, but it is a try. Well, the mouse will be indefinitely alive if we assume that the cat and the mouse have some distance between them and they are running in the same direction. Let us think of some cases wherein the cat can catch over the mouse. case1: When the cat changes direction, even a little bit, the cat will be able to cross the path of the mouse at one point of time and it can have its dinner. case2: If the cat runs in the path which is parpendicular to the path of the mouse, then the cat has quite a good chance of catching the mouse. case3: We have assumed that both the cat and the mouse are running at top speeds. If the cat changes its speed (a speed which is not equal to the speed of the mouse), then also it has a good chance of catching the mouse. Quote Link to comment Share on other sites More sharing options...
0 itachi-san Posted December 21, 2008 Report Share Posted December 21, 2008 This might not still be the complete picture, but it is a try. Well, the mouse will be indefinitely alive if we assume that the cat and the mouse have some distance between them and they are running in the same direction. Let us think of some cases wherein the cat can catch over the mouse. case1: When the cat changes direction, even a little bit, the cat will be able to cross the path of the mouse at one point of time and it can have its dinner. case2: If the cat runs in the path which is parpendicular to the path of the mouse, then the cat has quite a good chance of catching the mouse. case3: We have assumed that both the cat and the mouse are running at top speeds. If the cat changes its speed (a speed which is not equal to the speed of the mouse), then also it has a good chance of catching the mouse. think of a race track. the cat could take the inside 'lane' Quote Link to comment Share on other sites More sharing options...
0 Shakeepuddn Posted December 21, 2008 Report Share Posted December 21, 2008 (edited) The cat's ability to catch the mouse is a question of agility and size and unknowns. But if the riddle presupposes the two circling a round room along the perimeter or otherwise, the cat will win out in the end as its size allows it to hold an "inside track" to eventually gain on Mickey the Munchable. Edited December 21, 2008 by Shakeepuddn Quote Link to comment Share on other sites More sharing options...
0 Prof. Templeton Posted December 21, 2008 Report Share Posted December 21, 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 cat eventually wins its bigger has slightly smaller inside diam of round room, catches mouseTabby 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 21, 2008 Report Share Posted December 21, 2008 If there are only walls, it doesn't matter how fast or strong the mouse can be, he will eventually become dinner, because he cannot kill the cat (unless mighty mouse). Besides, cat can easily trick mouse into thinking he is tired so mouse slows down, and eat him. The issue for the cat is not allowing his meal to burn many calories. For the cat, it would be best to play dead, eventually the mouse would try to eat him and save him the trouble? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 Maybe the cat can cut across the ballroom to get the mouse. I really don't know. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 If the cat is running behind the mouse it cannot catch... but if the mouse is along the perimeter and cat in radial directions or in straight lines, it can have its dinner. (eg. mouse travels semi-circle perimeter of PI*R, cat travels 2*R) Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 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. You are thinking perhaps of a logarithmic or hyperbolic spiral? If so, I don't think either one saves Mickey. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 If the cat starts from the center of the circle facing the mouse, any move to the right or left by the mouse would be on the path of an outer circle relative to the cat's inner circle. Being perpetually in a smaller circle, the cat could always match the mouse's lateral moves degree for degree and still have a little extra distance to use for closing in on the doomed little mouse. If the mouse runs inward, the cat can just open his mouth, and if the mouse runs away from the center, he'll hit the outer wall eventually. Sorry, mouse. You're cat food. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 Its not like a horse and cart, the cat can close the gap by zig-zagging keeping the mouse to the wall and eventually use either paw no matter which way the mouse runs Either that or the mouse stops for dinner and becomes dinner! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 Actually, I think the cat gets a dinner. Mice are rodents, and need to gnaw constantly to keep their teeth from overgrowing their mouth. Also, since mice have a higher metabolic rate than cats they need to eat more often. What will probably happen in this scenario is the mouse will collapse from hunger and a jaw dragging the floor from not gnawing for an extended time. Dinner is served. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 I think we're all in agreement of the same thing. Tabby gets his dinner. Mouse gets his funeral. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 but can't mice climb walls? Maybe not the walls in question, but somebody has to give ol' Cat-Bait a chance! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 I don't think mice can climb walls... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 I don't think mice can climb walls... The experts disagree. Look here. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 The experts disagree. Look here. what if it's not a house mouse? Quote Link to comment Share on other sites More sharing options...
0 Guest 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? say, since it is a circular room, that both the cat and the mouse were running round by the wall in circles. if the cat was smart, it could stop and turn around so the mouse would run into it. if the mouse was also smart, when the cat stopped it would also stop. whether the mouse gets eaten or not depends on the IQ of the cat and mouse. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 21, 2008 Report Share Posted December 21, 2008 say, since it is a circular room, that both the cat and the mouse were running round by the wall in circles. if the cat was smart, it could stop and turn around so the mouse would run into it. if the mouse was also smart, when the cat stopped it would also stop. whether the mouse gets eaten or not depends on the IQ of the cat and mouse. But, a mouses instincts would still cause it to run, as the cat is behind him. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 22, 2008 Report Share Posted December 22, 2008 If they're free to run radially inside the circle: Start from the worst position for cat. It's at the one wall, mouse is at just opposite side. When the cat moves towards the mouse,(into the circle), whatever mouse behaves, after a few second they will be at the opposite sides of a new imaginary circle thats diameter is smaller than the first circle. Afterwards, cats make a second move, and the circle gets smaller...so on. Mouse has no chance. I see that everybody agree this. But what would you answer thar mouse is fast than cat in a rate of x. What is the minimum value of x for mouse to stay alive? Quote Link to comment Share on other sites More sharing options...
0 Yoruichi-san Posted December 22, 2008 Report Share Posted December 22, 2008 say, since it is a circular room, that both the cat and the mouse were running round by the wall in circles. if the cat was smart, it could stop and turn around so the mouse would run into it. if the mouse was also smart, when the cat stopped it would also stop. whether the mouse gets eaten or not depends on the IQ of the cat and mouse. restriction that the cat and mouse have to run along the wall...if the mouse runs along the arc of the circle, all the cat has to do is run along the chord to that arc and it will catch the mouse. It seems that if the mouse runs along any curved path, the cat wins by running along the straight path. If the mouse runs in straight paths it will eventually reach a wall and have to turn around at an angle, and the triangle inequality says the cat can find a shorter path... Basically, it seems the easiest thing to say is that the cat wins as long as it is always running directly at the mouse...(which is always the shortest distance at the time b/w the cat and the mouse). Worst case...they get in a logarithmic spiral...but it has finite length and at positive speed the cat will eventually catch up to the mouse. -Says Schrodinger's Cat...who was born in the year of the Rat Quote Link to comment Share on other sites More sharing options...
0 Yoruichi-san Posted December 22, 2008 Report Share Posted December 22, 2008 thinking about what Prof. T said...I guess if they start out infinitesimally close...the mouse might make it... Otherwise, it seems as long as the cat is running directly at the mouse, the distance b/w them can't increase...and it will stay the same only if they both run along the same vector...but when the mouse hits a wall, as d3k3 pointed out, the distance closes. Quote Link to comment Share on other sites More sharing options...
0 Prof. Templeton Posted December 22, 2008 Report Share Posted December 22, 2008 This problem goes back to the 1920's (R. Rado's Lion and Man) and the agreed upon answer for 30+ years was that the Cat always got the Mouse if it used it's optimal stategy of staying on the radius between the circles center (O) and the Mouses current position (M). In the 1950's a Russian mathematician (Besicovitch) gave a contradictary answer that proved the Mouse could always get away. I'll try to paraphrase the solution. The mouse follows a polygonal path M0M1,M2...such that MnMn+1 is straight and MnMn+1 is perpendicular to OMn. There's some bits added about the length of MnMn+1 so that the Mouse's path stays within the circle, but I'll forgo that. So for every n the Cat is on the radius OMn the Mouse will not be caught if he stays on the path MnMn+1 which is perpendicular to OMn and the length of the path is infinite so the Mouse stay alive for an infinite amount of time. While I get the gist of the solution, I can't claim to understand it fully how it was written, but I can provide a link to the PDF version of the book it appeared in, in 1953 here. The last problem on page 135. 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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