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 Urns of Infinity Goto page Previous  1, 2, 3, 4, 5  Next
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Rhino
Daedalian Member

 Posted: Fri May 19, 2000 12:42 am    Post subject: 121 Doggonit, extro..., you beat me to the chase. The "empty-bucketers" are asking for the number of a ball that has not been removed. My problem with that question is this: we supposedly have an infinite number of balls but each has a finite number on it. How can you name the last ball that was added? How can you name the last ball that was removed? You can't. There is no last ball for either. Sheesh, I feel like neo..."there is no last ball!" Maybe these demons actually stopped time when they did this trick, and the sorceror was just messing with them so that he would never have to sign their time card. Just a thought [This message has been edited by Rhino (edited 05-18-2000).]
ZenBeam
Daedalian Member

Posted: Fri May 19, 2000 1:07 am    Post subject: 122

Wonko:
 Quote: The way you just described it (add nine balls and don't take any out) does not describe the puzzle.

In my post at 2:43pm, I said "At each step, where a step is putting in ten balls, taking out one, the number of balls increases."

Extro:
 Quote: Before midnight, at the n-th step, 10*n balls have been placed in, and n balls have been removed. 10*n - n is 9*n. But that only works before midnight, when n is some finite number. It fails at midnight.

It becomes infinite at midnight. I don't see any justification for saying the number of balls gets smaller. The only steps the demon ever takes results in the number of balls increasing by nine. Never decreasing.

Dave1000:
 Quote: "I have an object in this bag. It is completely blue (and no other color). It is round. It is completely red (and no other color). What color is it?"

Purple

Extro:
 Quote: Was the demon ordered to instantly put an infinite number in at midnight?

Following his orders, in any finite length of time beginning before midnight and ending at midnight, no matter how short, the demon must add an infinite number of balls. You wouldn't argue that if the demon only added the ten balls per schedule, but never removed any, that the number of balls would be finite at midnight.

------------------
It is so clear, and so it is hard to see.
dave10000
Tinhorn

Ghost Post
Icarian Member

Posted: Fri May 19, 2000 2:54 am    Post subject: 124

ZenBeam:
 Quote: It becomes infinite at midnight. I don't see any justification for saying the number of balls gets smaller. The only steps the demon ever takes results in the number of balls increasing by nine. Never decreasing.

But that line of reasoning doesn't get you anywhere when there are an infinite number of steps. Suppose there were demon A and demon B, working side by side. Demon A puts in 1-10, then 11-20, then 21-30, etc... Demon B puts in 1, then 2, then 3. At midnight, whose urn has more balls? Demon A put in 10 times as many as Demon B did at each step, but they end up with the exact same set of numbers on the balls in their urns (i.e. the set of all positive integers).

We are dealing with infinite sets here. If you start with an empty set, add 10 items an infinite number of times, and remove 1 item an infinite number of times, it's not necessarily the same as adding 9 items an infinite number of times.

Here's another variant: Demon #2 adds 1-10, removes 1, adds 11-20, removes 2, etc... Demon #3 does something similar, except he starts with his urn filled with an infinite number of balls, labelled with all the positive integers (1,2,3,...), and only adds a ball with a number if it is not already present. He removes ball n at step n, just like demon #2.

Now look: At each step, demon #3 must have all the same numbers that demon #2 has (plus many more). Whenever demon #2 adds a ball, either demon #3 already has it, or he adds it. Now, you would agree that demon #3, at midnight, has an empty urn. But every time demon #2 added a ball, demon #3 already had it. How does demon #3 end up with nothing, but demon #2 end up with something?

Also, you are making my point regarding my question "Was the demon ordered to instantly put an infinite number in at midnight?" It relates to Green Dragons statement "No one of us is denying the power of the demon, saying that it cannot empty the urn instantly, exactly at midnight. But that's just not a step that the wizard has ordered him to take."

Obviously, he adds an infinite number of balls, and removes an infinite number. The two numbers are equal (aleph-null).
Rhino
Daedalian Member

 Posted: Fri May 19, 2000 3:00 am    Post subject: 125 Yup, I just talked to the sorceror and we shared a good laugh. Those demons never finished...they're still playing with the balls and it's been 11:59:59.99999...9 in their little room for quite some time now. They'll never get to 11:59:59.999... er 12:00:00, though. (just being silly here; please don't abuse me for it )
Ghost Post
Icarian Member

Posted: Fri May 19, 2000 3:13 am    Post subject: 126

 Quote: Since he (hypothetically) finished his task, he must have done one of these things last.

No. The steps he performs can be numbered.
2 - write 10
3 - Remove ball 1
4 - write 9
6 - write 19
7 - Remove ball 2
8 - write 18
etc...

For ANY step, there is a next step. Whatever step you want to call his last step, it was followed by an infinite number of other steps.

One last variant: The demon has two urns. There are an infinite number of balls, labeled 1,2,3,..., with no duplicates.

Step 1: Add balls 1-10 to urn A, then move ball 1 from urn A to urn B.
Step 2: Add balls 11-20 to urn A, then move ball 2 from urn A to urn B.
Step 3: Add balls 21-30 to urn A, then move ball 3 from urn A to urn B.

Same as before, except every ball removed is placed in a second urn, urn B.

Now, at midnight, what is in urn B? 1,2,3,...

Since the balls are labelled with finite integers, and urn B contains all of them, and there are no duplicates, what can be in urn A?

 Quote: THE ONLY WAY for the bucket to become empty is for the Demon to remove more than one ball without adding more. (If you disagree, would you care to explain how?)

That's what happens when you somehow finish performing an infinite number of steps. He does add more when he removes any ball, but he later removes those also.
dave10000
Tinhorn

Aarondalf
the original GL stud

 Posted: Fri May 19, 2000 9:40 am    Post subject: 128 You guys have been arguing for ages about pretty much nothing, as you actually CANT HAVE AN ANSWER, so just give up. For Dave1000, you say that the people who think there is an empty bucket are wrong because the demon couldnt put down his last step, and has to be finished. Yes??? But did you ever think about your own solution of infinity balls??? You want there to be a finished demon and a last step, NEVER GOING TO HAPPEN. Listen up, it is a question involving infinity so he can not ever write down his last step so please dont keep asking people to answer a question like that. I also think there will be 0 balls left and the way he takes out his balls is completely different than Demon one. First Demon one: Suppose we split his task among two other demons, one demon put in balls numbered 1-OO Then another Demon takes out all balls divisible by 10. We have 9x-x as x--->OO = OO There is a major difference between this and what Demon 2 does, and again we will split the tasks. One Demon puts in balls 1-OO Another takes out balls 1-OO which is x-x=0 The fact that there is no last step is obvious, but how in hell does this prove that there are infinite balls in the Urn. ------------------ You can't chainsaw a duck.
Tom
Daedalian Member

 Posted: Fri May 19, 2000 11:51 am    Post subject: 129 I've written far too much already, but seeing as there is something of a spirit of "us versus them" theme here, I thought I'd just say I agree with everything extro... has written so far .. I'm on his side. I think the "Was the demon ordered to instantly put an infinite number in at midnight?" is an absolutely perfect point, by the way. A similar argument to the "9 more balls each step argument" might go .. At each step, the number of balls in the urn is finite. How can the number of balls suddenly become infite at 12:00. But I'm just repeating extro..., really. dave10000 - this is going back a bit, but I don't think the premises are inconsistent. I see what you're saying, but I think my line of argument is right, and the other line is wrong. I do think I've spent as much time trying to show what I think is wrong with other peoples reasoning, as supporting my own. The "last number written down" problem is the same as your demon who stops when there is only one ball left, I think .. there is no last step. The balls really do all disappear at once at 12:00. Wonko - I agree with most of what you're saying, too .. I had y ending up with any number of balls, because I didn't want to assume he took one out at random .. for all we know, he might take out the 1st, 2nd, etc he puts in. We don't know. I'm also inclined to side with extro... that the urn will not be empty, but that's not really the issue. Just out of interest, is anyone actually going to believe Minotaur if he disagrees with what they already think anyway?
Impartial Observer
Guest

 Posted: Fri May 19, 2000 12:13 pm    Post subject: 130 Yes, I agree Tom, that makes a great deal of sense. Those who disagree with you are obviously fools, aren't they?
Tom
Daedalian Member

 Posted: Fri May 19, 2000 12:33 pm    Post subject: 131 I didn't write that! I did write this, though.
Ghost Post
Icarian Member

Posted: Fri May 19, 2000 12:42 pm    Post subject: 132

dave10000:
 Quote: If there is no last step, then the Demon has NOT finished his task.

OK, forget the demons. Let's come up with a numbering for the decreasing length time intervals before midnight.
Interval 1 = 11:59 to 11:59.5
Interval 2 = 11:59.5 to 11:59.75
Interval 3 = 11:59.75 to 11:59.875
etc...

By your argument, since there is no last interval before midnight, midnight never arrives.

But it does.

By your argument, you can forget about the demons, infinite supply of balls, etc..., and by just considering a perfectly well defined way of labelling time intervals between 11:59 and 12:00, your conclusion must be that it will never be midnight.

My argument is that each of those time intervals has a finite non-zero length, during which the demon can perform the required step. During interval X, he places balls 10X-9 through 10X in the urn, and removes ball X. He begins step X after interval X has started, and finishes before interval X ends. It's a fixed amount of work, and the interval get's smaller, but never reaches 0, so he's never working infinitely fast (though he approaches that). IF midnight arrives, the urn will be empty.

But the demons work does not stop midnight from arriving.

[This message has been edited by extro... (edited 05-19-2000).]
Tom
Daedalian Member

 Posted: Fri May 19, 2000 12:55 pm    Post subject: 133 dave10000, in response to your demon #5. Suppose you have a demon (#6) that just adds a (numbered, in order) ball at each step, and ends up with an urn with an infinite number of balls in it. Do you allow this? I guess really, if you don't, then we have nothing to argue about .. you are just saying "infinities make no sense", we are saying they do, and that's it. We can't really argue against each other. But, presuming demon #6 can do his task. Have demon #7 (with 1 ball and 2 buckets, EVEN and ODD) put the ball in EVEN when #6 puts in an even ball, ODD on an odd ball. Where is the ball at 12:00? Does this prevent #6 being able to complete his task? #2's task is more complicated than #5's .. but it's convergent, and #5's isn't. [This message has been edited by Tom (edited 05-19-2000).]
dave10000
Tinhorn

Ghost Post
Icarian Member

 Posted: Fri May 19, 2000 2:41 pm    Post subject: 135 dave10000: But you argued that the demon never finishes the task if, at midnight, there was no last step that he performed. That was the point I was addressing with the time intervals. If midnight arrives, he has finished, and there was no last time interval, so no last step.
Tom
Daedalian Member

dave10000
Tinhorn

Kevin
Icarian Member

 Posted: Fri May 19, 2000 5:14 pm    Post subject: 138 This discussion is getting way to deep. My mind is spinning in circles. But I need one of you smart guys to answer a question. Earlier I stated that there are an infinit amount of numbers between 11:59 and 12:00. But, no one of you will dispute that 12:00 comes and goes each day. Well given that there are an infinite amount of numbers to keep us from reaching 12:00, How in the world does 12:00 come? Or even better, HOW IN THE WORLD DO THE DEMONS FINISH THEIR JOB WITH AN IOTASECOND TO SPARE? PLEASE HELP!
ZenBeam
Daedalian Member

Posted: Fri May 19, 2000 5:48 pm    Post subject: 139

 Quote: You guys have been arguing for ages about pretty much nothing, as you actually CANT HAVE AN ANSWER, so just give up.

Aarondalf, I and at least a couple others arguing here are in the "undefined" camp, not the "infinity" camp. I'm not sure who is in the infinity camp any more.

Extro:
 Quote: But that line of reasoning doesn't get you anywhere when there are an infinite number of steps.

Even with an infinite number of steps, if a function has a value at some time T0, and is always increasing with increasing T, it can't have a smaller value at some time T1 > T0. There is no mechanism for a decrease specified in the problem statement.

 Quote: Also, you are making my point regarding my question "Was the demon ordered to instantly put an infinite number in at midnight?"

I'm not certain what point you are trying to make here. When you integrate 1/X from 1 to zero, you get infinity, even though 1/X is always finite over the open interval (1,0). just like here, if you break it up into sub-intervals (1,1/2), [1/2,1/4), [1/4,1/8), ... each subinterval integrates the same value.

In order for the urn to be empty, an infinite number of balls would have to be removed precisely at midnight. This is a larger discontinuity than adding a finite number of balls every 60 / 2^N seconds. It is also a discontinuity which is not specified in the problem.

 Quote: Obviously, he adds an infinite number of balls, and removes an infinite number. The two numbers are equal (aleph-null).

Infinity - infinity is undefined.

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It is so clear, and so it is hard to see.
Green Dragon
Daedalian Member

 Posted: Fri May 19, 2000 6:03 pm    Post subject: 140 I think that I'll just stop arguing. None of us (on either side) are conceding our points, or being convinced by the others. We are just arguing the same points, over and over, and it's usless. ZenBeam is right, that the probles in in some ways unsovleable. I believe, now, that this was a badly worded puzzle, and it is really quite usless to keep disscussing it. I am waiting (as we all must be) for the minatour's solutionn to be posted, but I will not continue disscussing it. There are diffrent ways to look at this puzzle, and until these are clarified, bye to all.
Ghost Post
Icarian Member

Posted: Fri May 19, 2000 6:19 pm    Post subject: 141

ZenBeam:
 Quote: Infinity - infinity is undefined

But set subtraction, for infinite sets, is very well defined. That's what we're dealing with here. You keep ignoring a method of analyzing this problem that works, and pointing at one that doesn't.

dave10000: You said the demon never finishes his task.
 Quote: If there is no last step, then the Demon has NOT finished his task.
He does. Now you say it doesn't converge.
 Quote: By the "fundamental theorem of limits" (in pre-calculus), the content of the Demon's bucket is undefined at midnight, whereas the time-series approaching midnight IS defined at midnight (because it is convergent). If you want to use a special analysis that differs from the fundamental theorem of limits to support your position, then I can't argue against it, but that position would be strictly NON-STANDARD.

What is non-standard is attempting to use calculus to solve a simple problem dealing with discrete sets of integers. It doesn't work, as you've pointed out, which is why I'm using simple logic, which does work. That is: Each ball is labelled with a finite integer. Each ball is placed in the urn before midnight, and is later, but still before midnight, removed, never to be placed back in. That's inescapable. EVERY ball is placed in, then later removed, before midnight. The urn is left empty, as it was to begin with.

Borodog
Daedalian Member

 Posted: Fri May 19, 2000 6:54 pm    Post subject: 142 Somebody was getting at this before; I'm not sure who. Both of the following positions are defensible. You really can't argue with the logic of either one: 1) Every ball that gets put in gets taken out later. The urn ends up empty. 2) Every time you take out a ball, you put nine more in. The urn ends up infinitely full. The problem is that these two propositions, are mutually exclusive, even though both follow the wording of the puzzle. The puzzle really and truly is NOT WELL FORMED. It has no "solution." You might as well say something like, "There's a ball in a bucket. Take the ball out, but when you do, leave it in." Now, is the ball in the bucket or not? Neither, because the problem has ceased to have any *meaning*. Before I was firmly an "empty-bucketer" (good phrase, whoever that was). But now I'm with Veridisaurus Rex. There REALLY IS NO SOLUTION, because there REALLY ISN'T A WELL FORMED PUZZLE. ------------------ Insert humorous sig here. [This message has been edited by Borodog (edited 05-19-2000).] [This message has been edited by Borodog (edited 05-19-2000).] [This message has been edited by Borodog (edited 05-19-2000).]
Wonko the Sane
Daedalian Member

 Posted: Fri May 19, 2000 7:19 pm    Post subject: 143 I haven't had time to read through most of the argument, but I'd like to respond to extro. You're still misinterpreting what I said. The probabilities don't add up. They change. After 3 iterations (well, 2.5, because the demon hasn't removed a ball yet), there will be 3n + 1 or 28 balls in the urn. EACH ONE has a 1/28 chance of being removed. After step 4 there are 4n+1 or 37 balls in the urn. EACH ONE has a 1/37 chance of being removed. By addding balls, the demon is changing the probabilities of EVERY ball being removed. If you take 1/37 and add it together 37 times you get 1, right? That's what I was trying to say.
dave10000
Tinhorn

Ghost Post
Icarian Member

Posted: Fri May 19, 2000 7:56 pm    Post subject: 145

Wonko:
 Quote: If you take 1/37 and add it together 37 times you get 1, right?

Yes, but what does that mean? It means that if there are 37 balls and you remove 1 at random, there is a probability of 1 that it was one of the 37.

Let's say the balls are numbered. Balls 1-10 are added, and a ball is removed at random. Balls 11-20 are added, and a ball is removed at random. This continues forever. Balls 1 through 10 each have less than a .314 chance of EVER being removed. For other balls, it is less. Multiply all these (an infinite number of values less than .314), and you get 0, the probability that all balls get removed.
ZenBeam
Daedalian Member

Posted: Fri May 19, 2000 8:37 pm    Post subject: 146

 Quote: The puzzle really and truly is NOT WELL FORMED. It has no "solution."

Now is a good time to point out, for anyone who may have lost track, that the puzzle does NOT ask "Is the urn empty, or full, or what?" Rather, it asks if the sorceror can "tell that the Demon had cheated?" A question which, I believe, is easily answered "yes" without deciding what is in the urns at the end.

Of course, I find this "digression" much more interesting than the original puzzle.

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It is so clear, and so it is hard to see.
Ghost Post
Icarian Member

Posted: Fri May 19, 2000 8:45 pm    Post subject: 147

dave10000: OK, regarding the claim you made (again) that the demon does not finish his task(and ignoring the issue of convergence for the moment): You agree that midnight arrives. The demon performs each step during a specific (finite, non-zero length) time interval between 11:59 and 12:00. Every step performed by the demon is performed during a time interval that ends before 12:00. So every step is finished before 12:00. If each and every step is finished before 12:00, then at 12:00, all steps have been finished. That, again, is indisputable.
 Quote: The response "He does [finish]" is hardly an argument. I guess my only response to that is "Does not!"

I think it's quite obvious that "He does finish" is not the argument, but the conclusion to the above argument, which I spelled out before, right after you first claimed he does not finish. You've said he does not finish, without argument, or explaining how that could possibly be, given the obvious (that each and every step is completed before 12:00).

If you want to claim that standard limits can be used here, try being formal about it, and you'll see that it fails. We are dealing with SETS of integers, and infinite sets at that.

You pointed out (not directly, but by your examples above) that:
(10-1)+(10-1)+(10-1)+(10-1)+ ...
is undefined. It can be taken as:
(10+10+10+...)-(1+1+1+...)
which, if it can be taken as anything, would be infinity minus infinity (which is undefined).

That's not quite correct actually. 1+1+1+... really does not have a sum. There is no limit. You might consider it to be infinite, but infinity is not a number.

But {1}U{2}U{3}U{4}U... (U being set union) does have a well defined value. It is the set of all positive integers. So you can do things with sets that you can't do with integers (like have some that are finite, and some that are infinite and still well defined, and subtract an infinite one from an infinite one, and get either a finite or infinite one, depending on which infinite sets you started with). They didn't teach you that in pre-calc?

The SET
((... ((({1..10}-{1})U{11..20})-{2}) ...))
is equal to
({1..10}U{11..20}U...) - ({1}U{2}U...)
which is equal to
{1,2,3,...} - {1,2,3,...}
which is the empty set.

It's not my "own personal logic".

Your whole argument is: You can't determine the answer using limits as taught in pre-calc, so the answer is undefined. Try standard set theory and simple logic (which is a branch of formal mathematics). It doesn't just "seem logical". It is.

("It is" is not my argument, but my conclusion. Some details of the argument may have been left as an exercise for the student.)

Ghost Post
Icarian Member

 Posted: Fri May 19, 2000 9:24 pm    Post subject: 148 Another PROOF: There are two possibilities at midnight: 1) The urn is empty, or 2) The urn contains at least one ball. It must be either 1) or 2), and not both. I'll assume 2), derive a contradiction, therby proving 1). If the urn contains at least one ball, let X be the lowest number labelling a ball in the urn (every ball is labelled with a positive integer, and every non-empty set of positive integers has a smallest one). The ball labelled X was removed from the urn at 1/(2^(X-1)) minutes to midnight.
dave10000
Tinhorn

 Posted: Fri May 19, 2000 9:45 pm    Post subject: 149 extro: You continue to attribute positions to me that I did not take, to deny math the way I learned it, at least, and to use math in ways that I learned, at least, were incorrect. I see no reason to continue to debate under such circumstances, and I choose not to. Life is too short. If you (or others) wish to consider this a victory for the "empty-bucketers," so be it. The half of this bulletin board that does in fact dispute what you proclaim to be "indisputable" (which shows that we are arguing from different premises), will draw their own conclusions. ------------------ Eternity is a long time; especially the end. -- Niels Bohr
Wonko the Sane
Daedalian Member

 Posted: Fri May 19, 2000 11:50 pm    Post subject: 150 Extro, I see what you're saying, but I don't agree. But seing as neither of us are going to agree and it's not particularly important to the puzzle anyway, it's a moot point and I'm just going to drop it.
Ghost Post
Icarian Member

 Posted: Sat May 20, 2000 12:14 am    Post subject: 151 I don't mean to be argumentative, but I can't see where I attributed something to you that you did not say. You said "If there is no last step, then the Demon has NOT finished his task." I addressed this, showing that he DOES finish his task, although there is no last step, just as midnight arrives, although there is no last interval before midnight. You say I deny math the way you learned it, and I suspect that's also not the case. All you have said is that using limits, you can't conclude what's left in the urn. I agree. Using limits, you can't conclude what's left, just as you can't use limits to evaluate the infinite sum 1+1+1+.... I'm not using limits. The fact that use of limits does not provide an answer does not mean there is no correct and well defined answer. You say I use math in ways that you learned were incorrect, but you haven't said where. You haven't ever objected to my arguments, except to imply that because mine arrive at a definite conclusion, whereas yours do not, mine must be wrong. I doubt you were ever taught that anything I've used in my arguments is incorrect. It's just that you are using limits, which are undefined in this case, whereas I am using well defined finite and infinite sets, and operations on them that are equally well defined, and yeild well defined results. In the proof I give above, it should be easy to show my error if there is one. A) I assume a non-empty urn, and show a contradiction, so conclude an empty urn. You might dispute that that's a valid principal (proof by contradiction), but do you? B) So, did I really show a contradiction? Well, I assume the labels on the balls in the non-empty urn are positive integers (you might dispute that), and that any set of positive integers has a least element (you might dispute that). So I take from the urn the ball with the least label, X, and tell you it was removed from the urn 1/(2^(X-1)) minutes before midnight (you might dispute that that's correct). And 1/(2^(X-1)) is greater than 0, so it was removed before midnight (you might dispute that). So there's a contradiction - the ball was still in the urn after it was removed and never replaced (you might dispute that that's a contradiction). I've think I've pointed out every point that anyone might dispute (you might dispute that), just to make it easy. Perhaps I'm dense, but I can't see how any of those are disputable.
stoatboy
Guest

Ghost Post
Icarian Member

 Posted: Sat May 20, 2000 12:01 pm    Post subject: 153 Just for the record: Demon #1 is left with an infinite number of balls (labelled with all the integers except those divisible by 10). Demon #2, without cheating, is left with 0 balls. Demon #2, with cheating, is left with an infinite number of balls (labelled with infinitely long strings of digits, which don't represent integers since they are not finite). Now, demon #1 and demon #2 (not cheating) do, at each step, add 10, then remove 1. And 10 minus 1 is always 9. And at any moment before midnight, they have the same finite number of balls in their urns (some multiple of 9). The "trick" is that demon #1 and #2 each remove an infinite subset of the infinite set of balls, but demon #1 is removing a proper subset, whereas demon #2's subset is equal to the original infinite set. Regarding the rope, one guy tosses all the rope (in 1 foot sections) into the black whole - an infinite amount. The other tosses 1/9th (or 1/10th) of his infinite rope into the black whole - also an infinite amount. But he is left with an infinite amount. And you can't map an infinite amount onto nothing. The point being, you can start with an infinite amount, and remove an infinite amount, and, depending on how you did the removing (and not how much you removed - it's always equally infinite), you can end up with nothing, a finite amount, or an infinite amount.
Wonko the Sane
Daedalian Member

 Posted: Sat May 20, 2000 1:39 pm    Post subject: 154 Thank you extro, that was very well put. I've been looking for a mathematically good way to put it (instead of the conceptual way of saying that it isn't the amount, but the order that matters). What all of us have to realize is that we're dealing with two infinites here. Most people have a hard enough time dealing with one of them, but trying to jam two of them together is even harder. We've got a combination of Zeno's Paradox and a relatively simple logic problem. Let's reduce the number of balls and restate the problem. Two demon are each given 500 numbered balls. Demon 1 is told to add 10 balls to his urn and remove the 10th one until he runs out of balls to add AND runs out of balls to remove. Demon 2 is told to add 10 balls to his urn and remove whatever ball in the urn has the lowest number until he runs out of balls to add and or runs out of balls to remove. Now, what will their urns look like in the end? Demon #2's urn will be empty and he'll have 500 balls on the floor, demon #1's urn will have 450 balls in it and he'll have 50 of them outside the urn on the floor. What about the cheater? The cheater will add 9 balls, hide the 10th, and paint a 0 on the first ball. He'll end up with 450 balls in the urn as well. However, he wont have any balls on the floor, the 50 that should have been there will be under the rug. He'll also have different balls in his urn than demon #1 did before. Extend this to infinite and you can get a better look at the situation. Demon 1 will have and infinite number of balls in the urn and an infinite number of balls on the floor Demon 2's urn will be empty and there will be an infinite number of balls on the floor The cheater's urn will have an infinite number of balls in it, but there will be none on the floor because those will be jammed under the rug. At least it seems that way to me.
Ghost Post
Icarian Member

 Posted: Sat May 20, 2000 5:10 pm    Post subject: 155 I think it's about time Minotaur posted the REAL solution so that we can finally end these (essentially) pointless arguments, which don't seem to be convincing anybody of anything they don't already believe. And maybe a solution to Monkey #3 would be nice too? btw... Sorry, I didn't mean to offend back on page 2!
Tom
Daedalian Member

 Posted: Mon May 22, 2000 9:29 am    Post subject: 156 I just wrote a big post, and just deleted it. It was more argumentative I'm right/you're wrong rubbish. It seems this all got a bit heated, hopefully we can all stay friends . By the way dave10000, the reason you got so much argument back is, I believe, because you write nice, clear posts that are good to argue against (as did a lot of other people, but dave10000 started to sound a bit annoyed). I have a lot of respect for your intellect and ideas; you argue well, it's just that I happen to disagree with you here. Well, I would if extro.. didn't always disagree with you first. I'm guessing there's a few maths degrees flying around here, and none of us are fools, obviously. There doesn't seem to be any point arguing any more (but not because the question badly defined .. that is what we're arguing about!) as no one is going to change anyones minds .. including Minotaur when he puts up his answer, I suspect. Could anyone get in touch with a respested mathematician about this? That said, they all got Monty Hall wrong, so who can we trust?
Tom
Daedalian Member

 Posted: Mon May 22, 2000 9:32 am    Post subject: 157 Oh, and Mathieu, I think it was me who got stroppy on page 2 .. I was in the middle of arguing and took it personally. Don't worry about it.
ZenBeam
Daedalian Member

Posted: Mon May 22, 2000 12:35 pm    Post subject: 158

I shouldn't post this, because I, too, am getting tired of this. Nevertheless...

quote:
There are two possibilities at midnight:
1) The urn is empty, or
2) The urn contains at least one ball.
It must be either 1) or 2), and not both.

There is a third possibility, that the number of balls is undefined. Thus, a contradiction in 2) doesn't prove 1).

 Quote: No one is going to change anyones minds .. including Minotaur when he puts up his answer, I suspect.

Almost certainly true.

------------------
It is so clear, and so it is hard to see.
Gandalf the White
Property of Luna

 Posted: Mon May 22, 2000 1:27 pm    Post subject: 159 Mathmematicians got Monty hall wrong? Who? I'd never seen or heard of "Let's make a deal", but when someone explained the problem to me for the first time, thankfully explaining it right, I understood that I should be switching straight away and I'm terrible at mathematical reasoning. ------------------ How many Grey Labryntians does it take to change a lightbulb? Precisely 0.99999...
Tom
Daedalian Member

 Posted: Mon May 22, 2000 2:42 pm    Post subject: 160 I founf it, finally. Here you are. Read the "'So Easy' to Blunder" section.
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