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Marbles Bonanza (Posted on 2003-09-08) Difficulty: 4 of 5
You have an empty container, and an infinite number of marbles, each numbered with an integer from 1 to infinity.

At the start of the minute, you put marbles 1 - 10 into the container, then remove one of the marbles and throw it away. You do this again after 30 seconds, then again in 15 seconds, and again in 7.5 seconds. You continuosly repeat this process, each time after half as long an interval as the time before, until the minute is over.

Since this means that you repeated the process an infinite number of times, you have "processed" all your marbles.

How many marbles are in the container at the end of the minute if for every repetition (numbered N)

A. You remove the marble numbered (10 * N)

B. You remove the marble numbered (N)

See The Solution Submitted by levik    
Rating: 3.6154 (13 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
circles | Comment 45 of 83 |
To SK and BW;
I have not attempted to find the hole in your reasoning or math because, I don't have the background. Why is it so hard to accept that I've never taken anything on set theory? My math classes were sooo long ago that I really don't recall a lot of the intricacies required to have a debate of a definitional nature. Does that mean that that other methods I conceive are necessarily faulted - no. Because I can't personally find the flaw in your math does that make it correct - no.

I really believe that the discrepancy lies in the question itself. You simply can't label marbles from 1 to infinity. Further, having stopped processing your marbles, there must have been a last iteration (Both of these important points are being ignored by your math. This is the very part of the problem I'm referring to as the suspension of disbelief (its in the earlier posts - look for yourself). Is this a hard concept to grasp - definitely. But until you can show how my reworking of the problems mechanics is flawed, then clearly there is an error. Might it be in my version - sure. But you've made no attempts to ever dispute the reasoning Iíve come up with (the sum of an infinite number of nines cannot equal zero), so further defense of your methods are pointless. I've made no alterations to any meanings nor have I altered the process involved Ė indeed this seems to me to mean that I am working on a problem that has an identical solution. I have not violated or misused the term infinity in my method. Each discreet step produces a easily quantified increase in the number of marbles in the jar. An infinte sum of discreet values is no contradiction. Sometimes it may even end up as a finite number, but certainly not in the case where (both) no successive member of the set is valued less than its predecessor and any term is positive.

I understand your reasoning. I see the math. I don't dispute that I can't number a marble that is still in the jar. It is a struggle to grasp how that results in a situation where there are any marbles left there at all, but lo and behold, there must be, or else at some point we've begun taking more marbles away than we have addded, which isn't stated as a process and therefore violates the situation.

You have considered that infinity is a number that can be worked with. Not so. Infinity is not a number, it is an idea. There are plenty of numbers larger than infinity. Its just that they are also called infinity. And unlike in the case of finite number where x=x, 4=4 and pi=pi, infinity does not (necessarily) equal infinity. There are an infinite numbers in fact, both greater than and less than infinity, and wouldn't you know it - there all called infinity. So the last marble taken out is indeed the infiniti-eth one, but there are still an infinite number more marbles in the jar.

Mathematics is simply a toll the humans use to gain more concrete understanding of the world around them. Many times there have been very good mathematical models that adequately explain certain processes in our world but are later discovered to be incorrect. What do we do when this happens? We either discard the existing theory and replace it with one that doesnít produce inconsistencies, or we adapt the theory so it includes the new situational data. We do not attempt to change the world to match our model, nor do we ignore that these inconsistencies exist. And sometimes when this happens, it takes the smartest people in the world millions of dollars and tens of years to do, but that doesnít make the inconsistent method any more correct in the mean time. Aside from what has previously been stated, I simply cannot flaw your arguments. Unfortunately, that doesnít take the inconsistency away.

To answer the question when there is an infinte number of marbles initially in the bag which are removed one at a time Ė this is in fact a different problem, and Iíll happily agree that after an infinite time you can remove all the marbles. Lets leave the semantics of removing an infinte number of items out of this, or else really this problem becomes a paradox as well (removing the last implies that there was never an infinite number to begin with...). The reason that these problems are different is direct to the point that infinity does not equal infinity. The way this problem has been remade, the origin of the infinity is different, and not linked to the number of marbles entering the jar, while originally it was quite dependant on the it.

So then, I request of you no further defences of the all marbles removed argument. If you wish to convince me, then find the hole in my reasoning.

Finally, just to appease you, (which shall continue in the next post as this has become too long) I have come up with several possibilities for the inconsistency were experiencing. Iíll state right out that these may be in left field, these are just some thoughts so you can continue to think Iím not mathematically educated enough to disprove a result.
  Posted by Cory Taylor on 2003-09-16 18:46:11
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