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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.)
re(3): more | Comment 41 of 83 |
(In reply to re(2): more by Cory Taylor)

Here are your statements with responses:

... the "value" of infinity is static
I'm not at all sure what that means. (And I'd love for you to explain it.) But, if by that you mean that the members of the set REPRESENTING a particular infinity is static, then YES.... It is static. e.g., The members of the set containing all even integers is static. It is not growing, it is not declining, it remains constant (over time?)... :-)

the value of [the set of integers] is twice that for [the set of positive integers]...
At least that's what I *think* you meant by your sentence. Again, I'm not sure what you mean, but I *think* you are saying the number of members is twice the size in one set as the other. However, I don't understand that line of reasoning at all. It is trivial to show a one-to-one mapping from one set ONTO the other set. And by your own (faulty) argument, if we can add one marble and remove a corresponding marble for EACH iteration, maintaining the empty status, then "at the end" it must remain empty. Well... I don't agree with your line of reasoning, but if you follow it, your reasoning produces inconsistencies.

I'm not going to debate with you the intricacies of the equations or the definitions of limits
Well, that's your prerogative, but then why do you post on this thread? And isn't this exactly what we're all doing?

An infinite number of nines summed cannot equal 0.
Ummm... I think I agree.... but we didn't sum an infinite number of nines.
Cory, I beg of you... *please* read what I and Brian posted earlier, because I'm getting the sense you didn't read them.

You are confusing TWO different issues about infinity. There is the SIZE of the set and the MEMBERSHIP of the set.

And you're not being consistent about your use of "value" (as in your comment about the "value" of infinity being static).

Now, I'm sure you'll agree that given the two sets {1, 2, 3} and {2, 4, 6}, that the two sets do not have the same elements (and therefore, they do not have the same value). But they ARE of the same size.

By extension, {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} are of the same size. And if we look at what the results are incrementally, and then expand these infinitely as you so wish to do.... then we see that the positive integers have THE SAME SIZE, the SAME number of elements, as the positive EVEN integers. And at the SAME TIME, all the elements of one are in the other but not vice versa. This is not a logical inconsistency (as you would have us believe). (By the way, the size of both of these infinities is called aleph null.)

Now, if you don't agree with this last line of reasoning, please explain where you find the problem.

--- SK
Edited on September 15, 2003, 3:38 pm
  Posted by SilverKnight on 2003-09-15 15:34:46

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