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)
I just ran across this problem--I wish I'd seend it earlier, since it's almost 2 years old, and the most recent comment is 7 months old. But here's my 2 cents.
How can you accept a solution that says in one case, you have an infinite number of marbles left, and in the other you have no marbles left? You're doing the SAME EXACT THING at each step in both cases: putting in 10 marbles and taking out 1 marble. Who cares how they're marked? The resulting count should be the same!
I think the correct answer is that the problem as stated has no meaningful answer, because it requires us to subtract two infinities, which is just as meaningless as dividing by zero. Here's a similar problem: Hotel Infinity has an infinite number of rooms all full - one guest in each room. An Infinity Bus arrives with an infinite number of people--all requiring a room. Does the clerk send them away because there's no vacancy? Of course not. He simply turns on his Infinity PA system, and instructs each guest to double his or her room number and move to that room. Now the new arrivals can move into the now-vacant odd-numbered rooms. (For those who want to talk about cardinality, both infinities are "countable" or aleph-null, or in 1-1 correspondence with the rational numbers.)
Does this prove that zero = infinity? Well, not really. I think the point is, when you're dealing with infinities, ordinary arithmetic can break down, and unless you're careful, it's easy to contrive situations (such as this problem) that have no meaningful solution. In this case, and in the original problem, we've shown that infinity minus infinity can be zero or infinity, or any other number you like (just like dividing 0 by 0). So, we have to discard the question as meaningless.
The solution isn't...
