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)
Defining Infinity: Regarding Brian Smith's comments, I have no comment
except to say that Silver Knight made a good reply.
Silver Knight: your one to one mappings have to be ONTO, which I'm sure
was just a minor oversight on your part. As you point out, if you can
find any such mapping (one to one and onto) then the sets are of the same
size (cardinality). The fact that some mappings can be found that are
one to one and not onto (as DJ seemed to be having a problem with) is not
relevant.
Sam: The puzzle does end, because each step takes only half as long as the
other. But I agree that the apparent paradox is a result of physically
impossible premises set out in the problem. But if mathematicians (or
philosophers) had to restrict themselves to physical reality, their jobs
would be so much more boring. And there is a lot of useful mathematics
that would never have been discovered (calculus, complex numbers, etc).
Fat Boy: the problem is clearly impossible in the real world, but is
a conceptual/theoretical problem. Since there is no marble that is
not processed within the minute, you do in fact process them all. That
is the point of halving the time each iteration: there are an infinite
number of steps, but they get completed within the minute.
Respectifully I disagree with most (save perhaps SK and Sam)
