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)
Sat here many hours debating whether to continue this thread.
Even went to the point of conferring with peers who've expertise in this subject (set theory).
None (well, neither) have found error with the method I've produced (sum of nines).
One faults the question as being imprecise enough to allow the interpretational deviations experienced, while the other outright denies the analysis of a zero result as being wrong by being incomplete and defiant to the particulars of the situation.
I don't mean to create any sort of stir here, nor do I do this as a result of mathematical ego as I have none (I work in road construction). I am just being thorough (as an engineer should be).
I apologize that I simply cannot accept your statement that my method is un-allowed without a plausible explanation (and both experts I've consulted have agreed with my interpretation - and they've both seen the question originally - not any adaptation I've made).
I will concede that the second aforementioned expert, while certain the math was flawed, couldn't specifically flaw your argument any better than I (he mentioned that there are numbers, and therefore marbles, greater than the infinity referred to in the "removal" process) he was going to double check his theories to be sure. But as a basis, he was able to confirm, in a way, that the process is identical to increasing the set size by nine infinitely (his particular method was to graph the set size versus the repetition number, which of course results in a straight line not passing through zero any point but the origin (and you'd better not tell me that the x-axis has a finite boundary)).
I will apologize for my lack of future response. For a time, I considered that this problem was not worth the effort it apparently required to convince of you of the error, and thereby wasted possibly one or two returns. As it happens, I am set to return to work tomorrow (I've just finished 5 days off) and won't return to internet capability for 4 (possibly, but more likely 2) weeks (check my forum posts or my personal description to verify this). I will say though, that I have great confidence in my experts, and its certainly going to take one heck-uv-an argument from you to refute them. By all means feel free to email me (though put "to cory" or some such in the subject or it'll just get deleted), or continue on as you see fit, but I'm now gone till seasons end (typically mid-October).
And I now wonder what monster I've created.
Edited on September 19, 2003, 2:37 am
Edited on September 19, 2003, 2:39 am
whee
