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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.)
No Subject | Comment 48 of 83 |
ok, when Ive asked you to dispute my reasoning, you have misunderstood my request, and disputed my disputes, not my completely separate solution. I will make this easy. Your solution determines that there are no marbles in the jar. Fine.

My solution is as follows: At any point, the total number of marbles in the jar is equal to 9*n, where n is the problem defined repetition number. The only time that, assuming this is correct, that there are no marbles in the jar is before you've started.

Dispute this, or agree that we've proven that zero=infinity.

As I have attempted to explain before to you, and will try from yet another angle, this is not true. Math is a tool that we use to simplify things and thereby better understand them. All branches of math serve this purpose. One of the greatest things, and the most basic reason that I like math is that the process of working through a problem is irrelevant. If one makes no false assumptions, errors or ommisions, then then "answer" to a problem is unique (note that there are problems that some would say have multiple solutions as in 'what are the roots of an equation', but in these cases the "correct" solution is only when you've included all the correct answers.. but this is another discussion). There are plenty of calculus problems that have multiple {forms} to their solution, a simple example being that when one evaluation method produces 1 and a different evaluation method produces sin2 + cos2. I'm certain you know these are in fact equal, and what this process has done is in fact proven that they are equal. I'm afraid that that is exactly what we've accomplished here - your method, allowing for the inconsistncies you're still trying to explain away concludes an answer of zero. My method, allowing for any inconsistencies it contains (of which I am not aware or we wouldn't still be debating this) produces an answer of infinity. If neither method is flawed then we've found a rigid proof that 0=infinity. Hopefully I'll get no argument from you that this proof cannot be, so therefore, by deduction, one method must be incorrect.

To strengthen my approach, I can use another method. Proof by contradiction (in this case more appropriately disproof by contradiction, but we'll get to that some other time). Assume that at time 61 seconds there are no marbles in the jar. Clearly at time 40 seconds there are 9 marbles in the jar. As marbles are a discreet, not a continuos variable, at some point, to end with zero marbles then, the last marble must have been removed (we won't even discuss that there never was a last, or wait is that your point...). However, with each iteration, before a single marble is removed, ten are added, negating the possibility that the last is removed.

And, I will restate that I do see why you're struggling here. All of the marbles, by the definition of the set they come from have been removed. Clear as can be. Yet, even clearer is that fact that we cannot have it both ways.
  Posted by Cory Taylor on 2003-09-17 22:54:54
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