After processing an infinite number of marbles, not once but twice in solving the puzzle Marbles Bonanza, you've grown rather tired of moving all these marbles around. Nevertheless, it is your duty to press on forward and try a third experiment. This time, though, you decide that you'll cut down on the amount of work by not removing any balls ever.

At the start of the minute, you put marbles 1-9 in the bag, and then add a 0 to the end of marble 1 (so that you now have duplicate marble 10s, one in the bag that you just modified, one out). Therefore you now have marbles numbered 2-10 in the bag, as in scenario B of the previous puzzle, and marble 10 outside the bag as in scenario A. 30 seconds later, you put marbles 11-19 in the bag, and add a 0 to marble 2, so that now you have two marbles numbered 20 - one in, one out. You continuously repeat this process, with each interval half as long as the one before. In general, for the nth operation, you put marbles 10n-9 to 10n-1 in the bag, and add a zero to marble n in the bag, so that it becomes marble 10n in the bag.

• How many marbles are in the bag at the end of the minute?
• What are the numbers on the marbles ?
• Is the situation inside the bag identical to either of the previous two problems after 31 seconds? 50 seconds? at the end of the minute? How about the situation outside the bag?

(In reply to re(4): No Subject by Ken Haley)

Yes, I agree with the first paragraph.

I disagree with the second pararagraph, and agree with the third paragraph.


4th paragraph;


The labeling is very relevant;  it is what  gives the problem logical  sense.  If you gave me two infinite sets of unlabled objects and told me to take one away from the other I would have no idea what to do!.

I don't understand why you insit that the number of marbles after an algorithm on the set of marbles should be invariant under relabeling. True, relabeling does not change the number of original marbles, but it will change the way the algorithms act on the marbles.

Let's say I have an alogirthim that says "remove all even balls from the bag". Acording to you, this algorithm is ill-defined, unphysical, senseless, etc (just like A and B), because I can get any final answer I want by relabling the original set of marbles. For example, if originally I had four marbles labeled {1,2,3,4} and applied the alogrithm I would be left with three marbles. If I relabled the marbles {2,2,2,2} and then applied the algorith, I would be left with no marbles.

Under your definition of what algorithms are acceptable, only those that say "remove n elements" would be stisfactory. In particular, one would never be able to make operations on infinite sets.

To me, any algorithm that aplied to any specific set gives a clear and single answer is acceptable. The answers do not have to all be identicall, even if the different sets are related through relableling.

I think that we are getting to a point where we have different definitions of what self-consistancy is. You require extremely strickt  standards, while I prefer a definition that admits more types of algortighms, so I'm afraid will never be able to agree.

Posted by ajosin on 2005-06-25 06:53:05