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(10): No Subject by Ken Haley)

Mr. Ken Haley, here is a list of your objections and answers to the official solution of the first marbles bonanza problem,

1- Objection: After each setp one has 9n marbles in the bag. As n goes to infinity the number of balls MUST go to infinity as well.

1.- Answer: While it is true that each finite step results in 9 marbles being added to the bag, it is an unjustified assumption that extrapolating this behaviour to infinity describes the operations done by the algorithm on the infinite set. For example, the operation {1,2,3,4,...} - {1,2,3,4,...} = {} can be described very differently by finite steps, but the end result is allways the same.

2: Objection: The infinite operation {1,2,3,4,...} - {1,2,3,4,...} = {} is not consistent because if I relable the first set I can get any answer I want.

2: Answer: Yes, but relabeling the set changes the initial set; you would be solving a different problem altogether. While it is true that relabeling does not change the number of marbles, it changes the way the algorithms act on the set, and the answer is expected to change (think of the alogrithm "remove all even" for example). There is no inconsistancy here!.

3.- Objection: I can define a function, f(n) = 9n that describes all the finite steps and therefore describes the algorithm. It is well known that this function goes to infinity as n goes to infinity, so the algorithm will do the same.

3.- Answer: (This answer is identical to Answer 1) While f(n) = 9 is a well defined function, it only describes the algorithm in it's finite regime (which we established that can be described in many different ways). To understand what happens to the complete oredered set more information is needed.


As far as I can see the following statements,

A) {1,2,3,4,...} - {10,20,30,40,...}  = {1,2,3,4,5,6,7,8,9,11,12,13,....}

B) {1,2,3,4,...} - {1,2,3,4,...} = {}

Hold their truth and logic under all the objections presented so far.

Posted by ajosin on 2005-06-27 21:54:30