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(9): No Subject by ajosin)
To both ajosin and Tristan:
The mathematical definition of a function is:
A relation between the elements of one set, X (the domain), and the elements of another set, Y (the range), such that for each element in the domain X there's only one corresponding element in the range Y. (see http://www.lhup.edu/~dsimanek/glossary.htm)
So, in my restatement of the problem, I have described a well-defined function: For any element in the domain (that is, any set of marbles in the bag) you can determine a corresponding element in the range (that is, the set of marbles in the bat at the next step in the sequence). Yet, you both assert that you don't have enough information to solve it. What's missing...from a mathematical standpoint? (After all, we all know that there's no physical way to represent this puzzle...we can only think of infinities abstractly--in this case, mathematically.)
re(10): No Subject
