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Problem #7: Boxes and Marbles

Silly Druid

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You have some numbered boxes, their number doesn't really matter that much, but let's say there are 1000 of them, numbered from 1 to 1000. Now, in each "phase", you put marbles in specific boxes:

In phase 1, you put a marble in boxes 1, 2, 3, 4, ...,1000 (so in all of them).

In phase 2, you put a marble in boxes 2, 4, 6, 8, ..., 1000.

In phase 3, you put a marble in boxes 3, 6, 9, 12, ..., 999.

And so on. In the last 500 phases, you just put a single marble in boxes 501 to 1000.

The following questions apply to the situation after all 1000 phases are completed:

1. Which box(es) have exactly 1 marble?

2. Which box(es) have exactly 2 marbles?

3. Which box(es) have an odd number of marbles?

4. What does the number of marbles really represent in relation to the number of the box?

For questions 1-3, you don't need to list all the boxes with a specific number of marbles, just identify some common property they have.

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I'll try to answer in such a way that you know that I know, but without spoiling the fun to others ;)

1. That's a rather trivial one ;)

2. That's a prime example of what a certain Eratosthenes would ponder upon while carrying a sieve ;)

3. I think that it's perfectly possible to square that circle :)

4. This question might be quite divisive, depending on whether one likes a proper answer or improper :umad: I'd go with improper though.

OK, but to add some more fun, let me add another question:

5. Which box will contain the biggest number of marbles and why? :grin2:

Edited by SasQ
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A box B gets a marble if the phase P is a factor of the box, right? If B/P = integer, then that box gets a marble.

The box that gets the most marbles would be the one with the most factors, not just prime factors. So box 16 = 2*2*2*2 has factors 2, 4, 8, 16. I would say that the box with the most non unique prime factors would have the most. We aren't talking about the size of the factor, just the number of different factors that can be made. That's my first guess.

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