divisibility

1. Emperor's Banquet

You have been invited to the emperor's banquet. The emperor is a rather strange host. Instead of sitting with his guests at a large round dining table, he walks around the table pouring oatmeal on the head of every other person. He continues this process, pouring oats on the head of everyone who has not had oats until there is only one person left. (Remember he is skipping every other person who has not been hit as he goes around.) You will know when you arrive where the first seat is to get the oats and that the emperor always rotates to the left with his oats. You do not know in advance how many people are there. Devise a system that will tell you in what seat to sit in order not to get "oatmealed" once you know how many people are at the banquet.

3. Boxes

You have n boxes to deliver to a series of stops. At the first stop you are to leave 1 box. At the second stop you are to leave two more boxes than at the first. At the third stop you are to leave two more than at the last stop, and so on. At what stop will you not have enough boxes to make a delivery?

4. Prove: If r divides d (r is a factor of d; or r 'gazinta' d) and r divides y (again, r is a factor of y), then r divides cd + xy for any c, x in Z.

5. Magic Dough

A baker has perfected magic bread dough that, when placed in the oven, doubles in size every minute. He has further determined a special measure of this dough that will exactly fill the oven in n minutes. If the baker puts two special measures in the oven, when will the oven be full? (If you do this problem and it takes you 2 seconds, you might have missed something.)