While tutoring the other day I came across a puzzle I don't know how to do. This isn't exactly noteworthy, as I can't do most puzzles. But it seemed like an interesting enough puzzle, so I thought I'd give it broader attention:
50 watches are placed at random on a table. Show that at some time the sum of the distances from the center of the table to the ends of the minute hands is greater than the sum of the distances from the center of the table to the centers of the watches.
The problem didn't say anything about the sizes of the watches (like, if they were all the same). Also, I guess we should say that they are all facing up (does it matter?). I think the problem did mention that the watches weren't all necessarily all set to the same time, just that they were all running at the same, accurate, pace.
So, any thoughts?