Harvey Mudd Professor Francis Su has put together a very nice collection of what he calls "Mudd Math Fun Facts," interesting puzzles that he uses to spice up his undergraduate math classes. As he says, they are "ideas that change the way you think." His facts are labelled by difficulty (easy, medium, or hard) and by the area of mathematics from which they are drawn (algebra, geometry, number theory, probability, calculus/analysis, and "other.")
Although designed for undergraduates, there are many gems in his collection that would be perfect for Albany Area Math Circle students who are working as MATHCOUNTS coaches or mentors to our motivated middle school students. What you might want to do is select 10 or so of these that you think your students might enjoy, print them out, cut the facts into separate strips, and put them in a bag or a box. Then, when your students need a quick change of pace, just draw one out at random, and get your students thinking about it. Make sure to think about these ideas yourself before you put them in the bag--you'll probably find you deepen your own problem-solving skills, especially if you think of extensions or generalizations. You might want to generate some of your own fun facts from cool puzzles you run across in other contexts.
Here's an example of a medium Mudd Math Fun Fact. You can find the answer here.
One hundred ants are dropped on a meter stick. Each ant is traveling either to the left or the right with constant speed 1 meter per minute. When two ants meet, they bounce off each other and reverse direction. When an ant reaches an end of the stick, it falls off.
At some point all the ants will have fallen off. The time at which this happens will depend on the initial configuration of the ants.
Question: over ALL possible initial configurations, what is the longest amount of time that you would need to wait to guarantee that the stick has no more ants?