Fun Facts are designed for, and work best in, lower-division college courses where the typical student is not a math student. The point of Fun Facts is to generate interest, arouse curiosity, and cause such students to look at mathematics differently!
I do one per class, usually at the beginning of the hour. Some are shorter, some are longer, but try not to spend more than 5 minutes presenting one, since it can eat up valuable class time. The purpose of the fact to only to pique their interest in math; you can go into more detail after class for those interested.
However, do not be afraid to give them a glimpse of some advanced mathematics. For instance, they will enjoy hearing the word “fundamental group” even if they don't understand what it means. If all students do is pick up the “buzzwords”, it still helps to popularize mathematics. For instance, how many of us know what “string theory” really is, but doesn't its mystique give physics a lot of glamor?
Sometimes I choose Fun Facts related to the theme of the current or upcoming class. However, it is also good to do several that are unrelated to the course material, for variety. If a Fun Fact works in well with the lesson, I'll do it in the middle of the hour. If I want them to think about a Fun Fact, I may begin one at the end, give them a day to think about it, and then finish it up the following lecture. Sequences of Fun Facts that relate to each other are fun to present over several days.
Also, feel free to link the “Random Fun Fact” URL to your web page as a fun way for your students to learn some Fun Facts at their leisure.
The Fun Facts in this database are ones which we have collected, but I'm sure you can think of many more—so feel free to be creative. And if you have good ideas, consider submitting them!
p.s. Please see the disclaimer.