There is really only one hard and fast rule when it comes to choosing suitable material for a math circle. Speakers should select topics which they find to be fascinating. The kids will quickly pick up on the extent to which a speaker is interested in the proceedings and react accordingly. It is not even necessary that the speaker be especially familiar with the topic, at least prior to preparing for the big day. In fact, there is some merit to selecting a topic which is not within one's standard repertoire, particular for speakers with a strong general mathematical background. Not only will the person leading the session enjoy learning a topic about which she has always wanted to know more, but she will also be much better able to view the material from the perspective of a student who has not encountered it before. The net result is a more enthusiastic and accessible presentation. To be sure, there are disadvantages to delivering new material, the most obvious being greater prep time, more hunting necessary for assembling a set of good problems, and a lesser degree of expertise when responding to questions. The latter objection is not really an issue (it's very informative for students to observe how mathematicians respond to questions to which they don't already know the answer), and the first two can be overcome in a couple hours time.
There are ways for adapting almost any topic so that it will work well within a math circle setting. One of the more common types of modification necessary is that of bringing an advanced topic within reach of the target audience. This is the dilemma faced by a research mathematician who wishes to promote his specialty to a general audience. However, there are beautiful motivational ideas behind even advanced material that can form the basis of an accessible math circle session. Moreover, it is important for students to begin to gain a sense of the various fields of mathematics even at the middle school level. Thus a topologist could assemble an entertaining collection of concrete ways to tell the difference between a torus and a sphere. (For example, given five points on the surface of a torus, it is possible to connect every pair of them with a path such that none of the paths cross. This cannot be accomplished on a sphere.) An analyst might spark a rousing debate by asking whether it is possible to cover an open segment with closed segments, a discussion which leads naturally to concepts of infinity, measure, and the Cantor set. Group theory can be brought within reach of eighth graders via symmetry groups. And there are many more variations on this theme.
Another standard challenge facing speakers concerns how to handle a topic which, despite being readily accessible to students, is much too broad to communicate in one sitting. Topics such as counting, probability, graph theory, number theory, and geometry all fall into this category; topics on which entire books have been written. The temptation to be avoided in these cases is to cover too much material in the given time. Students will either become bored or lost; neither state is optimal. A better approach would be to tightly focus on one or two inviting problems within the subject that will hold students' attention, then fill in necessary background material when they are motivated to absorb it. For example, rather than lay out the foundations of vector geometry, have the group puzzle out the following fact. If one were to begin at any of four desks in the room, walk halfway towards another desk, continue from there a third of the way towards a third desk, and then head a quarter of the way towards the final desk, one would arrive at the same finish point, regardless of the order in which the desks were chosen. This strategy of focusing on specific, captivating problems is employed in the subsequent chapters on Sneaky Segments and Making Change.
Finally, many math circles devote at least several of their sessions to preparation for various regional and national mathematical events. For example, most circles near San Francisco take time to go over general problem solving strategies and proof techniques shortly before the Bay Area Math Olympiad. Other math circles might spend a day helping students get ready for the AMC contests, if there is enough interest. Still other math circles choose to motivate their material solely through the intrinsic beauty of the subject without reliance upon an element of competition.
Over the years an informal canon of math circle topics has gradually developed. Each of these topics is accessible to middle and high school students but can be taken well beyond their standard curriculum. They encompass a multitude of fascinating problems and lend themselves well to engaging presentations. While no list could claim to be comprehensive, this one attempts to include all of the more popular math circle topics. A few general topics are shown in boldface, followed by related subtopics. Further information on them is spread across dozens of sources, but a couple of references cover a significant percentage of them; these are mentioned following the list.
Math Circle Topics  



Resources
The book Mathematical Circles (Russian Experience) by Fomin, Genkin, and Itenberg, published by the American Mathematical Society, walks through many of the topics listed above and contains an extensive sampling of problems. Some math circles post handouts at Resources their web sites; the Berkeley Math Circle has a particularly large collection available at http://mathcircle.berkeley.edu/. Tom Davis, who has been involved with math circles in the Bay Area for years, has posted notes on many topics at http://www.geometer.org/mathcircles/. There are many other useful books and web sites; for a more complete list of resources visit the new web site (still under development as of this writing) at http://legacy.mathcircles.org/.
Purpose of the modules
Each chapter in the second part of this handbook expands one of the above topics into a fullfledged session. There are detailed presentation notes, two of the sets of problems to accompany each topic, background, and hints/answers. The purpose of this compilation is twofold. Most importantly, these chapters should give speakers a sense of how to transform a promising topic into a lively exploration for a group of motivated kids. The modules span a range of difficulty levels and subject material in order to be as helpful as possible in this regard. On the other hand, they are not meant to serve as a sample syllabus for a semester's worth of meetings.
Of course, in a pinch they can also serve as readytogo presentations; do not hesitate to mimic the suggested dialogue or use the problems as they stand  they were assembled partly with this purpose in mind. (Just please mention the source!) Most of the sessions described will need editing before use, depending on the audience, time available, and leader's mathematical expertise. Select the most appropriate problems and pick and choose from the presentation notes; there is usually more material than will fit into a single meeting. The stars underneath the titles gauge the approximate level of mathematical maturity necessary on the part of the participants. One star topics are suitable for most middle school groups, two star topics could work for an advanced middle school or normal high school crowd, and three star topics should be reserved for a strong high school audience.
All these sample presentations and more should soon be available at the web site http://legacy.mathcircles.org/ maintained by the Mathematical Sciences Research Institute. In particular, the LATEX code for the problem sets appearing in the upcoming chapters may be downloaded, which should save typing time. The goal of that web site, similar to that of this handbook, is to serve as a resource for coordinators and leaders of math circles. The specific mission and function of the site will undoubtedly evolve over time, so please visit the web site for more information. And now, on to the mathematics!