The discussion so far has centered primarily about the fundamentals of math circle operation, such as scheduling, publicity, and funding. But now we interrupt the relentless stream of logistics to consider some of the less essential facets of a math circle. There are any number of ways that coordinators can add their own personal touch, ranging from a clever logo to a lending library. These extras contribute far more to a circle than the length of this section might suggest. They are likely to represent the coordinator's favorite aspects of the undertaking; these are the elements of the math circle which make organizing it a delight rather than a burden. Below is a cross-section of ideas taken from various math circles.
• The Berkeley Math Circle holds a monthly"Winner's Handicap" contest, run by a graduate student, in which participants are given four weeks to compose solutions to a set of five problems. The coordinators evaluate all the papers and award prizes to the top students. Then in subsequent months the stronger students are handicapped based on their scores on previous contests to help level the playing field and encourage the less experienced students to continue to work diligently on the problems. The Berkeley Math Circle has also created a nice logo involving a verdant tree featuring the digits of pi superimposed across its leaves with the Golden Gate Bridge in the background. This graphic appears at the top of their web site and on other materials associated with the math circle.
• The St. Mark's Institute of Math puts out a monthly newsletter featuring an introductory mathematical morsel, an interesting exposition, and a variety of questions (even open problems!), all centered about a common theme. The newsletter is published in hardcopy form only; the purpose is to populate the coffee tables in teachers' lounges across the greater Boston area with these brightly colored sheets of paper in the hopes that they will engage many more people than appear on the mailing list. The newsletter is currently mailed out to approximately 600 teachers and students.
• The Washington University Math Circle likes to send all participants home each week with an item related to the math that has just transpired. This might be a worksheet of problems for further investigation, or it could be some other type of souvenir, such as a set of linked Mbius strips resulting from an investigation in topology.
• The Stanford Math Circle presents warm-ups during the first ten minutes of each meeting. These might range from digit puzzles ("How many ways are there to make a perfect square with exactly four 1's?") to more meaty mathematical morsels ("How is it possible to fill all of three-dimensional space with non-overlapping circles?) In this manner the meeting can begin promptly while still allowing leeway for latecomers to hear the entire presentation.
• The San Jose Math Circle wheels in a large collection of math texts and problem books each week written for secondary students. Students can browse titles during any free moments and then check out books on the honor system. The coordinator also regularly distributes other items of interest, such as past issues of Math Horizons, the MAA journal produced for undergraduates.
• The Mobile Math Circle holds an internal olympiad in early March as a culmination of the many problem solving sessions they conduct throughout the year. The organizers recognize all the students who made significant progress, presents book awards to a couple of high scoring participants, and rewards their top two students with an all expenses paid trip to the Colorado Math Olympiad event later that spring.
• The San Francisco Math Circle hosts a year-end picnic. Families of all the participating students are invited to gather at Golden Gate Park to spend an afternoon together. Besides engaging in the expected outdoor activities, kids also take part in an informal ceremony recognizing outstanding participants.
• The Sudbury Math Circle, which is based at a private school in Sudbury, Ontario, devotes an entire school day to holding their annual "Math Challenge Day" for their own students and any others who are able to attend. The students take a math contest in the morning, followed by a math circle event from an invited speaker. A local pizza sponsor provides lunch, after which the students have recess and hear a second presentation by the speaker. Parents arrive, prizes are awarded, a reception is consumed, and everyone returns home exhausted but elated.
• The San Diego Math Circle runs math games as part of each meeting. Students attend a presentation during one half of the Saturday meeting and take part in an exciting, high energy team event during the other half. There is a snack break separating the two halves of the morning. This format permits speakers to work with each age group. The San Diego kids also sport cool T-shirts.