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What sort of collections of circles fit into a larger circle without overlapping? This question has motivated mathematicians for millenia. Here at UC Davis, Elena Fuchs has studied Apollonian circle packings and found that the curvatures appearing in such families of circles exhibit certain combinatorial features. Her results have led to stronger conjectures, widely accepted among number theorists, but one of these conjectures turned out to be false. This was discovered (via extensive number crunching) during a summer research project involving two students (one graduate, one undergraduate) led by Katherine Stange at the University of Colorado in Boulder. An article in Quanta magazine tells the story of the students' discovery and provides mathematical details.