How many simple closed curves can be drawn on a surface of finite type so that any pair of them intersect some specified number of times? We will introduce questions of this nature, explore some known cases in an interactive activity, and point to what is known and what is conjectured. We'll also reframe questions of this type in terms of graph-theoretic questions related to cousins of the curve graph. No prior knowledge of surface topology or graph theory is required, but curiosity and an appreciation of balloons are recommended. Speaker(s): Max Lahn (University of Michigan)