A graph is a thrackle if it can be embedded in the plane such that every edge crosses every other edge exactly once. Here, two edges cross if they overlap exactly at a shared endpoint, or if they do not share an endpoint but cross over each other in their interiors. Every tree is a thrackle, and so is every cycle except the four-cycle.
Conway conjectured that thrackles can have at most as many edges as vertices. This problem asks for a counterexample to Conway’s thrackle conjecture: a graph with strictly more edges than vertices that is a thrackle. Conway offered a prize of $1,000 for a solution to this problem.
Models supply a graph and a thrackle embedding of the graph, and the verifier checks that the embedding satisfies the edge crossing properties. The main risk for this problem is that Conway’s thrackle conjecture is in fact true, in which case no counterexample exists.