Given n matches (i.e., rigid unit line segments), find the number of topologically distinct planar arrangements which can be made. In this problem, two matches laid end-to-end with no third match at their meeting point are considered equivalent to a single match, so triangles are equivalent to squares, n-match tails are equivalent to 1-match tails, etc. Solutions to the match problem are planar topological graphs on e edges, and the first few values for e = 1, 1, 3, 5, 10, 19, 39, ... (OEIS A003055).

cigarettes | matchstick graph | planar graph | polynema | topological graph | unit-distance graph