What's the smallest shape that can cover every possible path of length one?
Imagine every conceivable curve, straight or wiggly, that is exactly one unit long. Moser's worm problem asks for the smallest-area shape that can be laid down to completely cover any one of those curves, in any position or orientation, without moving or rotating the shape once it's placed. Leo Moser posed the question in 1966 — can you find a shape this small, or prove how small it has to be?
Reveal the answer
It's still unsolved. The best known covering shapes so far are oddly shaped regions with area a little under 0.3 square units, built by chipping corners off and overlapping circular arcs, but no one has proven any of them is truly the smallest possible. Leo Moser posed the problem in his 1966 'Poorly Formulated Unsolved Problems of Combinatorial Geometry,' and it remains one of the most approachable open problems in geometry.
— Leo Moser, Poorly Formulated Unsolved Problems of Combinatorial Geometry — Mimeographed notes, 1966