# The number of worms (Moser's worm problem)

The Moser's worm problem [springer link] asks for the region of smallest area that can accommodate every plane curve of length 1. Curves can be rotated and translated and may be considered identical upto rotation and translation transforms.

The cardinality of the set of plane curves of length 1 is the same as that of the continuum. It is $\le$ the continuum because every such curve comes from a continuous function $[0,1] \to \mathbb{R}^2$, and such a function is determined by its values on the rational numbers. To see that it is $\ge$ the continuum, consider curves of constant curvature (sufficiently small such that they do not form a complete circle.)