  The "pyjama stripe" is the subset of $\mathbb{R}^2$ consisting of a vertical
strip of width $2 \varepsilon$ around every integer $x$-coordinate. The "pyjama
problem" asks whether finitely many rotations of the pyjama stripe around the
origin can cover the plane.
  The purpose of this paper is to answer this question in the affirmative, for
all positive $\varepsilon$. The problem is reduced to a statement closely
related to Furstenberg's $\times 2,\, \times 3$ Theorem from topological
dynamics, and is proved by analogy with that result.
