a planar set that contains a unit segment in every direction is called a kakeya set. these sets have been studied intensively in geometric measure theory and harmonic analysis since the work of besicovich (1919); we find a new connection to game theory and probability. a hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. at each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. thus they are engaged in a zero sum game, where the payoff is the capture time. we show that every rabbit strategy yields a kakeya set; the optimal rabbit strategy is based on a discretized cauchy random walk, and it yields a kakeya set k consisting of 4n triangles, that has minimal area among such kakeya sets. (talk based on joint work with y. babichenko, r. peretz, p. sousi and p. winkler).