The rules of a game of dice are extended to a ``hyper-die'' with \(n\in\mathbb{N}\) equally probable faces, numbered from 1 to \(n\). We derive recursive and explicit expressions for the probability mass function and the cumulative distribution function of the gain \(G_n\) for arbitrary values of \(n\). A numerical study suggests the conjecture that for \(n \to \infty\) the expectation of the scaled gain \(\mathbb{E}{H_n}=\mathbb{E} {G_n/\sqrt{n}\,}\) converges to \(\sqrt{\pi/\,2}\). The conjecture is proved by deriving an analytic expression of the expected gain \(\mathbb{E} {G_n}\). An analytic expression of the variance of the gain \(G_n\) is derived by a similar technique. Finally,  it is proved that \(H_n\) converges weakly to the Rayleigh distribution with scale parameter~1.