  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.
