  Let $a_n$ denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an $n \times n \times n$
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits
the following explicit expression in terms of a Gaussian hypergeometric
function: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.\]
