  How many mutually non-attacking queens can be placed on a d-dimensional
chessboard of size n? The n-queens problem in higher dimensions is a
generalization of the well-known n-queens problem. We provide a comprehensive
overview of theoretical results, bounds, solution methods, and the
interconnectivity of the problem within topics of discrete optimization and
combinatorics. We present an integer programming formulation of the n-queens
problem in higher dimensions and several strengthenings through additional
valid inequalities. Compared to recent benchmarks, we achieve a speedup in
computational time between 15-70x over all instances of the integer programs.
Our computational results prove optimality of certificates for several large
instances. Breaking additional, previously unsolved instances with the proposed
methods is likely possible. On the primal side, we further discuss heuristic
approaches to constructing solutions that turn out to be optimal when compared
to the IP. We conclude with preliminary results on the number and density of
the solutions.
