  We apply to the $n\times n$ chessboard the counting theory from Part I for
nonattacking placements of chess pieces with unbounded straight-line moves,
such as the queen. Part I showed that the number of ways to place $q$ identical
nonattacking pieces is given by a quasipolynomial function of $n$ of degree
$2q$, whose coefficients are (essentially) polynomials in $q$ that depend
cyclically on $n$.
  Here we study the periods of the quasipolynomial and its coefficients, which
are bounded by functions, not well understood, of the piece's move directions,
and we develop exact formulas for the very highest coefficients. The
coefficients of the three highest powers of $n$ do not vary with $n$. On the
other hand, we present simple pieces for which the fourth coefficient varies
periodically. We develop detailed properties of counting quasipolynomials that
will be applied in sequels to partial queens, whose moves are subsets of those
of the queen, and the nightrider, whose moves are extended knight's moves.
  We conclude with the first, though strange, formula for the classical
$n$-Queens Problem and with several conjectures and open problems.
