  We consider the problem of placing $n$ nonattacking queens on a symmetric $n
\times n$ Toeplitz matrix. As in the $N$-queens Problem on a chessboard, two
queens may attack each other if they share a row or a column in the matrix.
However, the usual diagonal restriction is replaced by specifying that queens
may attack other queens that occupy squares with the same number value in the
matrix. We will show that $n$ nonattacking queens can be placed on such a
matrix if and only if $n\equiv 0,1 \mod 4$.
