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 $n\equiv 0,1 \mod 4$.

Radio labeling is a variation of Hale's channel assignment problem, in which
one seeks to assign positive integers to the vertices of a graph $G$ subject to
certain constraints involving the distances between the vertices. Specifically,
a radio labeling of a connected graph $G$ is a function $c:V(G) \rightarrow
\mathbb Z_+$ such that $$d(u,v)+|c(u)-c(v)|\geq 1+\text{diam}(G)$$ for every
two distinct vertices $u$ and $v$ of $G$ (where $d(u,v)$ is the distance
between $u$ and $v$). The span of a radio labeling is the maximum integer
assigned to a vertex.

It has been conjectured that for knots $K$ and $K'$ in $S^3$, $w(K#K')=
w(K)+w(K')-2$. Scharlemann and Thompson have proposed potential counterexamples
to this conjecture. For every $n$, they proposed a family of knots ${K^n_i}$
for which they conjectured that $w(B^n#K^n_i)=w(K^n_i)$ where $B^n$ is a bridge
number $n$ knot. We show that for $n>2$ none of the knots in ${K^n_i}$ produces
such counterexamples.