  This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$
configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that
$\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is
odd and $n-1$ is a perfect square is reproved using the incidence matrix $N$
and analysing the form of $N^TN$. Then, for all $k$, configurations where
paralellism is a transitive property are considered. It is then analogously
established that if $n\equiv0$ or $n\equiv k-1$ mod $k$ for $k$ even, then
$\frac{n^2+n}{k}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{k}$ is
odd and $n-(k-1)$ is a perfect square. Finally, the case $k=3$ is investigated
in full generality.
