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This paper investigates the impossibility of certain (n²+n+k₍+₁) configurations. Firstly, for k=2, the result of gropp1992non that n²+n2 is even and n+1 is a perfect square or n²+n2 is odd and n-1 is a perfect square is reproved using the incidence matrix N and analysing the form of NTN. Then, for all k, configurations where paralellism is a transitive property are considered. It is then analogously established that if n0 or n k-1 mod k for k even, then n²+nk is even and n+1 is a perfect square or n²+nk is odd and n- (k-1) is a perfect square. Finally, the case k=3 is investigated in full generality.
Philbrook et al. (Fri,) studied this question.