All projective planes of order up to 10 have been classified. Here, some of these results are reproved. Specifically, it is shown that there is no projective plane of order 6 and that there is a unique projective plane of order 7. The existence of a projective plane of order \(n\) is equivalent to the existence of \(n-1\) mutually orthogonal Latin squares (MOLS) of side \(n\). This connects the current study to the nonexistence of 2 MOLS of side 6. That result was originally obtained by G. Tarry [Ass. Franç. Paris 29, 170–203 (1900; JFM 32.0219.04)] more than a century ago. About ten alternative proofs have later been obtained, including those by D. R. Stinson [J. Comb. Theory, Ser. A 36, 373–376 (1984; Zbl 0538.05012)], S. T. Dougherty [Des. Codes Cryptography 4, No. 2, 123–128 (1994; Zbl 0796.05012)], G. Appa et al. [Oper. Res. Lett. 32, No. 4, 336–344 (2004; Zbl 1052.05018)], and A. P. Burger et al. [Discrete Math. 311, No. 13, 1223–1228 (2011; Zbl 1229.05050)].
Appa et al. [loc. cit.] use linear programming (LP), which is also the approach of the current paper. The current study would hence have benefited from a comparison with those old results. One well-known difference between a search for a small number of MOLS of side \(n\) and a search for \(n-1\) MOLS of side \(n\) (that is, a projective plane) is that in the latter case one may actually start from a substructure coming from a Latin square of side \(n-1\) rather than \(n\); this observation is utilized by the authors. In some cases, a partial structure has to be extended somewhat, through a very small search tree, before LP gives the desired nonexistence result; also this is in line with the results of Appa et al. [loc. cit.].