We obtain a correspondence between pairs of $N\times N$ orthogonal Latin squares and pairs of disconnected maximal cliques in the derangement graph with $N$ symbols. Motivated by methods in spectral clustering, we also obtain modular conditions on fixed point counts of certain permutation sums for the existence of collections of mutually disconnected maximal cliques. We use these modular obstructions to analyze the structure of maximal cliques in $X_N$ for small values of $N$. We culminate in a short, elementary proof of the nonexistence of a solution to Euler's $36$ Officer Problem.

翻译：我们建立了$N\times N$正交拉丁方对与具有$N$个符号的置换图中不相连极大团对之间的对应关系。受谱聚类方法的启发，我们还得到了关于特定置换和式不动点计数的模条件，用于分析相互不相连的极大团集合的存在性。利用这些模阻性条件，我们分析了$X_N$中极大团在较小$N$值下的结构。最终我们给出了欧拉"36军官问题"无解性的一个简短初等证明。