Question: determine N(n) = the maximum number of mutually orthogonal Latin squares (MOLS) of order n. Known: N(n) ≤ n−1 always; equality iff a finite affine plane of order n exists (which requires n to be a prime power by Bruck–Ryser for most cases). For non-prime-powers, N(n) can be positive but is poorly understood.
Euler's 36-officers problem (1782) asked about N(6)=0, proven by Tarry (1900). Bose–Shrikhande–Parker (1960) disproved Euler's conjecture that N(n)=0 for n ≡ 2 mod 4, showing N(n) ≥ 2 for all n ≥ 3 except 6. The connection to finite fields gives N(q) = q−1 for q prime power. Wilson's 1974 pairwise balanced design constructions pushed lower bounds further, but the exact asymptotic behaviour for composite n remains one of design theory's deepest mysteries.
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