Orthogonal Latin squares: two n×n Latin squares L1, L2 where superimposing gives all n² ordered pairs. MOLS(n) = max mutually orthogonal set. For prime p: MOLS(p)=p-1 via finite fields. Euler's 36 officers: MOLS(6)=1 (no pair of orthogonal 6×6).
How to Use
Enter n:
- Pair: Two orthogonal squares
- Super: Superposition check
- MOLS: Maximum set size
Euler's Problem
Euler (1782): can 36 officers (6 ranks × 6 regiments) be arranged in a 6×6 square so each row/column has all ranks and regiments? He conjectured NO for all n≡2(mod4). Tarry (1901) proved n=6 impossible, but Bose-Shrikhande-Parker (1960) found solutions for all other n≡2(mod4), n≥10!
Bounds
MOLS(n) ≤ n-1. Equality when n=p^k (prime power). MOLS(2)=1, MOLS(6)=1, MOLS(10)≥2. For non-prime-powers: MOLS(n) ≥ n^{1/14.8} (MacNeish-Mann bound is not tight). The exact value of MOLS(10) is still unknown!
Step-by-Step Instructions
- 1Enter n.
- 2Generate MOLS.
- 3Check superposition.
- 4Count pairs.
- 5Compare bounds.