Parity of normalized determinants of even-order Latin squares

For even-order Latin squares, determine when the centered determinant has an extra factor of two beyond the baseline normalization. The scoped target includes binary-rank and adjugate criteria and an odd normalized quotient for every n≡2 mod 4, n≥6.

Why it matters and what counts as progress

Why it matters: It narrows the determinant’s possible divisibility while keeping the separate minimum-attainment question open.

What counts as progress: Determine the sharp centered-determinant divisibility and exhibit families attaining it; settling the minimum determinant requires additional work.

How the problem developed

3 Aug 2014 · Open question [1]
Peter Recorded the determinant divisibility bound for regular Latin squares and asked when its minimum is attained, noting exceptions at orders 4 and 6.
18 Oct 2026 · Partial result [2]
Sabrina Babanskyy Reported binary-rank and adjugate criteria for an extra factor of two in even orders, and a family with odd normalized determinant quotient for n≡2 mod 4 and n≥6.

AI contribution

Determinant Divisibility of Centered Latin Squares: Binary Criteria, Universal Counterexamples, and p-Adic Bounds

Oleksiy Babanskyy

For even-order Latin squares, Babanskyy gives binary-rank and adjugate criteria for an extra factor of two in the centered determinant. An explicit family has an odd normalized determinant quotient for every n≡2 mod 4 with n≥6.

AI involvementAI-centered

ImportanceModest

Technical verificationFormal (exact computation + human)

Human understandingFull