Combinatorics and Graph Theory · Problem 4.3.9

Systems of Distinct Representatives — David Guichard

Mathematics — Combinatorics · latin-squares

Problem

The transpose

A^\top

of a Latin square

A

is the reflection of

A

across the main diagonal, so that

A^\top_{i,j} = A_{j,i}

. A Latin square is self-orthogonal if

A

is orthogonal to

A^\top

. Show that there is no self-orthogonal Latin square of order 3. Find one of order 4.