Guichard-Combinatorics-Graph-Theory · Problem 4.3.8

Systems of Distinct Representatives

Mathematics — Probability · latin-squares

Problem

n \times n

Latin square

A

is symmetric if it is symmetric around the main diagonal, that is,

A_{i,j} = A_{j,i}

for all

i

and

j

. It is easy to find symmetric Latin squares: every addition table modulo

n

is an example, as in example 4.3.6. A Latin square is idempotent if every symbol appears on the main diagonal. Show that if

A

is both symmetric and idempotent, then

n

is odd. Find a

5 \times 5

symmetric, idempotent Latin square.