Distances of group tables and latin squares via equilateral triangle dissections

Denote by gdist ( p ) the least non-zero number of cells that have to be changed to get a latin square from the table of addition modulo p. A conjecture of Drápal, Cavenagh and Wanless states that there exists c > 0 such that gdist ( p ) ⩽ c log ( p ) . In this paper the conjecture is proved for c ≈ 7.21 , and as an intermediate result it is shown that an equilateral triangle of side n can be non-trivially dissected into at most 5 log 2 ( n ) integer-sided equilateral triangles. The paper also presents some evidence which suggests that gdist ( p ) / log ( p ) ≈ 3.56 for large values of p.