3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with T⊆L. We say that T is a latin trade if there exists a partial latin square T′ with T′∩T=∅ such that (L⧹T)∪T′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m⩾3. This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three.