Critical sets in back circulant latin squares.
Minimal and minimum size latin bitrades of each genus
Suppose that and are partial latin squares of order , with the property that each row and each column of contains the same set of entries as the corresponding row or column of . In addition, suppose that each cell in contains an entry if and only if the corresponding cell in contains an entry, and these entries (if they exist) are different. Then the pair forms a . The of is the total number of filled cells in (equivalently ). The latin bitrade is if there is no latin bitrade...
Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results
We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of mutually orthogonal Latin squares of order to construct a set of mutually orthogonal Latin squares of order .
Distinct equilateral triangle dissections of convex regions
We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex...
Page 1