Coloring and boundary invariants for polyominoes (Q6546696)

The paper addresses the problem of coloring polyominoes by utilizing Gaussian integers. The study introduces a mapping \( f: P \to \mathbb{Z}[i]/(v) \), where \( P \) represents the set of all polyominoes, and \( v \) is a Gaussian integer. Several examples are considered where such a mapping provides necessary conditions for the existence of a tiling.\N\NAdditionally, a new type of invariant called a boundary invariant, is defined. This invariant remains unchanged under translation, rotation, and reflection. The idea of a boundary invariant is to take a sum of the \( m \)-th powers of the Gaussian integers corresponding to the sides of a polyomino.\N\NThe author extends classical coloring techniques used for simple tiling problems. The paper presents several theorems, including:\N\begin{itemize}\N\item Theorem 1.1: Specifies conditions for tiling rectangles with hexominoes, requiring the area to be divisible by 18 and both dimensions to be at least 3.\N\item Theorem 1.2: Establishes that the area of a rectangle tiled by a 15-omino must be divisible by 45.\N\item Theorems 2.3 and 2.5: Detail the criteria for tiling with a set of hexominoes or dekominos.\N\end{itemize}\N\NThese theorems are derived using the concept of boundary invariants.