Polyiamonds and Polyhexes with Minimum Site-Perimeter and Achievement Games

Abstract

An animal is an edge connected set of finitely many cells of a regular tiling of the plane. The site-perimeter of an animal is the number of empty cells connected to the animal by an edge. The minimum site-perimeter with a given cell size is found for animals on the triangular and hexagonal grid. The formulas are used to show the effectiveness of a simple random strategy in full set animal achievement games.

Metastability for the Ising model on the hexagonal lattice

“…To this end we provide some results concerning polyiamonds of minimal perimeter and maximal area, complementing those already present in the literature (e.g. [30,42,26]). In particular we show that quasi-hexagonal polyiamonds are the unique shapes that maximize the area for fixed perimeter and minimize perimeter for fixed area for different notions of perimeter.…”

Biased Weak Polyform Achievement Games

“…The exterior boundary E(A) of the animal A is the set of cells outside of A but adjacent to a cell of A. The site-perimeter of A is the number of cells p(A) := |E(A)| in the exterior boundary (see [8,25]). Let δ be the size of the site-perimeter of a single cell.…”

“…Unbiased polyiamond games are studied in [4,5,8]. In this section we find the threshold sequences of polyiamonds up to size four.…”

“…(7) If the position has fewer than 3 unmarked goal cells, then the maker can win since he is allowed to mark 2 cells. (8) This means the breaker strategy might fail, so we stop. Note that the breaker strategy might actually work but require a more sophisticated analysis.…”

“…In this case P ositions is left empty. (8) We update the set of breaker marks with the current defensive marks. (9) We remove the current defensive marks from the set of empty cells.…”