Solitaire of Independence

TL;DR

This paper develops a general theory of a reversible puzzle-like process called solitaire on groups, focusing on the plane, providing algorithms, characterizations, and connections to symbolic dynamics.

Contribution

It introduces a comprehensive framework for solitaire processes on groups, analyzes specific cases on ^2, and links these to subshifts and independence properties.

Findings

Polynomial time algorithm for normal form in ^2

Complete characterization of line orbit via fill matrix

Cubic diameter of the line orbit graph

Abstract

In this paper, we study a reversible process (more precisely, a groupoid/group action) resembling the classical 15-puzzle, where the legal moves are to ``move the unique hole inside a translate of a shape $S$''. Such a process can be defined for any finite subset $S$ of a group, and we refer to such a process as simply ``solitaire''. We develop a general theory of solitaire, and then concentrate on the simplest possible example, solitaire for the plane $\mathbb{Z}^2$, and $S$ the triangle shape (equivalently, any three-element set in general position). In this case, we give a polynomial time algorithm that puts any finite subset of the plane in normal form using solitaire moves, and show that the solitaire orbit of a line of consecutive ones -- the line orbit -- is completely characterised by the notion of a so-called fill matrix. We show that the diameter of the line orbit, as a graph with edges the solitaire moves, is cubic. We show that analogous results hold for the square shape, but indicate some shapes (still on the group $\mathbb{Z}^2$) where this is less immediate. We then explain in detail the connection of the solitaire to TEP and more generally permutive subshifts. Namely, the solitaire is a closure property of various sets of subsets of the group that can be associated to such a subshift, such as the independence, spanning and filling sets.