Triangle solitaire

TL;DR

This paper introduces a polynomial-time algorithm for normalizing finite plane subsets using a triangle-shaped solitaire, characterizes the line orbit with fill matrices, and analyzes the orbit's diameter.

Contribution

It develops a new polynomial-time normalization algorithm for triangle solitaire and characterizes the line orbit using fill matrices, providing insights into its diameter.

Findings

Polynomial-time normalization algorithm for triangle solitaire

Complete characterization of the line orbit via fill matrices

Cubic bound on the diameter of the line orbit

Abstract

The solitaire of independence is a groupoid action resembling the classical 15-puzzle, which gives information about independent sets of coordinates in a totally extremally permutive subshift. We study the solitaire with the triangle shape, which corresponds to the spacetime diagrams of bipermutive cellular automata with radius 1/2. 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 fill matrix. We show that the diameter of the line orbit under solitaire moves is cubic.