Morpion Solitaire 5D: a new upper bound of 121 on the maximum score

TL;DR

This paper establishes a new upper bound of 121 moves for Morpion Solitaire 5D using line-based analysis, improving upon the previous bound of 138 derived from potential-based methods.

Contribution

It introduces a novel line-based analytical approach to determine upper bounds in Morpion Solitaire 5D, surpassing prior potential-based bounds.

Findings

Proves an upper bound of 121 moves for 5D variant

Introduces line-based analysis method

Improves previous upper bound of 138

Abstract

Morpion Solitaire is a pencil-and-paper game for a single player. A move in this game consists of putting a cross at a lattice point and then drawing a line segment that passes through exactly five consecutive crosses. The objective is to make as many moves as possible, starting from a standard initial configuration of crosses. For one of the variants of this game, called 5D, we prove an upper bound of 121 on the number of moves. This is done by introducing line-based analysis, and improves the known upper bound of 138 obtained by potential-based analysis.