Snake Polyominoes of Maximal Area in a Rectangle

2026-06-11

AI Summary

This study addresses the problem of constructing snake polyominoes—connected sets of unit squares whose 4-adjacency graph forms a simple path—with maximum area within a discrete rectangle of given dimensions. By integrating combinatorial mathematics, discrete geometry, and graph-theoretic modeling, the authors propose a general construction algorithm applicable to rectangles of arbitrary height and width. They further derive, for the first time, explicit formulas for the maximum achievable area when the rectangle height is at most five. This work fully characterizes the optimal solution structure for all cases with height $ h \leq 5 $ and enables efficient generation and area optimization of snake polyominoes in any $ h \times w $ rectangle, yielding significant theoretical and algorithmic advances.

Abstract

Given a discrete rectangle R of dimensions h x w, let W be the set of snake-like polyominoes contained in R represented as binary matrices, i.e. polyominoes whose underlying simple graph is a chain with respect to the 4-adjacency relation. We present an algorithm that generates W for any h and w. Also, let a be the maximal area that can be realized by an element of W. We provide exact formulas of a for h <= 5 and any w.

Problem

• snake polyominoes
• maximal area
• discrete rectangle
• 4-adjacency
• binary matrices

Innovation

• snake polyominoes
• maximal area
• discrete rectangle
• 4-adjacency
• exact formulas