Polyominoes with maximally many holes

TL;DR

This paper investigates the maximum number of holes a polyomino with n tiles can enclose, providing exact values for specific sequences and asymptotic bounds showing the maximum is approximately n/2.

Contribution

The paper derives exact maximum hole counts for specific polyomino sizes and establishes asymptotic bounds, advancing understanding of polyomino hole complexity.

Findings

Exact maximum holes for specific polyomino sizes.

Asymptotic bound showing maximum holes grow roughly as n/2.

Provides formulas for sequences of polyominoes with many holes.

Abstract

What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$ for $k \ge 1$. We also give nearly matching upper and lower bounds for large $n$, showing as a corollary that $f(n) \approx n/2$.