Bounding the number of holes required for folding rectangular polyominoes into cubes

TL;DR

This paper investigates the minimal number of holes needed in rectangular polyominoes to ensure they can be folded into a cube using grid-line creases, revealing bounds based on hole types.

Contribution

It characterizes the minimal sets of holes guaranteeing cube-foldability, showing bounds for uniform holes and unboundedness when multiple hole types are allowed.

Findings

Minimal sets of same-type holes have size at most 4.

Different hole types can lead to arbitrarily large minimal sets.

Holes of sufficient size guarantee cube-foldability.

Abstract

We study the problem of whether rectangular polyominoes with holes are cube-foldable, that is, whether they can be folded into a cube, if creases are only allowed along grid lines. It is known that holes of sufficient size guarantee that this is the case. Smaller holes which by themselves do not make a rectangular polyomino cube-foldable can sometimes be combined to create cube-foldable polyominoes. We investigate minimal sets of holes which guarantee cube-foldability. We show that if all holes are of the same type, the these minimal sets have size at most 4, and if we allow different types of holes, then there is no upper bound on the size.