Tilings of $(2\times2\times n)$-board with colored cubes and bricks

TL;DR

This paper investigates the enumeration of tilings of a 3D $(2\times2\times n)$ board with colored cubes and bricks, introducing recursive methods and identities that could aid mathematical education.

Contribution

It presents a recursive approach to count breakable and unbreakable tilings of 3D boards, extending tiling analysis into three dimensions with potential educational applications.

Findings

Derived recursive formulas for tilings of 3D boards

Established identities relating breakable and unbreakable tilings

Proposed methods useful for mathematical education

Abstract

Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three dimensions. After a short introduction and the definition of breakability we show a way to get the number of the tilings of an $n$-long board considering the $(n-1)$-long board. It describes recursively the number of possible breakable and unbreakable tilings. Finally, we give some identities for the recursions using breakability. The method of determining the recursions in space can be useful in mathematical education as well.