A Simple Approach to the Tiling Problem Using Recursive Sequence

TL;DR

This paper introduces a recursive sequence-based method to analyze tiling problems on rectangular boards, deriving formulas for various sizes and a variation called tatami tiling, aiming for simpler solutions in combinatorial mathematics.

Contribution

The paper presents a recursive approach to solve tiling problems, including non-recursive formulas and extensions to general cases and tatami tiling variations.

Findings

Derived recursive formulas for tiling counts of 2x n, 3x n, 4x n boards.

Extended the approach to general k x n boards for certain configurations.

Provided initial solutions for tatami tiling variations on small boards.

Abstract

The tiling problem has been a famous problem that has appeared in many Mathematics problems. Many of its solutions are rooted in high-level Mathematics. Thus we hope to tackle this problem using more elementary Mathematics concepts. In this report, we start with the simplest cases, with the smaller numbers: the number of ways to tile a $2 \times n$, $3 \times n$, $4 \times n$ rectangular board using $2 \times 1$ domino tiles, where the number of rows is fixed and we present a recursive formula based on $m$ and the earlier terms. This allows us to deduce the non-recursive formula for each case that is only dependent on $m$. For each case, we also expand and generalize the problem, not just for $2$, $3$, $4$ but for any positive integer $k$, for certain types of configurations of the board. We also focus on one of the famous variations of the tiling problem: tatami tiling, and present a solution for simple cases: $2 \times n$, $3 \times n$, $4 \times n$. In the end, we have managed to find a simpler solution for three different configurations of the board, with some we even deduced the non-recursive formula. We have also solved simple cases of the tatami tiling problem, with the hope to tackle the general case in the future. We realized that our method only works on a case-by-case basis, with little success in solving the general case. This approach is also applicable in many other counting problems which we wish to pursue for further research.