The Complexity of Rummikub Problems

TL;DR

This paper examines the computational complexity of forming optimal sets in Rummikub, presents a polynomial-time algorithm for the problem, analyzes its generalizations, and explores human play aspects through experiments.

Contribution

It introduces a polynomial-time algorithm for Rummikub set formation and analyzes how complexity changes with different input parameters.

Findings

The set formation problem can be solved in polynomial time.

Generalizations of the problem affect its computational complexity.

Experiments provide insights into human play strategies.

Abstract

Rummikub is a tile-based game in which each player starts with a hand of $14$ tiles. A tile has a value and a suit. The players form sets consisting of tiles with the same suit and consecutive values (runs) or tiles with the same value and different suits (groups). The corresponding optimization problem is, given a hand of tiles, to form valid sets such that the score (sum of tile values) is maximized. We first present an algorithm that solves this problem in polynomial time. Next, we analyze the impact on the computational complexity when we generalize over various input parameters. Finally, we attempt to better understand some aspects involved in human play by means of an experiment that considers counting problems related to the number of possible immediately winning hands.