The complexity of UNO

TL;DR

This paper models UNO using mathematical frameworks, revealing that single-player UNO is NP-complete while certain two-player versions are solvable in polynomial time, highlighting its computational complexity.

Contribution

It introduces formal models for UNO and analyzes their computational complexity, providing new insights into the game's algorithmic properties.

Findings

Single-player UNO is NP-complete.

Restricted cases of UNO are in P.

Two-player UNO can be solved in polynomial time.

Abstract

This paper investigates the popular card game UNO from the viewpoint of algorithmic combinatorial game theory. We define simple and concise mathematical models for the game, including both cooperative and uncooperative versions, and analyze their computational complexity. In particular, we prove that even a single-player version of UNO is NP-complete, although some restricted cases are in P. Surprisingly, we show that the uncooperative two-player version is also in P.