UNO is hard, even for a single player

Authors:
Erik D. Demaine;Martin L. Demaine;Nicholas J. A. Harvey;Ryuhei Uehara;Takeaki Uno;Yushi Uno
Affiliations:
MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA;MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA;Department of Computer Science, Faculty of Science, The University of British Columbia, 2329 West Mall, Vancouver, B.C., V6T 1Z4, Canada;School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan;National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai 599-8531, Japan
Venue:
Theoretical Computer Science
Year:
2014

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Abstract

This paper investigates the popular card game UNO^(R) 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.