A more colorful hat problem

Marcin Krzywkowski

Abstract


The topic is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. We consider a generalized hat problem with q >= 2 colors. We solve the problem with three players and three colors. Next we prove some upper bounds on the chance of success of any strategy for the generalized hat problem with n players and q colors. We also consider the numbers of strategies that suffice to be examined to solve the hat problem, or the generalized hat problem.

Keywords


hat problem

Mathematics Subject Classification


91A12

References


Aggarwal, G. et al. "Derandomization of auctions." Proceedings of the 37th Annual ACM Symposium on Theory of Computing. New York: ACM, 2005. 619-625,

Alon, N. "Problems and results in extremal combinatorics, II." Discrete Math. 308 (2008): 4460-4472.

Aspnes, J. et al. "The expressive power of voting polynomials." Combinatorica 14 (1994): 135-148.

Beigel, R. , L. Fortnow and F. Stephan. "Infinitely–often autoreducible sets." SIAM J. Comput. 36 (2006): 595-608.

"Berkeley Riddles." www.ocf.berkeley.edu/wwu/riddles/hard.shtml.

Bernstein, M. "The hat problem and Hamming codes." MAA Focus. November, 2001. 4-6.

Blum, W. "Denksport für Hutträger." Die Zeit. May 3, 2001.

Breit, M., D. Deshommes and A. Falden. "Hats required: perfect and imperfect strategies for the hat problem." manuscript.

Brown, E. and K. Mellinger. "Kirkman’s schoolgirls wearing hats and walking through fields of numbers." Math. Mag. 82 (2009): 3-15.

Burke, E., S. Gustafson and G. Kendall. "A Puzzle to challenge genetic programming." Genetic Programming. Lecture Notes in Computer Science. Springer, 2002. 136–147.

Butler, S. et al. "Hat guessing games." SIAM J. Discrete Math. 22 (2008): 592-605.

Cohen, G. et al. "Covering Codes." Mathematical Library 54. North Holland, 1997.

Ebert, T. "Applications of recursive operators to randomness and complexity." Ph.D. Thesis. Santa Barbara: University of California, 1998.

Ebert, T. and W. Merkle. "Autoreducibility of random sets: a sharp bound on the density of guessed bits." Lecture Notes in Comput. Sci. 2420. Berlin: Springer, 2002. 221-233.

Ebert, T. , W. Merkle and H. Vollmer. "On the autoreducibility of random sequences." SIAM J. Comput. 32 (2003): 1542-1569.

Ebert,T. and H. Vollmer. "On the autoreducibility of random sequences." Lecture Notes in Comput. Sci. 1893. Berlin: Springer, 2000. 333-342.

Feige, U. "You can leave your hat on (if you guess its color), Technical Report MCS04-03." Computer Science and Applied Mathematics. TheWeizmann Institute of Science, 2004.

Guo, W. et al. "The hat problem and some variations." Stat. Ind. Technol. Boston: Birkhäuser 2006. 459-479.

Immorlica, N. "Computing with strategic agents>" Ph.D. Thesis. Massachusetts Institute of Technology, 2005.

Lenstra, H. and G. Seroussi. "On hats and other covers." IEEE International Symposium on Information Theory. Lausanne, 2002.

Poulos, J. "Could you solve this $1 million hat trick?" abcNews. November 29, 2001.

Robinson, S. "Why mathematicians now care about their hat color." The New York Times. Science Times Section. April 10, 2001. page D5.


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