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1

On the hat problem on a graph

Krzywkowski M.

Opuscula Mathematica

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2012

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Vol. 32, no. 2

285-296

EN

The topic of this paper is the hat problem in which each of n players is uniformly and independently 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. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities.

2

The hat problem on a union of disjoint graphs

Krzywkowski M.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 2

167-176

EN

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. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem is known for cycles and bipartite graphs. We investigate the problem on a union of disjoint graphs.

3

A modified hat problem

Krzywkowski M.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 2

121-126

EN

The topic of our paper 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 a win. There are known many variations of the hat problem. In this paper we consider a variation in which there are n ≥ 3 players, and blue and red hats. Players do not have to guess their hat colors simultaneously. In this variation of the hat problem players guess their hat colors by coming to the basket and throwing the proper card into it. Every player has got two cards with his name and the sentence "I have got a red hat" or "I have got a blue hat". If someone wants to resign from answering, then he does not do anything. 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. Is there a strategy such that the team always succeeds? We give an optimal strategy for the problem which always succeeds. Additionally, we prove in which step the team wins using the strategy. We also prove what is the greatest possible number of steps that are needed for the team to win using the strategy.