This paper is devoted to investigation of the hat problem on graphs with exactly three cycles. In the hat problem, each of n players is randomly fitted with a blue or red hat. 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. Note that every player can see everybody excluding himself. This problem has been considered on a graph, where the vertices correspond to the players, and a player can see each player to whom he is connected by an edge. We show that the hat number of a graph with exactly three cycles is 3/4 if it contains a triangle, and 1/2 otherwise.