We consider the oblivious distributed guessing of a random variable in three scenarios (single guessor, single constrained guessor, multiple guessors). The optimal guessing schemes in each case are obtained by means of a probability distribution which the guessor(s) use in order to determine their sequence of guesses. Some of the optimal distributions obtained have links to Re´nyi's generalization of Shannon's entropy, while one of them is related to the rth power means from mathematical analysis.