Minimum Guesswork with an Unreliable Oracle

We study a guessing game where Alice holds a discrete random variable X, and Bob is trying to sequentially guess its value. Before the game begins, Bob can obtain side-information about X by asking an oracle, Carole, any binary question of his choosing. Carole's answer is unreliable, and is incorrect with probability ε. We show that Bob should always ask Carole whether the index of X is odd or even with respect to a descending order of probabilities - this question minimizes all the guessing moments for any value of ε. This in particular settles a conjecture of Burin and Shayevitz. We further count the number of optimal questions, and discuss some extensions including asymmetric channels from Carole to Bob and to multiple-choice questions.