On Card guessing after a single shelf shuffle

Abstract

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is shelf-shuffled exactly one time. One after the other a single card is drawn from the shuffled deck. The guesser makes has guess and the card is shown until no cards remain. We provide a distributional analysis of the number of correct guesses under the optimal strategy. We re-obtain the previously derived expectation and add a complete description of the distribution. We also obtain a central limit theorem for the number $n$ of cards tending to infinity. Furthermore, we discuss an unbalanced, biased shelf shuffle and show how to derive the extend our analysis, also adding the complete position matrix. Finally, a refined analysis of the number of correct guesses is carried out, distinguishing between pure luck guesses and certified correct guesses.