On card guessing game with one time riffle shuffle and complete feedback

Abstract: This paper studies the game of guessing riffle-shuffled cards with complete feedback. A deck of $n$ cards labelled 1 to $n$ is riffle-shuffled once and placed on a table. A player tries to guess the cards from top and is given complete feedback after each guess. The goal is to find the guessing strategy with maximum reward (expected number of correct guesses). We give the optimal strategy for this game and prove that the maximum expected reward is $n/2+\sqrt{2/\pi}\cdot\sqrt{n}+O(1)$, partially solving an open problem of Bayer and Diaconis.