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

TL;DR

This paper analyzes a card guessing game after a single riffle shuffle, providing an optimal strategy and asymptotic expected reward, thus addressing an open problem in the field.

Contribution

It introduces the optimal guessing strategy for the game and derives the asymptotic expected reward, partially solving an open problem by Bayer and Diaconis.

Findings

Optimal guessing strategy identified

Expected reward asymptotically $n/2 + ext{constant} imes \sqrt{n}$

Addresses an open problem in card shuffling theory

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.