No Feedback Card Guessing for Top to Random Shuffles

TL;DR

This paper investigates a card guessing game where cards are shuffled multiple times and guessed without feedback, providing strategies to maximize correct guesses after a certain number of shuffles.

Contribution

It introduces a new guessing strategy for top-to-random shuffles that optimizes expected correct guesses without feedback after many shuffles.

Findings

Optimal guessing strategy for m > 4n log n + c

Expected correct guesses increases with the number of shuffles

Strategy performance analyzed for large n and m

Abstract

Consider n cards that are labeled 1 through n with n an even integer. The cards are put face down and their ordering starts with card labeled 1 on top through card labeled n at the bottom. The cards are top to random shuffled m times and placed face down on the table. Starting from the top the cards are guessed without feedback (i.e. whether the guess was correct or false and what the guessed card was) one at a time. For m > 4nlog n+cn we find a guessing strategy that maximizes the expected number of correct guesses.