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November 1998 No-feedback card guessing for dovetail shuffles
Mihai Ciucu
Ann. Appl. Probab. 8(4): 1251-1269 (November 1998). DOI: 10.1214/aoap/1028903379

Abstract

We consider the following problem. A deck of $2n$ cards labeled consecutively from 1 on top to $2n$ on bottom is face down on the table. The deck is given k dovetail shuffles and placed back on the table, face down. A guesser tries to guess at the cards one at a time, starting from top. The identity of the card guessed at is not revealed, nor is the guesser told whether a particular guess was correct or not. The goal is to maximize the number of correct guesses. We show that, for $k \geq 2 \log_2 (2n) + 1$, the best strategy is to guess card 1 for the first half of the deck and card $2n$ for the second half. This result can be interpreted as indicating that it suffices to perform the order of $\log_2(2n)$ shuffles to obtain a well-mixed deck, a fact proved by Bayer and Diaconis. We also show that if $k = c \log_2 (2n)$ with $1 < c < 2$, then the above guessing strategy is not the best.

Citation

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Mihai Ciucu. "No-feedback card guessing for dovetail shuffles." Ann. Appl. Probab. 8 (4) 1251 - 1269, November 1998. https://doi.org/10.1214/aoap/1028903379

Information

Published: November 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0945.60002
MathSciNet: MR1661184
Digital Object Identifier: 10.1214/aoap/1028903379

Subjects:
Primary: 60C05 , 60J10

Keywords: Card guessing , dovetail shuffle , Riffle shuffle

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.8 • No. 4 • November 1998
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