No-feedback Card Guessing for Dovetail Shuffles

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 shuues 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 2 log 2 (2n) + 1 the best strategy is to guess card 1 for the rst half of the deck and card 2n for the second half. This result can be interpreted as indicating that it suuces to perform the order of log 2 (2n) shuues to obtain a well mixed deck, a fact proved by Bayer and Diaconis 3]. We also show that if k = c log 2 (2n) with 1 < c < 2 then the above guessing strategy is not the best.