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 shuffles and placed back on the table, face down. A guesser tri es 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 part icular guess was correct or not. The goal is to maximize the number of corr ect guesses. We show that, for k greater than or equal to 2 log(2)(2n) + 1, the best strategy is to guess card 1 for the first half of the deck and ca rd 2n for the second half. This result can be interpreted as indicating tha t 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 bes t.