A standard deck of 52 cards is shuffled and split into two piles of 26 cards each.
Two players cooperate, they must score as many tricks as possible.
The first player draw 5 cards from the first pile, he then plays one of them face down, and the card from the top of the second pile is played face down. The two cards are shuffled and turned face up. The second player must now guess which of the two cards the first player played. The first player indicate whether the guess was correct. If so, the players score the trick. The first player then draw a new card from the first pile, so that he has 5 cards in hand, and play a new card face down etc. The process is repeated until the second pile is empty and 26 tricks have been played. As the first pile runs out, the last 4 rounds are played with a reduced hand size.
The players have no way of passing information to one another during the game beyond what is explicitly stated above. They may however agree on a strategy before the game begins.
What is the optimal strategy for scoring as many tricks as possible? And how many tricks will this strategy score on average?
One could think that there would be a hard limit on the probability of scoring any trick at $5/6$, dropping to $4/5$, $3/4$, $2/3$ and $1/2$ on the last tricks, thus making the search for any strategy yielding more than $21.05$ tricks on average futile. One would be wrong.