This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
In Face-Up Poker, two players each select five cards from a face-up deck,
bet, discard and draw. Is there a winning strategy for this game? What if
the players select cards alternately?
If the first player draws four aces, the second player draws four
kings. If the first player keeps the four aces on the draw, the second
player draws a king-high straight flush, and if the first player
pitches the aces to draw a straight flush, the second player can always
make a higher straight flush.
Instead, the winning strategy is for the first player to draw four
tens. The second player cannot draw a royal flush, and in order to
prevent the first player from getting one, the second player must draw
at least one card higher than the ten from each suit, which means he
can't do better than four-of-a-kind. Then the first player wins by
drawing a straight flush from any suit.
If the cards are dealt alternately as in real poker, the second player
can always tie with proper strategy. The second player mirrors the
first player's selections in rank and color. For example, if the first
player picks up a red queen, the second player picks up a red queen.
When they are done playing, their hands will be identical except one
will have spades and hearts where the other has clubs and diamonds, and
vice versa. Since suits aren't ranked in poker, the hands are tied.
It is unknown if there is a winning strategy if the replacement cards
are dealt together as in real poker, as opposed to alternately.