## A Poker Puzzle

You need knowledge of poker hand values to access this puzzle. Here's a chart for the uninitiated:

Two players play a game. The first player picks any 5 cards that they like from the face up pack. The second player then takes whichever 5 cards they like from the remainder of the pack. The first player then may permanently discard any or all cards in their hand and replace them with what's cards from what is left of the pack. Finally the second player does the same, leaving both players with a 5 card hand. These hands are compared and the higher one wins. If both players have the same value hand then the second player wins the tie.

Which player has a winning strategy and what should they pick?

Answers below.

...

I came across this puzzle in a Martin Gardner book, but I'm not sure whether it was originally his. The first player does have a win, here's the logic. If the first player (let's say you) picks a royal flush straight away then player two picks a royal flush as well and they will win.

You need to pick a hand which will prevent the other player picking a royal flush on the first turn, so you are going to have to pick one from each suit. Every straight flush contains either a 10 or a 5, so if the first player picks all four 10s and a random extra card then player two is left with no good options. If they take a straight flush ending in at most 9 then you can gain a royal flush. However if they block your royal flush by taking all 4 jacks (or similar), then you can take a lower straight flush.

Notice that if the first player had aimed higher than 4 10s on the first turn, with say 4 aces, kings, queens or jacks then the second player can take the matching 4 directly below and claim themselves a straight higher than the first player's.