# Recreational Math

Posted on the 31 July 2011 by Erictheblue

Scraping around for a birthday present for one of my nephews, a college student majoring in computer science, I picked out a copy of Martin Gardner'My Best Mathematical and Logic Puzzles.  And since I'm pretty geeky myself, I couldn't help scanning it before the time came to wrap it in paper.  The first problem didn't interest me much.  Here's the second:

Two men play a game of draw poker in the following curious manner.  They spread a deck of 52 cards face up on the table so that they can see all the cards.  The first player draws a hand by picking any five cards he chooses.  The second player does the same.  The first player may now keep his original hand or draw up to five cards.  His discards are put aside out of the game.  The second player may now draw likewise.  The person with the higher hand then wins.  Suits have equal value, so that two flushes tie unless one is made of higher cards.  After a while the players discover that the first player can always win if he draws correctly.  What hand must this be?

That's down my alley.  I think I have an idea of how the first few games might go.  Since an ace-high straight flush is the highest hand in 5-card draw, the first player might in the first game thoughtlessly begin by choosing the ace, king, queen, jack and ten of a certain suit.  You can't lose with that.  But, according to the rules of this eccentric poker game, it quickly becomes evident that you can't win with it, either.  The second player need only choose the same  five cards of one of the other three suits.  It's easy to see how that game ends.  It's a tie.

The problem with the first player's first strategy is that it left the highest possible hand open to the other player.  The first player might prevent that in the second game by choosing, for his first hand, all four aces and one king.  Obviously the ace-high flush is now possible only for player one.  It turns out, however, that the first player's second strategy is worse than his first.  The second player should choose all four queens.  The highest possible hand available to player one is now a jack-high straight flush.  Meanwhile, player two is set up to fill in a queen-high straight flush on his second move.  The only way to prevent it is a defensive move that leaves player one with a hand lower than four queens.  Check mate.  Player two wins.

The way that hand works should, however, suggest a winning strategy to player one.  Picking four tens prevents player one from achieving an ace-high straight flush just as surely as picking four aces did.  Player two can now not make a hand higher than a 9-high straight flush, whereas player one is set up to fill in a 10-high straight flush on his second move.  Trying to block him by choosing four nines doesn't work (because player one, discarding three of his four tens, fills in an ace-high straight flush on his second move; player two cannot achieve a tie as none of the tens are available to him).  Of course, trying to block him by choosing four aces, or four kings or queens or jacks, is also useless (because player one uses his second move to complete a 10-high straight flush, and, as before, the highest hand availabe to player two is a 9-high straight flush).

Is there something a little off about the wording of the problem?  "What hand must this be?"--makes it sound as if there is only one hand that guarantees victory when actually four tens and any fifth card will do it.  And you know what?  The first player can guarantee victory by choosing just three tens on his first play, but in that case he has to exercise some care in the choice of the other two cards--the jack and the six, for example, from the suit in which he lacks the ten will work.  I'll leave it to my dweeby readers to  work out all the details.

If you find these kinds of problems difficult, here is an interesting conversation that should make you feel better.  Almost everyones' instincts are wrong when it comes to puzzlers of the kind that interest recreational mathematicians and, increasingly, neuroscientists.  Our minds, the product of evolution by natural selection, are built to cope with other challenges.