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### Math in Card Games

```
Date: 10/06/2000 at 10:11:25
From: Jack Joshi-Powell
Subject: Card games

How exactly is math involved in the playing, creating, and setting up
of card games?

I've checked many books and Web sites, but they talk about strategy,
not math.
```

```
Date: 10/06/2000 at 12:49:13
From: Doctor TWE
Subject: Re: Card games

Hi Jack - thanks for writing to Dr. Math.

Exactly what kind of card games are you talking about? I can think of
at least three categories: games using standard playing cards (4 suits
of 13 cards each) or similar decks (like Bridge, Cribbage, Poker,
etc.); games that use card decks specific to that game (like Old Maid,
Quartet, Set, etc.); and the new "customizable" or "collectible" card
games (like Pokemon, Magic: the Gathering, etc.) Whichever type of
card game you are referring to, they use math in many of the same
ways.

First, the designs of the cards are themselves an exercise in geometry
(along with a little bit of psychology and physiology). Designers must
consider the size of the cards. Too large or too small, and the cards
are impractical to hold in one hand.

The shape of the card is also a geometric consideration. Most cards
are rectangular with rounded corners, but what is a good height-to-
width ratio? Some games use other-shape cards, like square or round
cards, instead. Are these better or worse in terms of handling during
a game?

Then there's the symmetry of the cards. The faces of the cards in some
games (particularly the "customizable" card games) have no symmetry,
while others (particularly standard playing cards) have two-way or
near two-way symmetry, and yet others have four-fold symmetry. My wife
doesn't like playing with standard playing cards because she's
left-handed and the corner symbols on most playing card decks are
designed for right-handers. She spreads her card hand "backward" and
thus sees the blank corner instead of the card symbol. The card backs
are frequently symmetric geometric patterns as well.

Math is also involved in the design of the game in terms of winning
and losing. In many games, the designer wants the probability of each
player winning (assuming equally good strategies) to be equal, or as
nearly equal as possible. But determining whether the first player has
not be a very interesting game if the first player could always win.
For casino games like Blackjack, the house has an advantage. But if
the advantage is too large, players won't play the game; they'd lose
their money too quickly. Of course, if the house doesn't have the
advantage, the casino loses money and will go out of business. So
determining the probability of winning is an important step in the
design of a game.

Determining the best playing strategy also involves math. Knowing how
to determine the probability of the occurrence of random events can
help a player determine the best strategy for winning. The play of
many card games also requires basic arithmetic skills. In many games,
you have to add or subtract points. (For example, in Blackjack you
need to add the values of your cards and subtract it from 21.) Some
card games, like Twenty-Four, require the players to do mathematical
computations as part of winning the hand. (In the game Twenty-Four,
each card has 4 numbers on it. The first player to be able to make an
expression that equals 24 using the 4 numbers and basic arithmetic
operations wins the card.) Some "customizable" card games also have a
"casting cost" or equivalent requirement before a card can be put into
play. The player must determine what combination of cards (s)he can
afford to play on each turn.

Most card games also require some form of scorekeeping from round to
round or hand to hand. This often just involves simple arithmetic, but
that is math as well.

I hope this gives you some ideas as to where to start. Perhaps you can
then explore these areas in more depth for the particular game or type
of card game you're interested in.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
High School Symmetry/Tessellations
Middle School Probability

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