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 
an advantage or disadvantage is an exercise in probability. It would 
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/