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CHAPTER 3

MATH AND LOGIC

IN GAMES

Fortunately, the math used in game design is relatively simple: pri-

marily addition and subtraction with occasional multiplication. e

math used by programmers, however, is much more complex and in-

volves calculus, analytic geometry, quaternions, and other advanced

subjects. Even though a designer doesn’t need to be procient in

advanced math subjects, it does help to have a basic understand-

ing of them in order to converse knowingly with the programming

team during production. It is also important to have a good sense of

logic, especially Boolean logic, which is used in scripting languages

to change conditions on the playeld during play.

Probability and Statistics

One branch of mathematics a designer would be wise to study is

probability and statistics. Many game functions involve some kind

of random number generation, and a designer should understand

the basics of probability—how likely it is that a given result will hap-

pen. In some games where multiple random outcomes may result

from one game mechanic—for example, determining a critical hit

in a role-playing game—understanding the probabilities for possible

results is important to successfully balancing the game.

Statistics, which can be used to analyze results aer they happen,

is also an important design tool for balancing game play. If testing

reports that a critical hit against enemies happens multiple times per

combat encounter, then the probability for a critical hit is likely to

be too high. During the initial design process a designer can use a

paper prototype to test some of the more important game mechanics

to see if they are within the boundaries he or she imagined. Playing

Math and Logic in Games

36

with the numbers early on in the process can save considerable work

during testing and debugging.

One drawback in taking a class in probability and statistics is

that it usually assumes the student already has a basic understanding

of calculus, which is used to nd many solutions. However, there are

online websites that can oer some basic understanding of probabil-

ity without getting into too much detail.

Coin Flipping

e simplest probability is the ip of a coin, where there are only

two possible results (ignoring the probability of the coin landing on

its edge)—heads or tails. ere is a 50% chance of either result with

a ip. If the coin is ipped twice, there is still a 50% chance that the

result will be heads or tails. However, to see how oen either two

heads or two tails come up, a simple chart can be created with the

possible results (see Table 3.1)

ere is a 25% chance for any of these results. However, the like-

lihood of a coin coming up twice either heads or tails is only half that

of getting one heads and one tails. at is, there is a 50% chance of

tossing a heads and tails and only a 25% chance of tossing either two

heads or two tails.

As a coin is ipped more oen, it becomes less and less likely

to get all heads or all tails. With three ips, there are eight possible

results, and there is a two-in-eight chance (25%) to get all heads or

all tails, and a 75% chance to get a mixed result. Likewise, with four

ips, there are sixteen possible results, and there is a two-in-sixteen

chance (12.5%) to get all heads or all tails, and an 87.5% chance to

get a mixed result. Each time an extra ip is added, the results are to

First Toss Second Toss

Heads Heads

Heads Tails

Tails Heads

Tails Tails

Table 3.1. Results for two coin ips.

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