When I mention math to most college students who are interested in design, it comes as a surprise to them. Game design is a creative field, and creativity and math go together about as well as, um, chocolate and broccoli. Right?
Except that any time you hear the words "game balance" (as in, "this game isn't balanced" or "I need to tweak the balance of this level") you're really talking about math. Game balance means that the game is neither too difficult nor too easy, and while it can be done by trial and error, it goes much faster if the designer can put together some good mathematical models. Game balance is the designer's job, so a good designer needs to know at least enough math for that.
Two examples should suffice. In Civilization, each unit has its own cost, strength, defense, speed and several other numerical stats. If those numbers are out of whack, then players will find one particular unit type (or strategy) to be better than any other, and the game will quickly become boring. Similarly, in Final Fantasy (or any RPG of your choice), there's typically a huge database of numbers: monster stats, player stats, level progression charts, combat formulas and so on. Those are all math, and they all need to be designed.
So, what math does a designer need? I've found the following courses helpful:
Calculus. Calculus teaches you the math to describe and analyze how fast something is changing. It is therefore necessary if you’re trying to describe a variable in a game that changes over time. If you’ve ever heard talk of game “pacing” or the “difficulty curve” of a game, that’s calculus. Also, the entire field of Physics is based on calculus, so taking Calc will help greatly in your understanding of Physics (I'll talk more about Physics, and science in general, in a later post).
Linear Algebra. This gives you the tools to solve systems of equations using matrices, which is useful when you have several variables or stats in your game that you need to relate to each other. It’s also useful for solving certain types of game-balance problems, like Rock-Paper-Scissors-like ("intransitive") game mechanics. It’s also used in computer graphics for rotation and scaling, which you might encounter at some point.
Intro to Probability. The field of Probability was created to study games (gambling games in particular), so this shouldn't be much of a surprise. Any game with randomness requires probability to describe the exact nature of that randomness. Rating systems (or any other form of player ranking) also require probability, to show if they’re fair or not.
Intro to Game Theory. Sadly, “Game Theory” has very little to do with game design; it’s a branch of mathematics that deals with particular kinds of probability questions (particularly those that involve multiple players making simultaneous choices). If you’ve ever heard of the Prisoner’s Dilemma, that’s game theory. It’s useful when designing certain types of multiplayer dynamics, especially in boardgames or strategic computer games.