Richard Garfield

Summary: Mathematician Richard Garfield applied his understanding of combinatorics to create Magic: The Gathering, the game responsible for the collectible card game craze.

Richard Garfield was born on June 26, 1963, and is a mathematician and an inventor of card and board games. He holds a B.S. degree in computer mathematics and a Ph.D. in combinatorial mathematics and has worked as a professor at Whitman College. Garfield is probably best known for creating Magic: The Gathering, the first widely popular collectible card game, but he has also designed board games and many other card games—collectible or not.

It should not be surprising that a mathematical background would be useful in designing games. The connections between mathematics and music, architecture, dance, and other forms of art is somewhat well understood. The role of mathematics in the art of game design is at least as direct. The aesthetics of a game come from the dynamics and combinatorial interaction of its rules and various components. This principle is especially true of so-called collectible card games, such as Magic: The Gathering, in which each player’s creativity in designing his or her own deck is part of the game. The rules and the library of cards must have sufficient flexibility to accommodate a large variety of strategies and styles (keeping play interesting and dynamic), but sufficient control to prevent a single overpowered card or combination of cards from “breaking” the game.

Collectible Card Games

In a collectible card game, such as Magic: The Gathering, players buy packages containing random cards from a large universe of possible cards. Some cards are much more rare and others much more common. Players organize their cards into decks according to certain guidelines and play casually against friends or competitively at official tournaments. In order to keep the game dynamic, the universe of cards periodically grows as expansions are released and older cards are retired. Expansions are still developed today, though Garfield’s direct involvement is limited and intermittent.

Much has been written about the mathematics that underlies the gameplay of Magic: The Gathering. Many relevant mathematical ideas come from combinatorics and probability, and a recurring theme is trade-offs. Including many copies of a card increases the chances of drawing it at a crucial time, but at the expense of having a smaller variety of different cards to deal with unexpected situations. An important game mechanic in Magic is that cards come in five different colors. Different-colored cards tend to have different types of effects and require different “energy” in order to function. Multicolored decks can be much more flexible, but at the expense of being much more likely not to have the energy you need at a key moment.

Richard Garfield created several other collectible card games, including Vampire: The Eternal Struggle, Netrunner, BattleTech CCG, and the Star Wars Trading Card Game. Though these have enjoyed some success and favorable opinions from critics, none can boast the mainstream attention that Magic: The Gathering received. In Richard Garfield’s most famous board game, Robo Rally, players navigate robots around an obstacle course. Players construct programs for their robot out of instruction cards like “turn left” and “forward two spaces,” then all robots simultaneously attempt to execute their instructions. If a player has miscalculated, or if multiple robots attempt to use the same paths at the same time, the results can be unexpected. This is a challenging game, requiring players to develop skills in game theory, logic, sequential and spatial reasoning, and the basic concepts of computer programming.

Not all of Richard Garfield’s creations require the player to use mathematical skills in a conscious way. In The Great Dalmuti, players match cards from a special deck, racing to empty their hands and get the most prestigious status among the group. After each hand, players’ relative rank may change, causing changes to the seating order and their privileges within the game. The elegance of the game is that it is easier to advance to a higher rank than it is to consistently hold on to the highest rank. In this case, the mathematical structure operates “quietly in the background,” ensuring that the game remains dynamic and engaging.

In some cases, the mathematics in Richard Garfield’s games is more explicit. One such game is Complex Hearts, a complex-number-themed variation on the classical game of Hearts. As in the original game, players score points based on which cards they take. However, in this version, the scores can be positive, negative, or imaginary, depending on the cards and card combinations, so that each player’s total score is a complex number. The goal of the game is to keep the magnitude of one’s score as low as possible.

Bibliography

Bosch, Robert A. “Optimal Card-Collecting Strategies for Magic: The Gathering.” College Mathematics Journal 31, no. 1 (2000).

Prywes, Jon. “The Mathematics of Magic: The Gathering: A Study in Probability, Statistics, Strategy, and Game Theory.” http://www.kibble.net/magic.