The Operation of SET Explained
In the realm of board games, few have captured the hearts and minds of players quite like SET. First introduced to U.S. retailers in 1990, this intriguing card game quickly became a worldwide phenomenon, selling hundreds of thousands of copies since its debut.
At its core, SET is a game of patterns and combinations, where each card is characterized by four features, each taking one of three possible values. This setup is reminiscent of set theory, a branch of mathematics that studies sets and relationships between them. However, the mathematical structure of SET is more closely related to algebraic structures defined on sets, specifically vector spaces over finite fields, rather than directly through axiomatic or naive set theory.
Each card in SET can be thought of as a vector with four coordinates, where each coordinate is in {0, 1, 2} (mod 3). The defining feature of a SET is that three cards form a SET if and only if their component-wise sum is zero in the four-dimensional vector space over the finite field \(\mathbb{Z}_3\), denoted \(\mathbb{Z}_3^4\). This means that for each feature, the three cards either all have the same value or all different values, which algebraically means their values sum to 0 mod 3.
Another perspective is that if \(a, b, c\) are vectors representing cards, then \(a, b, c\) form a SET if they form an arithmetic progression in the space \(\mathbb{Z}_3^4\), that is, the difference vectors \(b - a\) and \(c - b\) are the same. This highlights the group structure underlying the game.
The concept behind SET can be generalized to other groups where sets correspond to tuples of elements whose combined group operation yields the identity.
While the term "set" in the game’s name refers to a subset of cards satisfying this algebraic condition, it is more a combinatorial or algebraic use of "set" than a connection to formal set theory, the foundational mathematical theory about collections of elements. However, the game inherently involves working with sets of cards and their combinations, which fits within the broad context of discrete mathematics where set theory is foundational.
SET is not just a game of chance; it challenges players to develop their pattern recognition, quick-thinking, and concentration skills. The game can be played solo or with multiple players, racing to find the most SETs in the least amount of time. The New York Times even offers an online multiplayer version of SET daily.
In case of multiple players, one point is awarded per SET found (and one may be subtracted for each invalid SET pointed out); the player with the most points at the end of the game wins. The game continues until the deck of 81 cards is depleted and all possible SETs are made.
SET was invented in 1974 by Marsha Jean Falco and has since become a staple in mathematics club meetings and classrooms, encouraging students to develop their thinking skills and gain hands-on understanding of mathematical theory. With its rich mathematical structure and engaging gameplay, SET continues to captivate players around the world.
In the realm of gaming, SET's popularity is evident, especially among mathematics enthusiasts, as it incorporates smart-home-devices-like patterns and combinations reminiscent of gadgets used in technology and algebraic structures. The game SET, despite its name, follows a combinatorial or algebraic use of 'sets' rather than formal set theory. However, it invites players to work with 'sets' of cards and their combinations, fitting within the context of discrete mathematics where set theory is foundational.
