# Exploration of the Game of SET and Possible Variations

#### Abstract

SET is a card game in which the deck consists of 81 cards, each with four characteristics, and for every characteristic there are three variations. The goal of the game is to create SETs by picking three cards that either match variations or all have different variations for each characteristic. Due to the nature of the game there are many different ways to look at it in a mathematical sense. We are trying to answer how many cards can there be without forming a SET, and how can variations affect this number. Any number of cards larger than this will guarantee a SET. It has been shown that this maximum for the standard game is 20 cards, but how do variations to the game of SET itself affect this number? For example, Lent Davis and Maclagan have shown that if there are 5 characteristics instead of 4, then that maximum number of cards jumps up to 45 (2003). With 6 characteristics it jumps up to 112 (Follett et al 2014). This paper will also explore other variations that affect these numbers such as the number of variations per characteristic or the number of cards used to make a SET. We will attempt to explore these problems using affine geometry and programming. By translating the cards into points in affine spaces we can see the relationship between the rules of affine geometry and the rules of the game. This relationship is that cards that are SETs are collinear points and collinear points are cards that are SETs. By making changes to the affine space we can simulate changes in the deck. With the code we take advantage of a relationship that the point values of any three cards, from a standard deck, will be equal to 0 when summed mod 3 when testing multiple cards and one characteristic. This is used to check every characteristic on the cards and compared with the other cards to see if the group of cards is a SET or not. We found a low estimate for the maximum size of the playing field with six characteristics in the deck. We find these results to be significant because it is the first time approaching this problem from a coding perspective.

*This paper has been withdrawn.*

Exploration of the Game of SET and Possible Variations

SET is a card game in which the deck consists of 81 cards, each with four characteristics, and for every characteristic there are three variations. The goal of the game is to create SETs by picking three cards that either match variations or all have different variations for each characteristic. Due to the nature of the game there are many different ways to look at it in a mathematical sense. We are trying to answer how many cards can there be without forming a SET, and how can variations affect this number. Any number of cards larger than this will guarantee a SET. It has been shown that this maximum for the standard game is 20 cards, but how do variations to the game of SET itself affect this number? For example, Lent Davis and Maclagan have shown that if there are 5 characteristics instead of 4, then that maximum number of cards jumps up to 45 (2003). With 6 characteristics it jumps up to 112 (Follett et al 2014). This paper will also explore other variations that affect these numbers such as the number of variations per characteristic or the number of cards used to make a SET. We will attempt to explore these problems using affine geometry and programming. By translating the cards into points in affine spaces we can see the relationship between the rules of affine geometry and the rules of the game. This relationship is that cards that are SETs are collinear points and collinear points are cards that are SETs. By making changes to the affine space we can simulate changes in the deck. With the code we take advantage of a relationship that the point values of any three cards, from a standard deck, will be equal to 0 when summed mod 3 when testing multiple cards and one characteristic. This is used to check every characteristic on the cards and compared with the other cards to see if the group of cards is a SET or not. We found a low estimate for the maximum size of the playing field with six characteristics in the deck. We find these results to be significant because it is the first time approaching this problem from a coding perspective.