**Author: Miriam Melnick, Bard College at Simon’s Rock**

- Introduction to the game by constructing the deck in a series of algorithmic steps.
- Discussion of how to play game and how to expand it from the standard 4 dimensions into n dimensions.
- Model of mathematical nature of SET using strings of 0's, 1's and 2's.
- Investigation of circumstances under which there are no matches visible.

"Pure mathematics is the world’s best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly." - Richard Trudeau, *Dots and Lines.*

**1 Introduction**

In the game of SET, players race to collect sets of 3 matching cards. Sometimes, they get stuck and the players claim there are no matches on the table. Is this true? Are there circumstances in which there are no matches visible? What conditions must be satisfied for this to occur? In this paper, we develop algebraic, geometric, and computational frameworks to answer these questions.

It is always important to have precise definitions for our terms. You are likely familiar with the common mathematical concept of a "set" as a collection of objects surrounded by curly brackets. Throughout this paper, we will use the following terminology to describe SET:

**set ** An unordered collection of objects. The traditional mathematical notion of a set. Denoted with curly brackets.

**SET** A card game played with a special n-dimensional deck.

**match**^{1} A set of three SET cards that conform to the SET rule as described below. Denoted using square brackets.

We introduce the game of SET by constructing the deck in a series of algorithmic steps (see Section 2.3). From there we will discuss how to play the game of SET and how to expand it from the standard 4 dimensions into n dimensions (see Section 2.5). Then we discuss the mathematical nature of SET and how we can model it using strings of 0’s, 1’s, and 2’s (see Section 3). We will investigate the circumstances under which there are no matches visible (see Section 3.4). We will model n-dimensional SET using the vector space Zn3 and discuss its algebraic structures (see Section 4). Then we will model n-dimensional SET using the projective affine space AG(n, 3) and discuss the geometric consequences (see Section 5). Finally, we will examine how many cards can be dealt and still have no match. We will approach this problem primarily from a computational direction (see Section 6).

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