**THE MAXIMUM NUMBER OF SETS FOR N CARDS**

**AND**

**THE TOTAL NUMBER OF INTERNAL SETS FOR ALL PARTITIONS OF THE DECK**

**By Jim Vinci**

**June 30, 2009**

**E-mail address:mathisfun@wowway.com**

**Introduction**

I never thought that a purchase of a card game would lead to any kind of mathematical adventure. However, it is evident, from the abundance of related articles and papers written, that SET has sparked an enormous amount of interest from non-mathematicians and mathematicians alike. Clearly, SET is no ordinary card game!

My first investigation into the mathematics behind SET began with a December 19, 1998 letter, written to the creators of the game, and concerning the odds statistics presented in the game’s instructions. Being an actuary by profession and an individual with a passion for mathematics, I was immediately drawn to the question of how to calculate the probability of no Set (the special meaning of this will be defined later) being present in twelve cards. This problem naturally led to the more general question of how to determine the maximum number of cards that could be chosen without a Set existing. I soon recognized that this was an extremely complex problem for which computer assistance would be required to evaluate the seemingly endless possible variations. Famous math problems, like the Four Color Theorem, were solved by the use of the computer, so why not this one? After all, counting problems are well suited to computer modeling!

But I also continued to optimistically hold on to the notion that the problem of determining the largest number of cards without a Set, and the possible maximum number of Sets for a given number of cards, could be formulated in algebraic terms. However, because other individuals, included the creators of the game, had already provided solutions to the original SET problem, I decided that it probably wasn’t worth any more investment of time.

Recently, my interest in the SET problem was rekindled after reading Mr. David Van Brink’s highly inspiring article on the subject. In that article, he demonstrates that a deck of N = 47 cards must contain a Set. While reading this article, it became apparent that the concept of parity was fundamental to formulating an alternate approach that might uncover the mathematical patterns behind SET.

This paper will cover the following topics:

- For the benefit of those new to SET, a brief explanation of the rules of the game and how Sets are formed.
- Development of a general formula for the total number of Sets that can occur when a deck of C
^{P}(P = number of properties in the deck, C = number of choices for each property) cards is partitioned into two piles, and Set counts are restricted to those that are found exclusively within each pile. - The thinking behind the clever visual solution for why any collection of 20 cards must contain a Set, including a discussion of the Set-blocking strategy.
- A proposed computer modeling method, referred to as the Consecutive Maximization Method, for use in identifying the largest possible Setless collection of cards from a P property deck.

Click on the attachment below to read the full article.