Academic Research | America's Favorite Card Games®

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# SET® AND MATRIX ALGEBRA

By Patricia J. Fogle, Ph.D., D.O.

How to use matrix algebra to build magic squares of SETs.

## By Patricia J. Fogle, Ph.D., D.O.

The use of statistics pervades the world in which we live. It is used arguably to defend positions in basic and applied scientific research, and ultimately affects all aspects of our lives. It therefore is important to understand the rationale and meaning of these "numbers" that affect our lives.

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

By Jim Vinci

This paper covers the following topics:

1. For the benefit of those new to SET, a brief explanation of the rules of the game and how Sets are formed.
2. Development of a general formula for the total number of Sets that can occur when a deck of CP (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.
3. 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.
4. 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.

Promoting Strategic Thinking Skills in Middle-School Students Using Set: The Family Game of Visual Perception®

Sylvia Sykes
Holy Names University, Oakland, CA

# Developing Mathematical Reasoning using Attribute Games

By Anne Larson Quinn, Ph.D., Associate Professor, Edinboro University, Quinna@edinboro.edu
Frederick Weening, Ph.D., Assistant Professor, Edinboro University, Fweening@edinboro.edu
Robert M. Koca, Jr., Ph.D.

Reproduced with permission from the Mathematics Teacher, copyright 1999 by the NCTM.

The game of SET® has proven to be a very popular game at our college mathematics club meetings. Since we've started playing, the membership has grown every month. In fact, one of our members brought her six year old son to a meeting, and he now looks forward to playing SET® with us every month. As a result of playing the game in our club and thinking about the results, we created and solved a variety of mathematical questions. For example, we wondered about possible strategies for winning and conjectured about phenomena that happened when playing. These questions involve a wide variety of traditional mathematical topics, such as the multiplication principle, combinations and permutations, divisibility, modular arithmetic, and mathematical proof.

# SET Recognition as a Window to Perceptual and Cognitive Processes

## MICHAL JACOB AND SHAUL HOCHSTEINHebrew University, Jerusalem, Israel

The Set visual perception game is a fertile research platform that allows investigation of perception, with gradual processing culminating in a momentary recognition stage, in a context that can be endlessly repeated with novel displays. Performance of the Set game task is a play-off between perceptual and conceptual processes. The task is to detect (among the 12 displayed cards) a 3-card set, defined as containing cards that are either all similar or all different along each of four dimensions with three possible values. We found preference and reduced response times (RTs) for perceiving set similarity (rather than span) and for including cards sharing the most abundant value in the display, suggesting that these are searched preferentially (perhaps by mutual enhancement).

# The Joy of SET

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

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

# Cognitive Modeling with SET

## How to Construct a Believable Opponent using Cognitive Modeling in the Game of Set

Niels A. Taatgen (niels@ai.rug.nl)
Marcia van Oploo (marcia@ai.rug.nl)
Jos Braaksma (j.braaksma@ai.rug.nl)
Jelle Niemantsverdriet (jelle@niemantsverdriet.nl)

Department of Artificial Intelligence, Grote Kruisstraat 2/1
9712 TS Groningen, Netherlands

Abstract

An interesting domain of application for Cognitive Modeling is the construction of computer opponents in games. We present a model of the game of Set. The model is sensitive to the difficulty of the situation in the game, and can explain the difference between beginners and experts. Furthermore, the model is used in a computer game in which players can play Set against various versions of the model.

# MATHEMATICAL FUN & CHALLENGES IN THE GAME OF SET®

By Phyllis Chinn, Ph.D. Professor of Mathematics
Dale Oliver, Ph.D. Professor of Mathematics
Department of Mathematics
Humboldt State University
Arcata, CA 95521

The Game of SET

# Mathematical Proof of Magic Squares

### By Llewellyn Falco

What you see here is a magic square, much like the addition and subtraction squares you may have used as a child.

These magic squares are even more talented, as they all follow the rules of the card game SET®. To learn how to make one with ease, read on.

SET® cards contain four properties: color, shape, number of objects, and shading. The rules state for each property, they must all be equal, or all different. For example, if we look at the top row of the square, we see three different colors, three different shapes, three different numbers, and three different types of shading within the objects. Need more examples? Any line on the magic square yields a set.