#
The Joy of Set

or
#
Some Combinatorial Questions Related to the

#
Configuration of Points in Z_3^k

Mike Zabrocki

York University

##
*Set*®

The official web page for the game of *Set*®
can be found at http://www.setgame.com
. This site has information on how to obtain the card game, rules
of the game, links to other web sites with information about the game,
some mathematics and some trivia.

##
Advanced Set

Play the game here

I have written a Java version of the card game that can be played
within a web
page here that should work on Internet Explorer with a reasonably updated
version of Java or downloaded in
zip
format for Macintosh OS 9 (or earlier?...It should *just* activate
Apple Applet Runner).
Other platforms may download the
source
in .tar.gz format and modify it so that it works on their system.

##
The Joy of Set

There is also an html
and pdf version
of a document that explains the game of set, the extended version that
I have written here, and some mathematics that is associated with the game.
This pdf document along with the program was written to be presented at
FPSAC
'01 at Arizona State University in May, 2001. Feel free to contact me and
comment about this document. The following changes need to be added if I were
to ever update the paper.

The open problem number 1 that is in this document is not really open. Diane Maclagan
showed me a proof at FPSAC '01. I didn't come up with a solution when I was
writing the paper so I pegged it for "very hard" when it was really somewhere
between "not obvious" and "difficult." I have also received other nice solutions
from people who e-mailed me. The answer is (from one of the last e-mails I
received)

|I_k|=3^k * Product for i from 0 to k-1 of (3^k-3^i)
Also, the maximal number of cards with no set is called 'a maximal cap in *A(3,n)*' where
*n* is the number of properties. I did not know that this language when
I first wrote this. The size of a maximal cap of *A(3,n)* for n equal to 1 through 5
is
2, 4, 9, 20, 45. I believe that the answer is currently not known for n=6 but that
it is known to be between 111 and 115. Computationally this question is hard.

Diane Maclagan and
Bejamin Davis
have written a paper for
The Mathematical Intelligencer about the Game of Set.

Check out the an
article by Ivars Peterson.

Some other links:

Observations on the game of set

The SET® Home Page.

I would like to thank Andrew Rechnitzer for helpful suggestions and
encouragement. I would also like to thank Carol Chang, Murray Elder,
Nantel Bergeron, John Wild, and Jamal Ahmed for their advice, support,
equipment use and suggestions.
Home page of Mike Zabrocki

email: zabrocki(at)mathstat(dot)yorku(point)ca