# Research progress for the game of Set (103 Section 11)

### Definitions, Rules, and Terminology

There are four categories of characteristics, each with three possible values:
• Color   (Red, Green, Purple - to be abbreviated as R, G, P)
• Shape   (Oval, Diamond, Squiggle - to be abbreviated as O, D, S)
• Fill   (Empty, Hatched, Full - to be abbreviated as E, H, F)
• Number  (1, 2, 3 - which obviously will be called 1, 2, 3)
A set is a collection of three cards that satisfies the following four rules:
• Rule C:   Each of the colors must be all the same or all different.
• Rule S:   Each of the shapes must be all the same or all different.
• Rule F:   Each of the fills must be all the same or all different.
• Rule N:   Each of the numbers must be all the same or all different.
A collection of 12 cards in play is called a deal.

A collection of some unspecified number of cards is called a party.

A collection of n cards is called an n-party. (Note that a "deal" is the same thing as a "12-party.")

A collection of cards in which each pair of cards is part of a set is called perfect.

Given a pair of cards, a card that makes a set when added to the pair is called a connector card for that pair.

An n-party that contains no sets is called an impossible n-party.

The game of Alpha-Set is an extension of the game of Set, where one extra category has been added (letters, with possible values A, B, and C). A triple of cards is an alpha-set if it is a set in the usual sense, with the added condition that the letters on the cards are either all the same or all different. All other definitions (such as perfect, impossible, etc) are generalized to alpha-sets in the obvious way.

Convention: Whenever we ask "how many sets are possible" in a particular context, we are allowing the sets to possibly overlap.

Convention: When we say that a set has "n characteristics in common", we mean that there are n characteristics for which all three cards in the set have the same value. (For example, a set all of whose cards are the same color, the same number, and the same shape is a set with three characteristics in common.)

(Note: The questions originally numbered 1, 2, and 3 did not end up being investigated by any group.)

4.   In a 12-card deal, how many sets are possible?

Suggested by David Centofante.   Researched by Group 4: Cassie Brown, Josh Brown, David Centofante, Jason Itam, and Lauren Ramsey.

5.   What is the minimum party size that guarantees that at least one set is present?

Suggested by Emily Belya.   Researched by Group 5: Emily Belya, Alicia Gore, Lindsay Hickman, and Monica Hinrichsen.

6.   In a full deck, how many sets are possible?

Suggested by Garret Johnson.   Researched by Group 6: Garret Johnson, Bridge Legler, Julianne Maguire, and Meg Moore.

7.   In the game of Set, what impact would a fifth category have on the game?

Suggested by Lori Pillichody.   Researched by Group 7: Anna Applegate, Philip Pierce, Lori Pillichody, and Alex Vaughn.

8.   Which set is more likely to occur: A set with at least one characteristic in common, or a set with no characteristics at all in common?

Suggested by Elyssa Rosenbaum.   Researched by Group 8: Lorayah Priester, Elyssa Rosenbaum, Tim Woodland, and Jen Yaho.

9.   In the game of Set, what impact would eliminating one of the three colors have on the game?

Suggested by Maria Hinrichsen.   Researched by Group 9: Matt Bosworth, John Cerminaro, Heather Cote, and Rachel Seda.

10.   Which method of play is the most effective strategy for finding sets?

Suggested by Paul Gleason.   Researched by Group 10: Paul Gleason, Malcolm Henderson, Christine Taglienti, and Evan Wilson.

### Established Theorems

Theorem 1.   There exists a 9-party containing 12 sets.   (Group 4)

Theorem 2.   There exists a perfect 9-party.   (Group 4)

Theorem 3.   If a perfect 9-party is extended to a deal, then that deal can have at most 14 sets.   (Group 4)

Theorem 4.   There exists a deal containing 14 sets. (Conjecture: 14 is the largest number of sets that could be found in any deal.)   (Group 4)

Theorem 5.   There are 216 possible sets with no characteristics in common.   (Group 6 and Group 8, independently)

Theorem 6.   There are 432 possible sets with one characteristic in common and three characteristics different.   (Group 6 and Group 8, independently)

Theorem 7.   There are 324 possible sets with two characteristics in common and two characteristics different.   (Group 6 and Group 8, independently)

Theorem 8.   There are 108 possible sets with three characteristics in common and one characteristic different.   (Group 6 and Group 8, independently)

Theorem 9.   There are 1,080 sets possible in a full deck.   (Group 6 and Group 8, independently)

Theorem 10.   A set with at no characteristics in common is more likely to occur than a set with at least one characteristic in common. Specifically, given a set, there is only a 1 in 5 chance that the set has no characteristics in common.   (Group 8)

Theorem 11.   Each pair of cards has exactly one connector card in the deck.   (Group 10)

Theorem 12.   There exists an impossible 18-party using only two colors that cannot be extended to an impossible 19-party.   (Group 5 and Group 9, independently)

Theorem 13.   There exists a 1-color impossible 9-party. (Conjecture: 9 is the largest possible size of a 1-color impossible n-party.)   (Group 9)

Theorem 14.   In the game of Alpha-Set, there exists a 9-party the contains 12 alpha-sets.   (Group 7)

Theorem 15.   In the game of Alpha-Set, there exists a perfect 9-party.   (Group 7)

Theorem 16.   In the game of Alpha-Set, if a perfect 9-party is extended to a deal, then that deal can have at most 14 alpha-sets.   (Group 7)

Theorem 17.   In the game of Alpha-Set, there exists a deal containing 14 alpha-sets.   (Group 7)

### Questions to think about before the exam

• What does each definition mean? What is each question asking? What is each theorem talking about? Can you give examples?

• For which of the original questions were likely answers found? Which of the original questions were answered and proved?

• For each Theorem that was presented in class, what types of arguments went into proving that theorem?