(Red, Green, Purple - to be abbreviated as*Color*,*R*,*G*)*P*(Oval, Diamond, Squiggle - to be abbreviated as*Shape*,*O*,*D*)*S*(Empty, Hatched, Full - to be abbreviated as*Fill*,*E*,*H*)*F*(1, 2, 3 - which obviously will be called*Number*,*1*,*2*)*3*

Each of the colors must be all the same or all different.*Rule C:*Each of the shapes must be all the same or all different.*Rule S:*Each of the fills must be all the same or all different.*Rule F:*Each of the numbers must be all the same or all different.*Rule N:*

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

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.)

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

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

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

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

**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?

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

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

**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)*