Research progress for the game of Set (103 Section 13)

Definitions, Rules, and Terminology

There are four categories of characteristics, each with three possible values: A set is a collection of three cards that satisfies the following four rules: A collection of 12 cards is called a block.

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

A collection of n cards that contains no sets is called an empty n-block.

Given a block of cards, a minority card in the block is a card that has at least one property that exists nowhere else in the block.

A wild card is a card you add to the game of Set with the property that two characteristics are fixed and two characteristics can take on any value. (For example, a "zebra diamond" card that can be used with value 1, 2, or 3 and with shape oval, diamond, or squiggle, as desired.)

A super wild card is a card you add to the game of Set with the property that all of its characteristics can take on any value. (In other words, a super wild card can be used as any card in the deck.)

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 the same," 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 3 characteristics the same.)

Questions to Answer

(Note: The questions originally numbered 6 and 8 did not end up being investigated by any group.)

1.   How many sets are possible?

2.   (a) How many sets are possible in a block where all cards are the same color? (b) How many sets are possible in a 27-block of the same color?

3.   How will the game change if various wild cards are added to the deck?

4.   What are some strategies one could utilize while playing Set?

5.   How many different sets are possible if you remove one of the shadings from the game?

7.   What is the largest empty n-block possible?

9.   Given any card, how many sets contain that card?

10.   Give an example of an empty 15-block. How many empty 15-blocks are possible? Are there any common features among empty 15-blocks?

Established Theorems

Theorem 1.   Given any card in the deck, there are a total of 40 sets containing that card. Moreover, these 40 sets overlap only at the original card.   (Group 9)

Theorem 2.   For any two cards in the deck, there exists only one third card that will complete a set with the original two cards.   (Group 4)

Theorem 3.   If a minority card differs from the other cards in a block by characteristic P, then any set in the block that contains the minority card must have characteristic P all different. Moreover, any set in the block that does not contain the minority cards must have characteristic P all the same.   (Group 4)

Theorem 4.   Five nonoverlapping empty 15-blocks can be constructed simultaneously from one deck of Set cards.   (Group 10)

Theorem 5.   In an empty 15-block, 9 cards is the maximum known number of cards that can share the same property (for example, all green, or all diamonds, etc). (Conjecture: There can be no more than 9 cards of the same property in a 15-block.)   (Group 10)

Theorem 6.   If one characteristic is removed from the game (for example, no color is considered, so only red cards are used), then there exists a block of 9 cards with no set. (Conjecture: 9 is in fact the most number of cards that contains no set under these circumstances. A website was found that claims that this conjecture is true: www.setgame.com/set/noset.htm)   (Group 7)

Theorem 7.   There exists a 20-block that contains no set. (Conjecture: 20 is in fact the most number of cards that can have no set. A website was found that claims that this conjecture is true: www.setgame.com/set/noset.htm)   (Group 7)

Theorem 8.   Within the 27 green cards is a full collection of 9 nonoverlapping sets. (And similarly true for any set of 27 cards that share one characteristic in common.)   (Group 2)

Theorem 9.   If only two shadings are used, there are 234 possible nonoverlapping sets. (And similarly true if only two colors are used, or only two shapes, or only two numbers.)   (Group 5)

Theorem 10.   There are 216 sets in which the characteristics are all different.   (Group 1)

Theorem 11.   There are 432 sets in which three characteristics are the same and one characteristic is different.   (Group 1)

Theorem 12.   There are 324 sets in which two characteristics are the same and two are different.   (Group 1)

Theorem 13.   There are 108 sets in which one characteristic is the same and three are different.   (Group 1)

Theorem 14.   There are 1080 total sets in the deck.   (Group 1)

Theorem 15.   A wild card can represent any one of 9 cards.   (Group 3)

Theorem 16.   A super wild card together with any pair of cards is always a set.   (Group 3)

Theorem 17.   There does not exist an empty 12-block that contains a super wild card.   (Group 3)

Theorem 18.   There exists an empty 12-block that contains a wild card.   (Group 3)

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?