The Game of SET

Dr Amie AlbrechtUniSA

Attributes

Each card has the following attributes:

Colour: purple, green, redShape: oval, diamond, squiggleNumber: one, two, threeShading: empty, filled, striped

Attributes

Question

A SET deck contains exactly one card of every possible combination of attributes.

How many cards in a deck?

The Multiplication Principle

If one event can occur in m ways, and a second event can independently occur in n ways, then

the two events can occur in mn ways.

How many McMeals?

How many South Australian licence plates?

How many South Australian postcodes?

Definition of a SET

A SET is a group of three cards where, for each of the four attributes, the features are

same across all three cards OR

different across all three cards

SET - Example 1

SET - Example 1

Colour: SAME (purple)

SET - Example 1

Colour: SAME (purple)Shape: SAME (diamond)

SET - Example 1

Colour: SAME (purple)Shape: SAME (diamond)Number: SAME (three)

SET - Example 1

Colour: SAME (purple)Shape: SAME (diamond)Number: SAME (three)Shading: DIFFERENT (empty, filled, striped)

SET - Example 2

SET - Example 2

Colour: DIFFERENT (green, red, purple)

SET - Example 2

Colour: DIFFERENT (green, red, purple)Shape: SAME (squiggle)

SET - Example 2

Colour: DIFFERENT (green, red, purple)Shape: SAME (squiggle)Number: DIFFERENT (one, two, three)

SET - Example 2

Colour: DIFFERENT (green, red, purple)Shape: SAME (squiggle)Number: DIFFERENT (one, two, three)Shading: SAME (filled)

SET - Example 3

SET - Example 3

Colour: DIFFERENT (green, red, purple)

SET - Example 3

Colour: DIFFERENT (green, red, purple)Shape: SAME (oval)

SET - Example 3

Colour: DIFFERENT (green, red, purple)Shape: SAME (oval)Number: SAME (three)

SET - Example 3

Colour: DIFFERENT (green, red, purple)Shape: SAME (oval)Number: SAME (three)Shading: DIFFERENT (empty, filled, striped)

SET - Example 4

SET - Example 4

Colour: DIFFERENT (green, red, purple)

SET - Example 4

Colour: DIFFERENT (green, red, purple)Shape: DIFFERENT (oval, diamond, squiggle)

SET - Example 4

Colour: DIFFERENT (green, red, purple)Shape: DIFFERENT (oval, diamond, squiggle)Number: DIFFERENT (one, two, three)

SET - Example 4

Colour: DIFFERENT (green, red, purple)Shape: DIFFERENT (oval, diamond, squiggle)Number: DIFFERENT (one, two, three)Shading: DIFFERENT (empty, filled, striped)

SET - Not Example 1

Colour: NOT ALL-SAME or ALL-DIFFERENTShape: SAME (diamond)Number: SAME (two)Shading: DIFFERENT (empty, filled, striped)

SET - Not Example 1

SET - Not Example 2

SET - Not Example 2

Colour: DIFFERENT (green, red, purple)Shape: SAME (oval)Number: SAME (two)Shading: NOT ALL-SAME or ALL-DIFFERENT

SET - Not Example 3

SET - Not Example 3

Colour: DIFFERENT (purple, green, red)Shape: DIFFERENT (oval, diamond, squiggle)Number: DIFFERENT (one, two, three)Shading: NOT ALL-SAME or ALL-DIFFERENT

SET - Not Example 4

SET - Not Example 4

Colour: SAME (green)Shape: DIFFERENT (oval, diamond, squiggle)Number: NOT ALL-SAME or ALL-DIFFERENTShading: SAME (empty)

SET - Exercise 1

SET - Exercise 1

SET - Exercise 2

SET - Exercise 2

SET - Exercise 3

SET - Exercise 3

SET - Exercise 4

SET - Exercise 4

SET - Grid 1

SET - Grid 1 - Set 1

SET - Grid 1 - Set 2

SET - Grid 1 - Set 3

SET - Grid 1 - Set 4

SET - Grid 1 - Set 5

SET - Grid 1 - Set 6

>

Question

Pick out two random cards from the SET deck.

How many cards, if any, can be paired with the first two to complete a SET?

Question

Consider the entire SET deck.

How many SETs are possible?

Playing SET

• Shuffle cards and lay twelve in a grid face up on the table.• Call out ‘SET’ if you see a SET and have the others check.

• If correct, keep the SET. The dealer replaces the three cards.• If incorrect, keep playing.

• If there is no SET on the table, deal three more cards.• At the end of the game, the winner has the most number of SETs.

Question

What is the best strategy when searching for SETs?Which type are you most likely to find?

Start by tallying your results from playing SET.

# diff # same # sets4 03 12 21 3

Possible further questions

• Can only three cards be left at the end of the game?• What is the largest number of SETs that can be present among a layout of nine cards?• What is the maximum number of cards that contain no SETs?• How does the analysis change if we play with two or three decks?• ...