The Game of Set

For the first evening meeting of 2020, Greg Fein led us through an exploration of the card game, Set.

The meeting began with Greg passing out a few decks of the game Set and giving time for all to explore the cards. Some of us were familiar with the game, but for others this was the first time seeing these cards. We then shared our observations of the cards in the deck.

The following observations came out:

• 3 colors: red, green, and purple
• 3 shapes: slug, diamond, and tic-tac
• 3 shades: filled in, partially filled in, not filled
• 3 numbers: 1, 2, and 3

We then shared a few things we wondered about the cards:

• How many cards are there in the game?
• Are the cards all unique?

Greg then presented us with the rules of SET:

The object of the game is to identify a ‘Set’ of three cards from 12 cards laid out on the table. Each card has a variation of the following four features: COLOR, SHAPE, SHADE, and NUMBER.

A ‘Set’ consists of three cards in which each feature is EITHER the same on each card OR is different on each card. That is to say, any feature in the ‘Set’ of three cards is either common to all three cards or is different on each card.

To solidify the rules, we spent a few minutes looking for “sets” together, justifying whether three cards were indeed a “set”.

For example:

This IS a “set” because all three cards are red; all are ovals; all have two symbols; and all have different shadings.

This IS a “set” because all have different colors; all have different symbols; all have different numbers of symbols, and all have different shadings.

This is NOT a set because all have different colors; all are diamonds; all have one symbol; however, two are open and one is not.

This is NOT a set because all are squiggles; all have different shadings; all have two symbols; however, two are red and one is not.

We then had a few minutes to play the game at our tables. After we had all found a few sets, we came back together to generate some mathematical questions we might like to explore based on the game.

Some questions that came up:

• What is the likelihood that there is at least one set in 12 cards?
• Will the game end with no cards?
• How many sets with all 4 traits different? 3 traits different? 2? 1? 0?
• How many different possible sets?
• If there are 2 sets in a hand of 12, does the order you pick them up matter?
• What’s the maximum number of cards with no set?
• What’s the minimum number of cards to guarantee there is a set among them?
• What is the most efficient strategy for identifying sets?
• What’s the likelihood that 12 cards have four non-overlapping sets?
• How many of each attribute are there?
• In 12 cards, what’s the most sets you can have (if sets can overlap)?

We then set off in our groups to explore one or more questions for the remainder of the meeting. The last few minutes were spent sharing what we had worked on.

The following gallery contains the posters of all the groups. WARNING: may contain spoilers!

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We explored many questions and found a few answers; much remains to be discovered!

In attendance: Macarena, Brigette, Benny, Attila, Audrey, Sophie, Kevin, Greg.