Set Game -- General Considerations

Version of April 23, 2002

The Pack. The standard pack has 81 cards, each bearing one or more symbols which we call tokens. On any one card, all the tokens are the same. Each card can be categorized by:
• The color of its tokens -- red, green or purple.
• The shape of its tokens -- oval, diamond or squiggle.
• The texture of its tokens -- solid, shaded or hollow.
• How many tokens there are -- one, two or three.
Color, shape, texture and quantity we call properties, and each property has the three values listed above. As an example, solid, shaded and hollow are the values of the texture property. For brevity, we say "a red card" when we mean a card with red tokens; similarly we could speak of an oval card, a shaded card, or a three card. Describing a card completely, we might call it a squiggle purple hollow two.

With four properties, and three values for each property, there are 81 = 3 x 3 x 3 x 3 possible cards, and that is exactly what is supplied with the standard pack.

Configurations of Properties. Within any collection of cards, we define:

• If no two cards disagree in a property, that property is uniform.
• If no two cards agree in a property, that property is diverse.
• A property that is neither uniform nor diverse is broken.
To speak of a property's configuration is to speak of whether it is uniform, diverse, or broken. Less precise, but more convenient, is to say that a property is uniform if all the cards are the same, and diverse if they are all different. Here are some characteristics of properties:
• Within a broken property, nonuniformity requires at least two cards that disagree, and nondiversity requires at least two cards that agree. To accomplish both of these calls for at least three cards.
• If a property is to be diverse, there can be at most three cards in the collection, because each property has only three different values.
• Uniformity might occur with any number of cards.

A collection's disposition is all of its configurations viewed as an aggregate.

Here is an example collection with two cards:

• green-oval-solid-two
The disposition of this collection is color diverse, shape uniform, texture diverse, number uniform. We can add a card:
• green-oval-solid-two
• red-oval-hollow-two
Now the disposition is color broken, shape uniform, texture diverse, number uniform. We add a further card:
• green-oval-solid-two
• red-oval-hollow-two
• green-oval-solid-one
Texture and number become broken.

A collection with only one card is regarded as both uniform and diverse in every property. Explanation: Because there is only one card, there is never a case where two cards disagree, hence the collection is uniform. Similarly, there is never a case where two cards agree, thus the collection is diverse. By the same line of reasoning, a collection with zero cards also is both uniform and diverse.

Whenever we add a card to a collection, there is a possibility that we will break a property that had been uniform or diverse. However, any broken properties will remain broken. In the other direction, deleting a card from a collection may transform a broken property into uniformity or diversity, but any properties already uniform or diverse will remain such. Beyond that, a property already uniform might remain uniform but additionally become diverse, and vice versa.

The Fundamental Criteria. While there are many variations on the rules, the heart of the game is to form a collection of three cards that satisfies all four of the following rules. Such a collection is called a set.

• All cards are equal color, or all are different colors.
• All cards are equal shape, or all are different shapes.
• All cards are equal texture, or all are different textures.
• All cards are equal number, or all are different numbers.
To say the same thing more succinctly,
• No property is broken.
Here are some examples. This is a set diverse in all four properties:
• green-diamond-hollow-three
• purple-squiggle-solid-two
This set is uniform in the texture property and diverse in the others:
• green-diamond-hollow-three
• red-oval-hollow-one
• purple-squiggle-hollow-two
This set is uniform in texture and number, diverse in color and shape:
• green-diamond-hollow-one
• red-oval-hollow-one
• purple-squiggle-hollow-one
This set is diverse in shape, and uniform in the other properties:
• purple-diamond-hollow-one
• purple-oval-hollow-one
• purple-squiggle-hollow-one
We cannot exhibit a set uniform in all four properties, because then the three cards would then be equal in all respects, and the standard Set pack does not contain duplicates.

Here is an example of a non-set:

• red-oval-solid-three
• green-squiggle-hollow-three
Two cards are red, and one card is green -- color is broken. We failed the fundamental criterion pertaining to color, and it makes no difference that we satisfied the others.

The elegance of the Set Game arises in part as follows. A collection with fewer than three cards cannot have a broken property, while a collection with more than three cards cannot have a diverse property. Hence three is the "ideal" number for the number of cards in a set, because uniformity, diversity and brokenness are all possible.

We caution the reader that what this game calls a set is not what mathematicians call a set. This distinction can give rise to confusion because mathematical sets are convenient in analyzing the game. We have used the term collection when we have had a mathematical set in mind.

Assured Unique Fulfillment. Given any two cards, there is always one ("assured") and only one ("unique") additional card that, in conjunction with the first two, will complete a set ("fulfillment"). For instance, consider these two cards: