Version of April 23, 2002

- 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.

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*.

- 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
- red-oval-shaded-two

- green-oval-solid-two
- red-oval-shaded-two
- red-oval-hollow-two

- green-oval-solid-two
- red-oval-shaded-two
- red-oval-hollow-two
- green-oval-solid-one

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.

- No property is broken.

- green-diamond-hollow-three
- red-oval-shaded-one
- purple-squiggle-solid-two

- green-diamond-hollow-three
- red-oval-hollow-one
- purple-squiggle-hollow-two

- green-diamond-hollow-one
- red-oval-hollow-one
- purple-squiggle-hollow-one

- purple-diamond-hollow-one
- purple-oval-hollow-one
- purple-squiggle-hollow-one

Here is an example of a non-set:

- red-oval-solid-three
- red-diamond-shaded-three
- green-squiggle-hollow-three

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:

- purple-squiggle-shaded-one
- red-squiggle-hollow-one

- Since the first two cards are purple and red, there is no way that all three cards in the ultimate set can be the same color; hence they must all be different. The only color different from purple and red is green, so that will be the color of the third card.
- The first two cards are already the same shape, squiggle, so once the set is completed it will be impossible for all three cards to be different. As a result, all three will need to be of the same shape, and that is squiggle.
- The first two cards are shaded and hollow, which are different from each other, so the third card will have to be something else, and that is solid.
- The numbers of the first two cards agree, being ones; the third card will have to be the same.

- purple-squiggle-shaded-one
- red-squiggle-hollow-one
- green-squiggle-solid-one