Set Game -- General Considerations
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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:
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
- 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
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
- If no two cards disagree in a property, that property is
- If no two cards agree in a property, that property is
- 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
- 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:
The disposition of this collection is color diverse, shape uniform,
texture diverse, number uniform. We can add a card:
Now the disposition is color broken, shape uniform, texture diverse,
number uniform. We add a further card:
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
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
To say the same thing more succinctly,
Here are some examples. This is a set diverse in all four properties:
- 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.
This set is uniform in the texture property and diverse in the others:
This set is uniform in texture and number, diverse in color and shape:
This set is diverse in shape, and uniform in the other properties:
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:
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
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:
Here is how we figure the third card:
The completed set looks like this:
- 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
- The numbers of the first two cards agree, being ones; the third card
will have to be the same.
The assurance of unique fulfillment is another reason behind the elegance
of the rules of Set.