Set Game -- Variations on the Pack

Version of April 9, 2002

Number of Properties. Because of the way that the number of properties influences the number of cards in the pack, it is natural to call the standard pack four-dimensional, because it has four properties thus 81 (three to the fourth power) cards.

The most frequently mentioned variation of the Set game reduces the number of properties from four to three, yielding the three-dimensional pack of 27 = 3 x 3 x 3 cards. This can be implemented by selecting one value of one property, and rejecting all cards that do not embody it. As a case in point, the factory instructions suggest to beginners that they use only the red cards, omitting the greens and purples. Principles of forming a set are the same, and the fundamental criterion pertaining to color will automatically be satisfied because all cards are uniform in color.

Further reducing the pack to two properties is possible, but with only 9 = 3 x 3 cards, an interesting game is difficult to devise. Toward the other extreme, the five-dimensional pack, with five properties and 243 cards, could be produced; each token would perhaps be printed horizontally, vertically, or diagonally. Instead of four fundamental criteria, there would now of course be five. Such a large pack might have trouble gaining favor with card players, as evidenced by the tiny percentage of card games that use more than 108 cards. Among other matters, 243 cards are difficult to shuffle.

Number of Values. What might happen if the number of values of a property is not three? In fact, there is no reason that all properties must have the same number of values. Consider the pack whose properties and values are thus:

• Color -- red, green, purple or brown.
• Shape -- oval, diamond or squiggle.
• Texture -- solid or shaded.
• Number -- two.
If fully implemented, there would be 24 (4 x 3 x 2 x 1) cards.
• Compared to the standard pack, collecting a set of uniform color in this special pack is more difficult; a set with diverse colors is easier.
• Shape, which still has three values in this special pack, exhibits no change.
• With only two texture values to pick from, there is no way to collect three cards of diverse texture, although the uniform and broken configurations can still be achieved.
• Number has only one value, so all the cards in any collection of three cards are automatically uniform in this property.

Number of Cards in a Set. If a set were defined to contain only two cards, the fundamental criteria would always be satisfied -- of any two things, either they are all the same or all different. Because the set of one card is trivial (and the set of zero cards absolutely so), the standard set of three cards turns out to be the smallest that gives interesting results.

A larger set, of four cards, gives nice results when used with a pack whose properties have four values. A pack of three properties each having four values would naturally have 64 = 4 x 4 x 4 cards, a convenient handful. The properties and values could be as such:

• Color -- red, green, purple or brown.
• Shape -- oval, diamond, squiggle or bowtie.
• Number -- one, two, three or four.
Suppose we have three cards, two brown and one purple. Under the standard rules, the color property is already broken, so no fourth card we can add will yield uniformity or diversity: we have lost the assurance of unique fulfillment. To remedy this problem, we can define an additional configuration, pairedness. The configuration list now appears thus:
• If no two cards disagree in a property, that property is uniform.
• If no two cards agree in a property, that property is diverse.
• If two cards agree at one value in a property, and the two other cards agree at a different value, that property is paired.
• A property that is neither uniform, paired nor diverse is broken.
Uniformity and diversity are the same as before, and brokenness remains the catchall for leftovers. Pairedness, however, differs somewhat in character from uniformity and diversity:
• Uniformity is meaningful to define when
• properties have any number of values, and
• sets have any number of cards.
• Diversity is almost as flexible, being a useful notion when
• properties have any number of values, and
• sets have any number of cards not exceeding the the number of values.
• Pairedness, by contrast, we have defined only when
• properties have four values, and
• sets have four cards.
If either of these fours is increased or decreased, then in most cases there is no simple extension of the definition of pairedness that will assure unique fulfillment.
Assured unique fulfillment is implemented in this 64-card pack as follows. We start with any three cards and seek the fourth; within each property,
• If the three cards are uniform, the fourth will agree with them.
• If the three cards are diverse, the fourth will be different from any of them.
• If two cards are of one value, and one card is of a different value, the fourth card will agree with the "minority" card.
Example. Starting with these three cards, we want to finish the set:
• red--bowtie--one
• brown--bowtie--one
• purple--bowtie--three
In color, the three we have are different from each other, so the fourth will have to be diffent from them; hence it will be green. In shape, the first three are bowties, and so will be the fourth. In numbers, we have one pair and one single, so we match the single (the three) making two pairs. The completed set turns out to be
• red--bowtie--one
• brown--bowtie--one
• purple--bowtie--three
• green--bowtie--three