Set Game -- Variations on the Pack

Home Page of This Site

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:

If fully implemented, there would be 24 (4 x 3 x 2 x 1) cards.

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:

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: 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: 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, Example. Starting with these three cards, we want to finish the set: 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