Figure 1. An example of three cards from a Set deck that form a set.
Abstract. The card game called Set® inspires combinatorial problems related to triplets of points in \Z_3^k, p_1, p_2, p_3, such that p_1 + p_2 + p_3 = (0^k).
Résumé. Le jeu des cartes qui s'appelle Set® inspire les questions combinatoires qui sont liés aux triplés des points de l'espace \Z_3^k, p_1, p_2, p_3, tel que p_1 + p_2 + p_3 = (0^k).
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The card game of Set® was invented by a population
geneticist by the name of Marsha Jean Falco in 1974. She was studying
the condition of epilepsy in German Shepherds and began representing genetic
data on the dogs by drawing symbols on cards and then searching for patterns
in the data. After realizing the potential as a challenging puzzle, with
some encouragement from friends and family she developed and marketed the
card game. It is now available in many gaming stores in the U. S.
and worldwide.
A deck of Set® has 3^4 = 81 different cards. Each card has a white background with either one, two, or three forms that are either an ellipse, a diamond or a squiggle (on any given card all of the shapes are the same). The forms on the card are drawn in either red, green or purple ink, and each of the shapes are either filled solid, shaded with lines, or just an outline. In other words, there are four distinguishing properties on each of these cards: shape, number, color and shading. For each of these properties, a card can take on one of three different values giving 3^4 different possibilities. Some examples of cards are shown in Figure 1.
The game is played by placing 12 cards from the deck face up on a table. The players scan the upright cards to locate a collection of three such that each of the four values for the properties on their faces are either all the same or all different. A collection of 3 cards with this property is called a `set.' The first player to locate a `set' removes it from the board and three more cards are dealt face up. The player with the most number of sets after all of the cards have been dealt and all sets removed wins the game.
Figure 1. An example of three cards from a Set deck that form a set.
With 12 cards on the table it is possible that no sets exist in the upright ones. After all players agree that there isn't a set on the table, three more cards are dealt. If again no sets are showing, three more are dealt until a set appears. The cards are not replaced if there are already twelve on the table.
A shareware computer version of the card game is distributed by the company SET Enterprises, Inc. and may be downloaded for free from their web site. The freeware part of the program deals cards that have only three properties. It also includes a demo version of the full computer game, but the demo version is rigged (i.e. the cards are not chosen at random from the deck). Both the simplified and demo versions are much easier than the original game, however the version available for purchase has more options than is possible with just the deck of cards.
There are two obvious generalizations to this game. The first would be to add more properties (where for a deck with k properties there are 3^k possible cards). The second would be to vary the number of values that these properties could take on (that is, if each property takes on p different values and there are k properties then there are p^k different cards).
After playing a few rounds of the game, several questions came to my mind about this game. The first was "What is the largest number of cards that one can expose that does not include a set?" Although it is easy to experiment with fewer than 4 properties, it is difficult to generalize the results with more.
This very question had been answered for a deck with kproperties where k is 4 or less by some recreational mathematicians and their results are available on the Set® web site. They had found by a computer exhaustive search that in a deck with 4 properties, it is possible to place 20 cards without finding a set. With a deck that has 3 properties, it is possible to have 9 cards that do not contain a set; with 2 properties, 4 cards; with 1 property, 2 cards.
The first attack to try to answer this problem is to look at Sloane's
On-line
Encyclopedia of Integer Sequences. The entry 2, 4, 9, 20 returns a
maximum number of responses, however none of them are suggestive of the problem that
is considered here. Four values is just not enough in this case to
determine if the sequence had been considered before in a different situation.
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