Set (game)

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This article is about the game known as Set. For the term used by the Ophidian 2350 card game for turning a card 90 degrees, see Set (gaming).

Three cards from a Set deck. These cards each have a unique number, symbol, shading, and color, and are thus a "set."

Set is a card game designed by Marsha Falco and published by Set Enterprises in 1991. The deck consists of 81 cards varying in four features: number (one, two, or three); symbol (diamond, squiggle, or oval); shading (solid, striped, or open); and color (red, green, or purple). Each possible combination of features (e.g., a card with three striped green ovals) appears precisely once in the deck. Set placed 9th in the 1995 Deutscher Spiele Preis.

1 Games
2 Variations
3 Basic Combinatorics of Set
4 Trivia
5 External links

[edit] Games

Several games can be played with these cards, all involving the concept of a set. A set consists of three cards which satisfy all of these conditions:

They all have the same number, or they have three different numbers.
They all have the same symbol, or they have three different symbols.
They all have the same shading, or they have three different shadings.
They all have the same color, or they have three different colors.

The rules of Set are most elegantly described thus:

Two are and one is not if and only if it is not a set.

Given any two cards from the deck, there will be one and only one other card that forms a set with them. One example of a set would be these three cards:

One red striped diamond
Two red solid diamonds
Three red open diamonds

In one game, the dealer lays out cards on the table until either twelve are laid down or someone sees a set and calls "Set!" The player who called "Set" takes the cards in the set and the dealer continues to deal out cards until twelve are on the table. If a player sees a set among the twelve cards, he calls "Set" and takes the three cards, and the dealer lays three more cards on the table. It is possible that there is no set among the 12 cards; in this case, the dealer deals out three more cards to make fifteen dealt cards, or eighteen or more, as necessary. This process of dealing by threes and finding sets continues until the deck is exhausted and there are no more sets on the table. At this point, whoever has collected the most sets wins.

[edit] Variations

One more common variation on classic Set is Chain Set. In Chain Set, one card from the previous set must be used to make a new set. This means that the set possibilities are different for each player and additional deals are much less likely. There is also Memory Set, where the cards are face down and three are turned face up at a time, as in the classic game Memory.

Other variations, invented mostly in Poland and Norway:

Super-Set involves finding two pairs of cards such that both pairs lack the same card to form a set with (see the Mathematics of Set below). Note that four cards can form a valid Super-Set or not, depending on how they are grouped into pairs.

Ultra-Set: the table consists of three separated groups, 6 cards each. A valid Ultra-Set is a normal Set, with position in groups as the fifth property, i.e. either all three cards lay in the same group, or each of them lays in another. Also, a more complicated variation of Ultra-Set is played, where nine groups of 3 cards are laid out 3×3. Here a valid Set must satisfy the Set condition both on rows and on columns.

Mega-Set is Set played with 3 distinguishable blocks of cards, i. e. with 243 cards. The blocks can be distinguished for instance by painting the card backgrounds with very light variants of 3 Set colors (red, green, violet). Typically 15-16 cards dealt.

Double-Set: all rules are standard, but only pairs of two disjoint Sets can be collected. For experienced players, the standard table of 12 cards should be enough. This variant can be obviously combined with any of above, e. g. Double Ultra, and this will be quite hard.

Eight-Set: yet harder than Double-Set. One can only collect groups of 3 or 4 Sets which don't have to be disjoint, but they must together consist of at least 8 cards (it is possible that only 7 cards form 3 different sets, and this is not permitted to collect). In this variant, 15-16 cards are dealt.

[edit] Basic Combinatorics of Set

Given any two cards, there exists one and only one card which forms a set with those two cards.

The largest group of cards you can put together without creating a set is 20.[1]

The probability of getting any given deal of 20 cards is

$\frac{1}{{81 \choose 20}} = \frac{20! 61!}{81!} = 2.1302 \times 10^{-19}$

(This is equal to: $\left(\frac{20}{81}\right)\left(\frac{19}{80}\right)\left(\frac{18}{79}\right)\cdot\cdot\cdot\left(\frac{1}{62}\right) = \frac{20! 61!}{81!} = 2.1302 \times 10^{-19}$ .)

When a game of Set is played correctly (i.e., no one accidentally takes a false Set) with a complete deck of 81 cards, it is impossible to end up with only 3 cards that are not a Set. Put another way, if a complete deck of 81 Set cards is partitioned into 27 piles of three cards, and 26 of the piles form Sets, the remaining pile must also form a Set.