Definitions

# Set (game)

Set! redirects here. Set! is also a special form in the Scheme programming language.

Set is a real-time 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 won American Mensa's Mensa Select award in 1991 and placed 9th in the 1995 Deutscher Spiele Preis.

## 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 summarized by: 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 twelve 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.

## 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, given four cards, it is possible that they form a valid Super-Set with one grouping, but not with another grouping.
• 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 lie 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.
• Ghost-Set involves finding 3 disjoint pairs of cards in which the cards that are missing from each pair also form a set amongst each other. It is termed a ghost set because the three missing cards do not have to be on the board.
• Super Ghost-Set involves finding 4 disjoint pairs of cards in which the cards that would complete the sets for 2 of the pairs form a super-set with the cards that would complete the sets for the other 2 pairs of cards.
• 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, obviously, can be 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 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 be collected). In this variant, 15-16 cards are dealt.
• Set-Do-Ku: the 81 cards of the Set deck can be grouped by any 2 properties (e.g. color and shading), giving 9 groups of 9 cards. Cards are then placed within a 9 x 9 grid to match a standard Sudoku puzzle by mapping a specific Sudoku digit to a specific card group (e.g. Sudoku digit 1 maps to any open-red card, Sudoku digit 2 maps to any striped-red card, Sudoku digit 3 maps to any solid-red card, etc.). After this initial setup the puzzle is solved using the remaining cards.

## Basic Combinatorics of Set

• Given any two cards, there exists one and only one card which forms a set with those two cards.
• Therefore the probability of producing a Set from 3 randomly drawn cards is 1/79.
• The largest group of cards you can put together without creating a set is 20.
• There are $frac\left\{3\right\} = frac\left\{81 times 80\right\}\left\{3!\right\} = 1080$ unique sets.

• The probability of getting any given deal of 12 cards is

$frac\left\{1\right\} = frac\left\{12! 69!\right\}\left\{81!\right\} approx 10^\left\{-14\right\}$

## Trivia

The game evolved out of a coding system that the designer used in her job as a geneticist

According to a 20 page article on the mathematics of Set by a couple of UC math professors, the inventor was studying the genetics of epilepsy in German Shepherd dogs and began taking notes on cards. She and her family saw the potential for a pattern match game and developed it to the current form.