of Set are summarized by: If you can sort a group of three cards into "two of and one of ", then it is not a set. For example, these three cards form a set: One red striped diamond Two red solid diamonds Three red open diamonds Given any two cards from the deck, there is one and only one other card that forms a set with them. In the standard Set 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. A player who sees a set among the twelve cards calls "Set" and takes the three cards, and the dealer lays three more cards on the table. (To call out "set" and not pick one up quickly enough results in a penalty.) There may be 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. Variants were included with the Set game that involve different mechanics to find sets, as well as different player interaction. Additional variants continue to be created by avid players of the game. Basic combinatorics of Set Given any two cards, there is exactly one card that forms a set with those two cards. Therefore, the probability of producing a Set from 3 randomly drawn cards from a complete deck is 1/79. A Cap set is a mathematical structure describing a Set layout in which no set may be taken. The largest group of cards that can be put together without creating a set is 20. Such a group is called a maximal cap set . Donald Knuth found in 2001 that there are 682344 such cap sets of size 20 for the 81-card version of Set; under affine transformations on 4-dimensional finite space, they all reduce to essentially one cap set. There are unique sets. The probability that a set will have features different and features the same is . (Note: The case where