Durak is a traditional Russian card game, the object of which is to get rid of all your cards: the last player left holding cards loses the game and is named the durak ('fool'). There is so much variation in the rules of durak that it might be better described as a family of related games, rather than a single, unified game.
In this series of articles, I will analyze durak in various ways using mathematical methods. Before doing any analysis, though, it's important to define, with the appropriate amount of rigor, the forms of durak that will be under investigation. Those familiar with the rules and variations of durak might only need to skim this article, but should refer back to it while reading the later articles whenever precise definitions or rules are in question. The rules are not described in rigorous mathematical terms, but are specific enough that mathematical models of game situations can be easily and naturally conjured when necessary.
Podkidnoy ('throw in') Durak may be the most common form of the game.
The game is played by 2-6 players with a 36-card deck, formed by removing the 2s, 3s, 4s, and 5s from a standard 52-card deck. To begin, six cards are dealt privately to each player. The remaining cards are left to form the draw deck, whose bottom card is flipped face up (while staying at the bottom) and left visible to all players. The suit of this face-up card will be the trump suit for the game. Cards with this suit are called trump cards. If there are 6 players, there will be no draw deck, and the trump suit may be picked in some arbitrary manner. The direction of play traditionally goes clockwise around the table.
The game consists of a series of bouts. In each bout, a certain player is the attacker. At the beginning of the game, the role of first attacker can be assigned randomly or by any method. The attacker begins the bout by placing down a card from their hand, face up, to be the initial attack card. The next player after the attacker is the defender, who can defeat any attack card by placing a card from their hand, face up, on top of it as a defense card. A card may only be used to defeat a particular attack card if it is either the same suit as the attack card but of a higher rank (aces are high); or if it is a trump card and the attack card is not. At any time during the bout, any player who is not the defender may add an additional attack card from their hand, so long it has the same rank as some attack or defense card already in play. The total number of attack cards in play must neither exceed six nor the number of cards the defender was holding just before the start of the bout—if either of these limits is reached, no one may add more attack cards. The bout ends in one of two ways: either the defender is unable or unwilling to defeat all the attack cards, and must take all the attack and defense cards in play into their hand; or the defender defeats every attack card and the other players are unwilling or unable to add more, in which case all the attack and defense cards are discarded. If the defender takes the cards, the other players may add new attack cards retroactively (following the usual rules and limits for legal attack cards), which the defender must also take into their hand.
After a bout, players with fewer than six cards left in their hand must draw from the draw deck until they have six cards or the draw deck is emptied. The attacker of the bout draws first, and then the other players, following the normal direction of play, except the defender, who is skipped and draws last. If the defender successfully defeated all the attack cards, they become the attacker for the next bout. Otherwise, the next player after them becomes the new attacker.
If a player has no cards at the end of a bout, and the draw deck is empty when it comes time for them to draw, they are out of the game and removed from the cycle of players. When there is only one player left, that player is deemed the durak.
Perevodnoy ('transfer') Durak follows the same rules as Podkidnoy Durak, with one additional rule: during a bout, if no defense cards have been played yet, the defender may "pass on" the attack by adding a new attack card, which must be of the same rank as the other attack cards. By doing this, they become the attacker, and the next player becomes the new defender. This cannot be done if it would violate the usual rules for the number of valid attack cards (i.e. the number of attack cards facing the new defender exceeds the number of cards in their hand).
Perevodnoy Durak is sometimes played with an additional way to pass on attacks: if the defender holds the trump card of the appropriate rank, and all the conditions for passing on the attack are met, they may pass on the attack by simply showing the trump card instead of putting it down as an attack card. Playing the trump card as an attack card to pass on the attack is still allowed, just not required. One may not pass on the attack using the same trump card more than once in a single bout.
Durak is ordinarily played with a deck with 9 ranks, 4 suits, and 2-6 players. Many results about the game can be generalized to a games that are played with the same gameplay rules, but with a different number of suits, ranks, and players. These games are not forms of durak proper, but can be considered forms of generalized durak. To keep things reasonable, we will require that there be at least two ranks and two suits, that no identical cards (i.e. two cards with the same rank and same suit) exist, and that there be enough cards to deal everyone in with six cards (i.e. the number of suits times the number of ranks is at least six times the number of players).
A game of generalized durak may follow any of the durak rulesets. Thus, we may speak of, for example, generalized Podkidnoy durak or generalized Perevodnoy durak.
The durak endgame is a form of durak in which:
The durak endgame is a game without chance or hidden information[2]. If players have perfect memory, then an ordinary game of durak must become a durak endgame before it can end. This is true because:
A durak endgame may be played with any of the durak rulesets, and with a non-standard deck. We might therefore refer to, for example, a Perevodnoy durak endgame, or a generalized Podkidnoy durak endgame. The endgame may be very brief—it need not even include a single bout.
back According to pagat.com's page on Perevodnoy Durak, the variation of the game with this rule is called Perevodnoy durak s proyezdnym in Russian, which is glossed as 'transfer fool with a pass card'. I will refer to it as 'Perevodnoy durak with the Proyezdnoy rule'.
back [2]: It might therefore be called a game of perfect information, but the exact definition of that term varies, so I have avoided using it.