### on 80 points (bashifen), aka tractor (tuolaji), aka double rise (shuangsheng)

This is a popular card game of the 5-10-K class played in China, the rules of which are described in English here, but not in a generalizable way. Another version on Wikipedia makes logical sense but isn’t how I’ve seen it in practice. In any case, like for many card and board games, the rules are not described in a proper way for the new player and that’s annoying (some rules are not important or obvious, other rules are important but obscure, etc.). I remember in the help files of say, MS Hearts, they follow a four-section template of: (0) basics, (1) goal of the game, (2) playing the game, and (3) strategy; in that order, which I think is the perfect logic that should be used for describing all games. Why nobody wants to explain a relatively simple game as this except by enumeration of examples is a mystery to me, so here goes:

__Basics:__

The game is played with two joker-included decks of cards. 5, 10, and K are point cards worth face value, except K is worth 10 points. It is a standard 4-player partnership game, so a person and his cross is a team. It is played in multiple rounds and each round there is a dealer (hence a “dealer” team and an opposing “grabber” team). Each team keeps a “level” represented by a card rank (i.e. 2,3,…,A), and each round is associated with the dealer’s level, whose corresponding rank cards are called the level cards.

__Goal of the game:__

A team’s goal in the game is to rise from the lowest level (2) through the highest level (A) before the other team. To rise to a higher level, the team must accrue enough points in a round by taking tricks with point cards in them. In usual score keeping, only the grabber team’s points are counted — it’s a zero-sum game. So the dealer team can be better thought of as seeking to prevent the other team from accruing points.

__Playing the game:__

Both teams start the first round at the lowest level 2.

Establishing trump and rank: Each round of play begins by dealing all but the last 8 cards face-down, hands-up, dealer first. During the deal, the trump suit is determined by revealing identical level card(s) of the same suit or identical joker(s). The player first to the table with the most identical cards at any given moment is said to govern the trump in that suit at that moment; except if another player reveals the same number of (or greater) identical jokers, in which case the first of the latter players to do so governs the trump and the round is notrump. A player who governs the trump at a given moment may only continue to reveal cards identical to the winning ones.

Once dealing completes, the last determined trump suit becomes official. The final card ranks in descending order are then defined as follows: color jokers, black jokers, trump suit level cards, other level cards, other trump suit cards in A…2 order, other rank cards in the leading suit in A…2 order, other rank cards of the other suits in A…2 order. There is a little bit of strangeness here: jokers and level cards are considered part of the trump suit for all purposes of play other than this initial rank definition.

Bottom pile: After dealing/trump determination, the dealer takes the last 8 cards into his hand. Then he takes 8 cards from this hand to leave face-down as the “bottom pile”. Point cards left in the bottom cards have consequences for scoring.

Playing tricks, leading: As usual, the dealer leads on the first trick and subsequently the winner of the prior trick leads. The lead can lay down cards with the following “patterns” (*to be a pattern, cards in the pattern must be in a single suit*): a single card, a multiplicity of *identical* cards (not just in the same “suit”, c.f. trump suit), “tractors”, and “tosses”.

Tractors are runs of the same multiplicity (>1) of identical cards, where “run” is determined by the final card ranks.

Tosses are combinations of singles, multiples, and tractors whose components are seperately unbeatable *in the suit*. It does not imply that a different outcome is impossible were the components played sequentially even in an optimal way (so tosses are advantageous, at least on that trick). However, it *would* be a degenerate contraction play if this were a one-suit game. Anyway, this is the most obfuscating aspect of the game.

Note that there is a recursive expansion of higher order patterns into lower order patterns. Let’s denote the patterns single, multiple, tractor, and toss respectively as S,M,T,X. Then the allowed expansion rules (from observed practice) are M=M*S*, T=M*, and X=T*M*S*. The X expansion is the unique maximal one given by its definition. The T expansion fully and uniquely expands the tractor into its component multiples. The M expansion is different from the rest because it is non-unique.

We can also define an algorithm for matching a subset of some set of cards to any pattern. It involves recursive expansion of the pattern and matching T and M, which are defined by their lengths. Note that T, M, or S are all degenerate forms of X, so without loss of generality we only describe the case for X. Follow this precedence:

1. Given X, the order is to match T* in the expansion, from long to short; each match removes a T from the pattern from further consideration;

2. For the remaining which we call X’, we expand X’=T’*M*S*=M*S* and match the M* from long to short; each match removes an M;

3. For the remaining which we call X”=M*S*, we take the longest remaining M and expand it in X” in all possible ways (which necessarily shortens some M) and sort the resulting list of expanded patterns of X” by longest M, then we discard all but the top of the list, and repeat from 2 on the top of the list; each successful match removes an M; we keep doing this until all the M’s are matched or all M’s are turned into S’s;

4. Match the remaining S* trivially.

To complete the algorithm, define “successful” match to a T or M of a certain length. There are two kinds. In “minimal” pattern matching, a successful match is between T or M in the set of cards whose *maximal* length is *the same as* that of the target T or M. In “maximal” pattern matching, a successful match is between any subset of cards that *can* form T or M of the target T or M length (i.e. they may be part of a longer T or M).

Playing tricks, ranks of same-pattern laydowns: When full pattern matching (i.e. the *maximal* pattern matching requires no T- or M-expansion) occurs between two laydowns, then we can determine rank between them. One multiple outranks another if its singles all outrank; one tractor outranks another if the lowest multiple of the first outranks the same of the second; one toss outranks another if there exists a correspondence of singles, multiples, and tractor components in the two tosses such that each component of the first outranks the corresponding component of the second.

Playing tricks, following: A follower must play the same number of cards as the lead. In selecting these cards, he must attempt to match the leading pattern out of his leading suit using the *minimal* pattern matching algorithm. If his leading suit is exhausted in the middle of the algorithm, the remaining can be made up without restriction using any card of any suit. A trick is won by the person first displaying a laydown that is not outranked. A laydown is not outranked if (1) it has the leading pattern (i.e. is a full pattern match to the leading pattern, and of course in some single suit); and (2) it is not outranked among laydowns having the leading pattern. Corollary: The leading laydown can only be outranked by the same pattern with higher rank in the leading suit or by the same pattern with any rank in the trump suit; if the latter is played, that can only be further outranked by the same pattern with a higher rank in the trump suit.

Scoring: The point card values won by the grabber team are added. If the final trick is won by the grabber team, then the bottom pile is revealed, with point card values in the pile generally counting as double value. If the total is <80, the dealer team wins the round, retains deal, and rises 1 level. If the total is >=80, the grabber team wins the round, and becomes dealer. If the total is >=120, the grabber team also skips a level (so rises by 1). These are the usual cases. Ostensibly each additional 40 points allows the winning team to skip another level, but I’ve not seen this in play.

__Strategy:__

Exercise for the reader.

Extra credit: The 3-deck, 5-person, dynamic partnership variant dubbed “finding friends” (zhaopengyou) can be easily described by diffing with the above description.

In this game, each person keeps his own level. During each round, one person is dealer and it is the dealer’s job to entice one other person to be a partner for the round, during the course of play. This is done by the dealer declaring an “identification card” on the first trick such that, when it is first played by anyone other than the dealer, that person is teamed with the dealer. Then the remaining three players become the grabber team, and they must score 130 points to win, otherwise the dealer team wins. Winning means the people in that team get to advance one level. If the grabber team reaches 195 points, the grabber team gets to skip a level. Losing means people in that team stay at the same level.

Notice that there is a bug here in that the dealer can declare in a way that no partner can be found… I guess that’s ok since it would be stupid to do so, but in other literature it is a strategy with point compensations.

Do you know where one could find some resources on strategy in the tractor game in English?

Cheers,

Neale