LISTING THE HANDS
JoŽl BRADMETZ



The complete pack

There are 52! ways of distributing a bridge pack, i.e. 8.065 * 10 e+67. There are not so many hands because all the permutations of the cards of rank modulo 1, 2, 3 and 4 leave the game unchanged. There are 13! such permutations for each rank i.e. 13! * 4 = 1.50356173 * 10 e+39 in all. The number of different possible distributions between four players is thus the total number for the pack divided by these permutations, i.e.: 8.065 * 10 e+67 / 1.503 * 10 e+39 = 5.364 * 10 e+28.
This result can also be reached in considering that there are C(13,52) * C(13,39) * C(13*26) possible distributions of the cards, i.e.: 5.364 * 10 e+28.
From the point of view of the play of the cards (but not that of the contracts and the bidding), this number can be further reduced because the suits are permutable. If one divides the number of possible distributions by 4! (the permutations of the suits), the final number of possible distributions is 13,411,184,439,699,948,178,299,803,550.
This number is very high, but there are more possible ways of distributing a 59-card pack than elementary particles in the whole universe...




A single suit

The treatment of a single suit often represents a segregated module, independent of the game as a whole, although no general theory explains how to put several plays together. If the same team kept the lead for the 13 tricks of the game, the problem would be trivial - that of playing four independent suits - and the only point to watch would be clumsy discards by a partner. The complexity of bridge only becomes evident with the changes of lead.
There are 4 e+13 possible distributions of one suit between the 4 players. If one considers the situation purely from declarer's point of view and hide the opposing hands there are 3 e+13 possible distributions (each of the 13 cards goes either to South, or to North or to the opposition. This number can be reduced by symmetry to (3 e+13 +1) / 2 (one distribution is not permutable: that where South and North are both void in the suit concerned). We thus arrive at the much more reasonable number of 797,162 and, by convention, South will always be attributed with a hand as long as that of North. There are 2 e+13 = 8192 overall different hands available to the opponents, within which one must consider all possible distributions between East and West.





CANONICAL REDUCTION OF HANDS

Many North-South hands are equivalent from the point of view of the play of the suit. When all the transformations and substitutions which are described below have been applied to the North-South hands, we can be said to have carried out a canonical reduction of the hands.





Hands which are identical by permutation

All permutations of adjacent cards between South and North give identical hands as long as the problem of regaining the lead does not arise. AX95-Q4 = AX94-Q5.
Removing theses hands reduces the number to 159,094.





Hands with cards in addition to useful cards

A possibility of reduction is offered by useful cards, that is all of South's strongest cards up to the number of cards held by East-West. For example: AQX83/J976 is played like AQX8/J942 (with the 3 being out of contention and 8 being the weakest of South's useful cards). AQX987-V432 = AQX765-V432.
Removing theses hands reduces the number to 127,842.





Hands which are identical by substitution

Certain cards are inert because they will never take a trick and we can substitute lower cards for them, which avoids gaps in the hand and facilitates computing. For example, AQ/3 can be reduced to AQ/2. These inert cards may have various origins, internal or external to the North-South partnership.





Hands with cards below a sequence in the short hand

Internal reduction is also possible when a hand contains nothing but master cards, let us say n. In this case, the n lowest of partner's cards will be discarded on these master cards and are thus inert. AX976/KQ = AX932/KQ. This reduction can only be applied to master cards because, if AX/KJ = A2/K3, AQ2/JX is different to AQ2/J3 (starting with JKA yields three tricks in the first case and only two in the second).
Removing theses hands reduces the number to 116,477.





Hands with Cards below the weakest useful card

Another internal case comes about when North (which is never longer than South by convention) or South, in the case where the hands are equal, possesses one or more cards which are weaker than the weakest card of his partner. For example, KJ9/753 becomes KJ9/432 or AQX/J96 becomes AQX/J32.
Removing theses hands reduces the number to 75,073.





Hands with Cards lower than the opponents' sentinel

External reduction of inert cards depends on East-West's sentinel. We call the sentinel of rank 1,2,..,n the weakest of the 1,3,..,2n-1 strongest cards of East-West. Consider AJ2/765. In the best possible situation for declarer (i.e. the worst for the opponents), the following sequence will occur: AKQ5 JX94 and the opponents hold the 8 which takes the third trick. The 8 is the sentinel of rank 3 and 7, 6 and 5 are inert cards which can be reduced. Thus: AJ2/765 = AJ2/543. The sentinel takes tricks even when playing to lose as many tricks as possible.
Removing theses hands reduces the number to 66,728.





Twin hands

Finally, a certain number of hands exist which may be called twins because they contain non-neighbouring cards which may be substituted on condition that one is a master and that the strength is concentrated in South by convention. For example AQ32/K54 and AKQ5/432 are twin hands.
Removing theses hands reduces the number to 66,141.





Empirical reductions


Trivial hands
So-called trivial hands are all those in which declarer achieves all his objectives at p = 1 or p = 0 (playing, nevertheless, in an intelligent fashion). A32/K54 is a trivial hand which makes always 2 and only 2 tricks provided that North does not discard K on South's A.
Hands with Single strategy
Certain hands develop one strategy (e.g. AQ/Ø) and may also be considered as trivial.