The Law of Symmetry

This is an excerpt from Ely Culbertson's 1954 "Contract Bridge Complete" regarding the frequency of balanced and unbalanced distributions. The chapter is still as relevant as it was 70 years ago. "Contract Bridge Complete" has since passed into the public domain.

To say that mathematics is only a small part of bridge is not to impugn the validity of mathematical law, but the laws of simple probabilities must be modified and corrected when applied to cards. In calculating the probabilities of various hand and suit distributions the mathematicians presuppose an abstract perfect shuffle which is nonexistent in practice. This fact renders many of the current mathematical tables, which are sacred to so many experts, at best of problematical value.

The conditions governing simple probabilities in bridge or poker are somewhat similar to those governing the fall of loaded dice. A deck of cards is similarly loaded by the artificial selection of longest suits for trump bids, of smallest cards for losing tricks, and of biggest cards for the winners. Suppose that pure chance has dealt to a player

If the card are not at all shuffled the next deal will be either extraordinarily normal or once more freakish. It will hardly be average and it will certainly not be due this time to pure chance. At a contract of six spades the player leads out seven rounds of spades. The first twenty-eight cards therefore form an artificial pattern, the characteristic feature of which is the abnormal condition that seven times in succession the first card of the trick is a spade. If the dummy holds four strong clubs, one of the opponents will cling to the bitter end with his four clubs, so in the last four tricks there will be at least eight clubs bunched together, and thus the artificiality of the pattern will be enhanced, unless the cards are shuffled to the point of wearing off their spots. Speaking scientifically, every deck of cards after a few, hours of selective playing becomes, in the parlance of card sharps, a cold deck. It is innocently stacked by the players themselves.¹

The factor X - the artificially formed patterns and the imperfect shuffle - must be seriously reckoned with in calculations, and forms the basis of the Law of Symmetry.

The Law of Symmetry is not a law at all in the sense of a physical law. It is rather a loose collection of trends and tendencies which are implicit in the artificial formation of a deck of cards. As an auxiliary to the laws of probabilities it attempts in a quick and practical manner to correct roughly the errors in the application of simple probabilities caused by the artificial suit formations of the play. The Law of Symmetry can be defined as a guide for judging the types (balanced and unbalanced) of suit and hand-patterns in the remaining three hands or, if two hands are seen, in the two unknown hands.

There are three ways of determining the distribution of the· unknown hands. First and most important, by means of card reading. Second, by means of percentages. Third, by means of the Law of Symmetry. The last method, though still imperfect, is the most fascinating of the three, for here a player attempts to penetrate the unknown hands by using only his own hand as a guide.

Thus, if you hold a 4-4-4-1 distribution you will consider it a very morbid symptom indicative of monstrous cancer-like suit growths around the table. To start with, there must be somewhere a long suit to balance your singleton. If it is a five-card length, it means that someone else will have four or five of the same suit - at any rate something unbalanced. Similarly, the remaining suits must, so to speak, entwine themselves around your 4-4-4-1 hand-pattern. And as they do so they will naturally be packed so as to fit in more easily. Thus, if your hand is lopsided or cross-eyed, then the entire deck is very probably lopsided or cross-eyed. The fruit does not fall far from the tree.

When, however, you hold a hand-pattern belonging to the respectable but rather bourgeois family of balanced patterns, say the prosaic 5-3-3-2, this pattern symptom is nothing that is really alarming. At least one of the unknown hands, and at least one of the suits, will also be balanced, and probably other hands and suits as well.

The cards have their own laws of gravitation and their natural trend is along the lines of least resistance: some violent attraction such as an eight-card suit will cause abnormal perturbation throughout the deck, pulling the cards out of their natural orbits even though they be governed by the laws of great numbers. There are thirty-nine suit or hand-patterns, starting with the most common, the 4-4-3-2, and ending with the patterns, such as 11-2-0-0, which, like the rarest of comets, gravitate on the outermost bounds of the distributional constellations - millions of light-deals away. In the Law of Symmetry we are not concerned, at least not directly, with the frequencies or probabilities of various suit or hand-patterns; we are mainly interested in their types. Thus the common balanced type of 5-3-3-2 is a blood cousin to the unusual but stodgy 7-2-2-2; while the unbalanced type 5-4-3-1 is intimately related to the swan-like grace of a 7-4-1-1. The balanced type is a hand or suit pattern which contains no singleton and usually no second long suit. The unbalanced type always contains a singleton and usually a second four-card or longer suit.

Mathematically speaking, a 4-4-4-1 and a 5-4-3-1 are miles apart, the former occurring in three percent of all cases and the latter in thirteen percent. Actually, however, they are far more closely related to each other than a 5-4-3-1 is related to its mathematical counterpart, the 5-3-3-2, which occurs only a little bit more frequently.

Any suit or hand-pattern contains a long suit and its remainders. The remainders of a balanced type are called balanced and those of an unbalanced type are unbalanced. For instance, the remainders of 6-3-2-2 are balanced, while the remainders of 6-4-2-1 are unbalanced. The even number 2 is the dominant characteristic of balanced remainders and the odder number 1 distinguishes the unbalanced.

The suit and hand-patterns of any deal can be expressed in a special table (of which an example is given on this page below) which shows the distributions of the four suits in horizontal lines, and the distributions of the four hands in vertical lines. We can, therefore, call these distributions, in the terms of their arrangement in the table, as horizontal (the suit distributions) and vertical (the hand-patterns).

There often exists a striking parallellism or affinity between two balanced, or two unbalanced suit and hand-patterns. Thus, if one hand is balanced it is probably that somewhere around the table there is another balanced hand. This does not mean that the correspondence is exact, but simply that the various types of kindred distributions, like gloves, are apt to come in pairs. So if you see a 5-4-3-1 in your hand, its brother or cousin hand-pattern is probably hiding around the corner.

What is far more extraordinary is that, in addition to the correspondence between two related hand-patterns, there is a close correspondence between at least one hand-pattern and a suit pattern. Thus, if you hold a 5-3-3-2 type of hand-pattern there is very probably somewhere a suit also distributed 5-3-3-2 or in a similar balanced fashion. It is also extremely likely that the balanced type is formed by the remainder of your own long suit. Again, if you hold a 6-4-2-1 hand-pattern, there is a suit somewhere which is distributed in a similar unbalanced manner, and it is quite likely that this affinity" suit is your own. In other words, the remainder of your six-card suit will probably be unbalanced.

The above facts may be summarized in the following two theorems:

1. A balanced hand-pattern is generally accompanied by at least one other balanced hand-pattern, and an unbalanced hand-pattern is generally accompanied by another unbalanced hand-pattern. The more unbalanced the pattern, the greater the expectation that another hand will be of the same type.

2. A balanced hand-pattern is generally accompanied by a similar balanced pattern in at least one of the four suits, and an unbalanced hand-pattern by a similar unbalanced pattern in some suit, usually the one in which the hand in question is the longest. The more freakish the hand-pattern, the greater the expectancy of a similar freakish distribution of the longest suit.

It will be noted that in the deal previously shown South’s hand is 4-4-3-2, and so is the club suit; West’s hand is 5-3-3-2, to which the distribution of the diamond suit conforms; the other two hands and the other two suits are 5-4-3-1. Here there is perfect symmetry; for every hand-pattern there is an identical suit distribution.

The Law of Symmetry is chiefly useful in teaching one when to expect the unexpected. In the play it is particularly valuable as a basis for deciding, in borderline cases, when to finesse and when to play for the drop, and, even more, when to plan the play on the basis of bad breaks. In bidding its greatest value is in serving as a warning to watch one's step when holding an unbalanced pattern, where the danger of unfavorable breaks is increased, while with a balanced pattern the expectancy of an average break is increased.

For instance, I pick up

It would be the height of naiveté to lean too heavily upon the expectation of average break of either of my long suits. I would accordingly conduct the bidding very gingerly; my spade suit may break evenly, but if the diamond suit breaks 5-5-2-1 then the spades will follow it. Take out this hand from the deck and deal the remaining thirty-nine cards, observing what happens.

If I hold a seven- or an eight-card suit I expect one or two other very long suits. With a hand-pattern 7-4-1-1 I am not so happy about my seven-card suit, for it is astonishing how often it will break 7-4-1-1. It is one thing to hold AKQ32 with a 5-4-3-1 hand-pattern and quite another thing if the hand-pattern is 5-3-2-2.

The hand-pattern 4-3-3-3, while it usually indicates a number of 4-4-3-2's, 4-3-3-3's, and 5-3-3-2's around the table, is sometimes the most deceiving symptom of all. It may be the tail end of violent distributional storms which are raging in other hands. It is the peculiarity of the 4-3-3-3 hand that it often serves as a joiner between the freakish and normal distributions. Hence in bidding I am always careful to step lightly with a 4-3-3-3 unless well heeled.

Once more the reader should be warned that these theorems merely express trends, and not certainties or even strong probabilities. To use the Law of Symmetry as the main guide in the bidding and play would be disastrous: but to use it as a prop, a practical aid to the bidding and playing inferences as well as to the theory of percentages, is definitely valuable.

The Law of Symmetry deals principally with distributions rather than with individual honors, and this aspect of it is the only one with which we shall concern ourselves in this volume. It may be well in passing, however, to note that there are also remarkable correspondences between honors, which suggest that the law may have a much wider application. For instance, if a player holds a singleton King it will happen much more often than probability warrants that another singleton King is in the offing.² Similarly with Queens and other singleton honors. Sometimes, by a strange quirk of suit formations, a singleton honor may evoke a singleton lower honor or vice versa. Even more extraordinary are the cases where a solid sequence in one suit evokes a solid sequence in another suit or a long suit, full of holes, in one hand is in touching affinity with an identical long suit, which is full of identical holes, in another hand.

¹ Two or three master card players in the world have penetrated at least subconsciously to the bottom of cards. They not only remember the cards played in the present deal, but store in their minds the sequence of events in the preceding deal. Then, projecting the lay of cards of the preceding deal into the present deal, they attempt to pick up the broken bits and fragments that remained apparently intact after the shuffle and apply that knowledge to the wizardry of play.

² I am aware that psychologically we are apt to be more struck by the unusual than by the usual and therefore to remember it better, while forgetting the tremendously more numerous instances which did not attract particular notice. To be certain of my conclusions I have for a period of years made actual tests, and am convinced that this is not a psychological illusion.