PERMUTATIONS OF SMALL CARDS

Some readers may be of the opinion that the foregoing theory of small-card systems is not comprehensive enough, as it does not embrace such ideas as odd-even signals, for instance. These are based on:

odd small card  = odd number of cards

even small card = even number of cards

However, we shall show that this is not the case, with some interesting consequences.

Odd-even Signals

These mean that:

having an even number of small cards, we play an even small card (2468)

having an odd number of small cards,we play an odd small card (3579)

But what if we have an even number of cards, all of which are odd ? Or vice versa ?

In that case, we must also take into account the rank of    the small card, and agree that, for example:

 H = even number of cards signal L L = odd number of cards

Next, we must ask ourselves: Having an even number of cards, can we lead any even one (assuming we have more than one)?

Similarly, having an odd  number of cards, can we lead any odd one? Obviously it would be wasteful to ignore any possibility of signalling, so we should use some length signal, say L (normal length). So this version of odd-even signals is in principle the system LL (length) with the following restriction:

If possible the first card played should be even when holding an even number of cards, and odd when holding an odd number of cards.

It is easy matter for the reader to determine that these are simply ordinary length signals, with the proviso that the rank of small cards is

86429753, or 8 > 6 > 4 > 2 > 9 > 7 > 5 > 3.

Applying the signal L (normal length) to this order of small cards, we see that:

Holding an even number of cards we have to play an even small card (as they are "higher" than odd ones)

Holding an odd number of cards we have to play an odd small card (as they are "lower" than even ones)

Lacking a suitable small card (or having a surplus of them) the signal L given will be identical to that given when the order of small cards is "normal", as 8 > 6 > 4 > 2 and 9 > 7 > 5 > 3.

Odd-even signals can, of course, be used in several versions (various combinations of L and L*) which correspond to the following orders of small cards:

86429753    97538642    35792468    24683579

86423579    97532468    35798642    24689753

Theoretically, all these versions are completely equivalent. Clearly, they are also as good as normal length signals (98765432) or reverse length signals  (23456789).

Permutations of small cards

The case of odd-even signals demonstrates that there is no need to slavishly adhere to the natural order of small cards: 98765432.

The numbers on small cards serve solely to distinguish between them, or, more exactly, to compare their rank. To this end, however, we can use any random order of small cards, for example: 72693548  93726458    24589763  etc.

Using any given order of small cards, we can play any small-card system,  for example:

"We lead classically, but the order of small cards is 49732856"

"We use normal length signal, but small cards rank as follows: 3 > 6 > 5 > 9 > 4 > 7 > 8 > 2"

"We play Combine, but the small card x is higher than the small card y if and only if |cos(x)} < |cos(y)|"

Obviously, these things have no intrinsic merit, their only good point being that they demand a lot of concentration from opponents unfamiliar with them.  They can, however, be used, as the Laws of Bridge have not, as yet, banned the use of conventions whose sole purpose is to confuse opponents and force them to expend extra mental effort.

Minimization of the first small card

An idea of Marek Dryanski caused me to reconsider my conclusion as to the intrinsic merit of permutations of small cards.It appears for the first time in this, the third edition of this book:

For any given small-card system, find a permutation of small cards such that the average rank of the first small card played is as low as possible!

Clearly, the smaller the card played to the first trick, the smaller the chance of it being a working card, thus reducing the risk of losing a trick.

Marek Dryanski uses the system MML (mixed) with this order of small cards:

3 5 7 9 10 8 6 4 2 where the lower small cards are positioned at the extremes of the order. This permutation of Dryanski's is excellent for systems using the highest or lowest small cards at the first trick (MM, LL, QQ), although it is not necessarily the best. Finding an optimum order of small cards for other systems would require a lot of work, and the resultant gain would be minimal.

Perhaps the fourth edition will have more to say on the matter.

Equivalence of systems

Two small-card systems will be termed equivalent when one of them applied to a given permutation of small cards is identical to the order.

For example, when the Reverse System is applied to the permutation  2 9 3 5 7 4 6 8 the result is an equivalent small-card system with, amongst others, these plays:

 9 3 = == 9 5 3 = == H 8 7 4 3

 8 6 7 5 3 H 7 6 5 3

This apparent lack of regularity disappears when the small cards in the  above holdings are written not in their natural order, but in order of the permutation 2 9 3 5 7 4 6 8:

 9 3 = == 9 3 5 = == H 3 7 4 8

 6 8 3 5 7 H 3 5 7 6

Each class of systems mutually equivalent contains as many systems as there are permutations of nine small cards (ie 9! = 3,265,920). Equivalent systems have, of course, the same informative properties and from that point of view are equally good.

Reverse Systems

One specific instance of the  equivalence of systems is the equivalence gained by applying the reverse order of small cards (2 3 4 5 6 7 8 9).

System A is the reverse of System B when it is created by applying the signals of System B to the reverse of the natural order of small cards, ie 2 3 4 5 6 7 8 9.

For example, using normal length signals (Low = odd) with the order  2 3 4 5 6 7 8 9 is the same as using reverse length signals (High  = odd) with the natural order, and the same applies to quality and mixed signals.

It is especially easy to work out a given play in a reversed system;it is enough to imagine that the small cards are written in reverse order, thus:

 x x = x x

 x x x x x x

 H x x x H x x x

... etc

The fact of equivalence (and especially reversal) saves us considerable effort when researching into small-card systems: with two reversed systems,  it is sufficient to analyse one, as the efficiency of the other will be the same. For example, in Problem 2 - 3 the systems below are reversed (and thus equally informative):

MUD:

 x x

 x x x

 H x x

Rev:

 x x

 x x x

 H x x

Reversed Combine

From an informational point of view, it is as good as normal Combine. In practice, it is somewhat worse, in view of the fact that an honour a high small card is led (which may be a working small card). Thus far theory of small-card systems has not found the answer to the question - which systems are best and why? So we have to find a method of assessing the quality of a small-card system. We must dismiss all criteria based on observation or the opinion of even the best players. These are false criteria, biased by years of habit and coloured by the chance effect of some fascinating deal. Equally useless would be citing numerous examples of a method's efficacy. Even were we to cite a hundred examples, a determined critic would give a thousand counter-examples. The only objective method of evaluation can be a statistical method, based on examining every possible significant occurrence and counting successes and failures. Such a method

was presented by the author in an article "Distributional Leads" ("Brydz", September 1974) where it was used to determine how to differentiate between a doubleton and a tripleton. In this book it has been extended to apply to length-quality problems.

evaluating

METHODS OF EVALUATING SMALL–CARD SYSTEMS

Thus far the theory of small card–systems has not found the answer to the question – which systems are best and why? So we have to find a method of assessing the quality of a small card system. We must dismiss all criteria based on observation or the opinion of even the best players.  These are false criteria, biased by years of habit and coloured by the chance effect of some fascinating deal. Equally useless would be citing numerous examples od method’s efficacy. Even were we to cite a hundred examples, a determined critic would give a thousand counter–examples. The only objective method of evaluation can be a statistical method, based on examining every possible significant occurrence and counting successes and failures. Such a method was presented by the author in an article “Distributional Leads” (“Brydz”, Sept. 1974) where it was used to determine how to differentiate between a doubleton and a tripleton. In this book it has been extended to apply to length–quality problems.

An example of comparison of small–card systems

Examine this defensive situation:

 xxxx Your partner, West, has led a small card against a suit contract. You have won the ace and return– ? AJ10 ?

the jack, which declarer won with the king. From the bidding it is clear that South has 2 or 3 cards in the suit (ie partner has 4 or 3), which means that there are three possibilities:

 xxx In the hidden hands (partner's and declarer's) there are four small cards.  It is relevant which ones?  No! 1.  xxx  2.  Qxx  3.  Qxxx AJxx 1.  KQx 2.  Kxx 3.  Kx

They can be any four small cards, as the important thing is solely the rank of small cards. So let us say that they are the four lowest: 5 4 3 2. Now let us check how certain systems perform in this situation. How often, irrespective of which small card declarer has played, will we be sure which holding (xxx, Qxx or Qxxx) partner has?

 West has: Cla MUD Rev BT Jou C 543 53 ● 45 43 45 35 54 ● 542 52 ● 45 42 45 25 54 532 52 ● 35 32 35 25 53 ● 432 42 ● 34 32 34 24 43 ● Q54 45 ● 45 54 ● 45 45 ● 45 ● Q53 35 ● 35 53 ● 35 35 35 ● Q52 25 ● 25 ● 52 ● 25 ● 25 25 ● Q43 34 34 43 34 34 ● 34 ● Q42 24 24 42 24 ● 24 24 ● Q32 23 23 32 23 ● 23 ● 23 ● Q543 34 34 34 ● 43 ● 43 ● 43 ● Q542 24 24 24 ● 42 ● 42 ● 42 ● Q532 23 23 23 ● 32 ● 32 ● 32 ● Q432 23 23 23 ● 32 ● 32 ● 32 ●

The above table lists all possible combinations of small cards in the West hand (14 possibilities). For every small–card system the card played by West have been given, and the symbol ● signifies that West's holding is known with complete certainty. As you can see, in this problem Combine has a 100% rate of success; this would also apply to the systems QM, LM, and MML.

An example of statistical analysis

We shall now conduct a detailed analysis (using the same defensive problem) of the MUD system.

First trick

Second trick

F

I0

I1

FS

II0

II1

II2

 x x x

543

4

L.

45

L.

 Small cards played by partner F = to the first trick S = to the second trick

 Symbols  I0  I1   II0  II1  II2 I  II = number of trick 0 1 2 = number of small cards revealed by declarer

 Information gained ● = certainty (1-way) M = Mixed (2-way) L = Length (2-way) Q = Quality (2-way) 3 = Triple (3-way)

542

4

L.

45

L.

532

3

3+

L

35

L

432

3

3+

L

34

3+

L

 H x x

Q54

4

L

45

L

Q53

3

3.

L.Q.

35

L.

Q52

2

Q

25

Q43

3

3.

L.Q.

34

3.

L.Q.

Q42

2

Q

24

Q

Q32

2

Q

23

Q

 H x x x

Q543

3

3

Q

34

3

Q

Q542

2

Q.

24

Q.

Q532

2

Q.

23

Q.

Q432

2

Q.

23

Q.

63.10%

1

1

14

M

L

3

5

5

6

Q

6

9

5

7

3

5

3

13

14

13

14

14

7

7

8

8

14

Private notes:

W oryginale oznaczałem następująco:

 dokładne info – duża czarna kropa mixed = kółko puste w środku length = kwadracik pusty w środku mixed = kółko puste w środku quality = trójkącik pusty w środku three–way = semafor ┐ kropka w środku = plusik w środku =

Dopisać objaśnienia w tabeli z prawej

Cards played by declarer

Let us assume that declarer plays the optimum cards for him, ie those which give us a minimum of information, and never plays an honour unless he has to. This assumption is biased in favour of the declarer, as he will often be either not good enough to play the concrete card or simply too lazy to do so. In theory, however, we have every right to assume that the declarer is infallible.

Four–way information

(ie a total lack of information) has no specific symbol; it is merely denoted by a blank space.

Simplifications

As we move from right to left along the columns, information from either trick generally increases, and never decreases.

Schematically:    I0I1          II0 II1 II2

This fact enables us to simplify our symboling:

1) If there is a blank space in any column (except I0  or  II0) it denotes the same information as in the preceding column.

2) If the symbol ● appears in column I0 then there will be a blank space in all subsequent columns.

To summarize:

a blank space denotes either a repeat of previous information or four–way information.

In the example table repetition of information occurs frequently (in columns I1, II2), but there is no case of four–way information as we have no considered the possibility of the lead being from xxxx, as it is only an example table. It may also be the case that declarer has at this disposal two optimum cards, and, depending on which one he plays, we will have different information. This situation  will be denoted by a double symbol (eg column I1, line Q43).

 Private notes: ♫ (było kółko z plusem) = ilość sukcesów (trafnych decyzji) zakładając że dwuznaczność wystarcza (?) ♪ (kółko z kropką) = ilość sukcesów (trafnych decyzji) zakładając że niezbędna jest jednoznaczność (?) ?? sprawa niejasna – jak to właściwie liczyłem?

Summary indicators

Totals of the various kinds of information are given at the bottom of the table in the lines marked with the symbols: ● M L Q 3.

Success indicators:  ♫ ♪

Summary indicators enable us to compare systems fairly well. But there are dubious situations where it would be better to use one figure rather than several summary indicators to describe the efficiency of a system in every informative situation. For a measure of quality we shall use the success indicators, which indicate how often we can make the correct decision on the basis of available information. These indicators (denoted by the symbols ) appear at the bottom of the table, and their exact meanings are:

= number of accurate decisions assuming 2–way information is sufficient

= number of accurate decisions assuming 1–way information is needed

An accurate decision will be denoted by placing an extra symbol after the symbols: M L Q 3  or  "blank":

a cross (+) for indicator

a dot (.) for indicator

When there is two–, three– or four–way information decisions will be based on the assumption that all possible distributions of small cards have an equal probability. For example, in the example table there are 14 equally probable possibilities:

543, 542, .....,Q54, Q53, ...., Q543, Q542, .....

Obviously, this assumption is a simplification of the true probabilities, but the error introduced by it is not significant, and the analysis can be done without the use of a computer.

Example: calculation ♪ in II0

A correct decision will occur only when partner has the particular holding (ie xxx, Qxx or Qxxx) we play him for, ie the most likely holding based on available information. Let us see how that works depending on the cards played by partner:

45:

Partner can have 543, 542 or Q54, which means we play him for xxx (odds of 2 – 1). We will be correct when he has 543 or 542, so we put a dot after the symbol L on the appropriate lines.

34:

Partner can have 432, Q43 or Q543, which means we have three–way information (xxx, Qxx or Qxxx). As all three possibilities are equally likely, we can play him for any of them – say Qxx. So we put a dot after the symbol L on line Q43.

and so on. Altogether it will turn out that in situation II0 we get 8 correct out of 14.

Example: calculation of ♫ in I0

This time, a correct decision will occur when partner has the more likely of the two possibilities when we have two–way information. Thus all correct decisions of the type are included in , which means we only need to consider three–way and four–way information.

In situation I0, the only time three–way information occurs is when partner leads the three, which means:

either  532     432  (xxx)

or      Q53     Q43  (Qxx)

or      Q543           (Qxxx)

So we should play him for the two–way "xxx or Qxx", which means we will be correct in four cases:  532  432  Q53  Q43.

Correct decisions 532 432 are denoted by a cross after the symbol 3.

Correct decisions Q53 Q43 need not be denoted by a cross as they have already been marked with a dot in the calculation of . (and each is included in ). So we see that the indicator is based on the assumption that two–way information is sufficient for a correct decision.

Efficiency of a system

In the bottom left–hand corner of the table is a percentage, which describes the overall efficiency of the system for that problem, or, more simply, the percentage of correct decisions. This is calculated as follows:

1) Calculate the average of correct decisions of the type using the following weighting:

 I0 = 7 I1 = 5 II0 = 2 II1 = 3 II2 = 1 18 18 18 18 18

2) Repeat the procedure for

3) The average number of total correct decisions is (8/18 • average ) + (2/10 • average )

4) The resultant number is divided by the number of possible distributions and expressed as a percentage.

How often will we correctly decide what partner has led from?

Equal two–way information

This occurs when we have to choose between two alternatives: partner is known to have either A or B, and both are equally likely.

Some examples:

Type 1:1 = H54 or 654

We can choose either Hxx or xxx, as the chances are equal (as all possibilities have an equal probability) and are 1/2 = 50%

Type 2:2 = H54 H53 or 654 653

We can choose Hxx or xxx, as both have an equal probability of 2/4 = 50%

Type 3:3 = H54 H53 H52 or 654 653 652

We can choose Hxx or xxx, both having an equal probability of 3/6 = 50%

In general: we may be dealing with equal two–way information of the type n : n.

Counting successes of type ♪ when dealing with equal information

If we were to rely solely on the equal probability of all possibilities, then when dealing with equal information of the type n : n the success rate would be 50 % (n successes out of 2n occurrences). But we know that apart from the information given to us by the small–card system we are in possession of other information (from the bidding and play), thanks to which the chance of resolving equal information rises to over 50%. This has been analysed in "Evaluation of signals", coming to the conclusion that the chance of resolving equal information depends on the type of signal, the percentage being:

50% for signal M0

56% for signal L

59% for signal Q

68% for signal M1

Thus the success rate when dealing with equal information of the type n:

n is the same as the percentages above. However, in the previous edition of this book, I assigned the following (incorrect) values:

L = Q = 50%      M0 = M1 = 67%

As a complete correction of the above error would require a vast amount of work, I have rectified it in the following manner:

1) For L and Q the value remains at 50%. As they are almost equivalent the error introduced by this is very small.

2) M0 becomes less than 50%. To be more exact, the value for M0 is reduced by as much as it has been increased for M1. This works

because the ambiguity of M0 is always linked with M1(in the same column). But M0 occurs very rarely anyway.

3) M1 (when not linked with M0) remains at 67%. For every three symbols relating to M1(with no dot in the middle and unreduced

by M0) one success of the type *. is added.

4) The role of extra successes (for M1 in the value of a small–card system is analysed in all the summaries.

5) Thanks to this it is possible to correct the excessive advantage of M1over L and Q, which amounts to 17% in the tables (even

though it should only be about 10.5%). The summaries take this into account.

Test problems

To work out the efficiency of a small–card system it is necessary to construct statistical tables for many different defensive problems.

In the next few chapters we shall test the workings of nearly all small–card systems in the following problems:

2–3 (4)    3–4 (4)    4–5 (5)    2–3 (5)    3–4 (5)    5–6 (6)

where the number in brackets refers to the number of hidden small cards (in partner's and declarer's hands).