PART TWO

SMALL–CARD SYSTEMS or signalling quality and length

 

 

WHAT A SMALL–CARD SYSTEM IS

 

What a small card is

The obvious formal answer is:   Small cards   98765432,    Honours  AKQJ10

However, we need a different definition – a more realistic one.

A small card is any card which can be interchanged with any other card  in the  same suit without affecting the play of that suit.

The above definition rightly differs from what the words "small card" and  "honour" bring to mind:

The rank of a small card has no bearing on its trick–taking capacity.

The rank of an honour, in contrast, has a definite bearing on  its trick–taking capacity

 

Some examples to illustrate this:

 

In a suit distributed as follows:

 

AK96

 

105

 

J83

 

Q742

 

all cards below the queen are small cards. Any of them can be interchanged  with any other, and the result will be the same: NS will make 4 tricks in  the suit. Thus the above diagram can be presented more clearly:

 

 

 

AKxx

 

xx

 

xxx

 

Qxxx

 

Each small card is denoted by the symbol "x",

as its rank is totally  irrelevant. 

 

But in this example:

 

A105

 

Q94

 

J82

 

K763

 

the small cards are all those lower than the 8. The 9 and 8 are not small  cards, as exchanging either for the 3 (for example) will enable NS to make 4 tricks in the suit if it is led by East or West. So this suit distribution can be denoted as follows:

 

 

 

A10x

 

Q9x

 

Wxx

 

Kxxx

 

 

The properties of small cards

From the examples given, and our definition of small cards, the following  facts emerge:

1) The boundary between small cards and honours is totally fluid and depends strictly on the distribution in that suit

2) All small cards are equal

 

What is a working small card?

The first fact is one which you have no doubt discovered the hard way, having thoughtlessly discarded a small card and found later that it  was,  in fact, an honour which would have taken a trick. So let us define  a "working small card" as one which may turn out to be a honour (even though it is formally a small card). Clearly, the higher the rank of a small card the greater the probability of its being working.

 

Small cards as sources of information

Taking the second fact, it follows that having decided to play a small card to a trick it is totally irrelevant which one we play, as none of our

small cards will takes tricks. However, small cards may be equal, but they do possess numbers and can be distinguished. This begs the question – how can we take advantage of the fact that they can be distinguished ? 

The answer is simple – to transmit information. If we play the lowest small card rather than the highest; if we play our small cards in ascending order rather than in descending order, then partner will be able to draw certain conclusions from the card we play. Obviously, this will only be so if we have agreed beforehand what these signal mean.

 

What is a small–card system ?

Any kind of information can be transmitted by means of small cards, eg hand pattern, number of aces and kings, number of cards in the majors, etc. However, we will not go into these interesting possibilities here; we shall limit ourselves to basic defensive problem:

How can one transmit information about the suit played by means of small cards ?

We shall call any such method of transmitting information a "small–card system". We will analyse traditional small–card systems and try to discover the optimum system.  NB The General Assumptions from Part One are still valid.

precision

PRECISION OF INFORMATION

Let us decide how to precise information transmitted as to length and quality of a given suit should be.

 

Length

In practice, on most hands the length of declarer's suit is known approximately, on the basis of the 26 cards visible (dummy and your hand) and the course of the bidding and play. An experienced player will know (and will rarely be wrong) that in any given suit:

declarer has 0 or 1 card

declarer has 1 or 2 cards

declarer has 2 or 3 cards  etc.

If he can deduce this about declarer's hand, he can do the same for partner's hand. So it should be sufficient to tell partner whether we have an odd or even number of cards in the suit played. He will be able to work out the exact number by considering the cards he can see, and the bidding and play.

 

Length problems

The deductions made so far mean that defensive problems can be classed as follows:

Problem 0–1

Problem 1–2

Problem 2–3 .... etc.

For example, problem 3–4 signifies the defensive situation when partner knows you have 3 or 4 cards in a given suit.

 

Quality

Obviously, length is not the only important factor. Suit quality (number of honours) is equally important. Practical experience has shown that partner cannot determine your suit quality with a sufficient degree of certainty on the basis of available information (dummy, his own hand, the bidding and play), so you must help him by signalling using small cards in a previously agreed way. Because you rarely possess two honours in a suit (the side playing the contract usually has the majority of honours), and because attempting to inform partner as to the rank of your honour would create too many difficulties, you have to be content with telling partner whether your suit is of bad quality (all small cards) or of good quality (any honour).

 

types

TYPES OF INFORMATION

 

Basic possibilities

So far we have singled out two types of information:

Length  = even or odd number of cards

Quality = no honour or one honour

This means that there are four basic possibilities:

Even number of cards and no honour

Even number of cards and one honour

Odd number of cards and no honour

Odd number of cards and one honour

 Here is a tabular representation of these possibilities applied to problem 3-4 (ie when partner knows you have 3 or 4 cards in the suit):

 

Quality

Length

bad

good

even

xxxx

Hxxx

odd

xxx

Hxx

where "x" = small card and "H" = an honour.

We assume that small cards are in descending order (from left to right), ie in the natural order you place them in the hand.

Signals

Clearly, it would be best to give partner exact information as to which of the four possible holdings you possess, eg "I have xxx", "I have Hxx" etc. Let us forget this ambitious undertaking for the moment and content ourselves with a more modest task: We will give partner ambiguous information of the form “I have xxx or Hxx" or "I have xxx or xxxx" etc. How many ways are there of achieving this?

The following diagram for problem 3 - 4 will show us:

 

 

So we see that there are three possible methods, which will be called signals.

 

Length signal (L) : information about number of cards

odd = xxx or Hxx

even= xxxx or Hxxx

Quality signal (Q): information about quality

bad  = xxx or xxxx

good = Hxx or Hxxx

Mixed signal (M) :  mixed information

?  = Hxx or xxxx       [  !? = odd number of small cards

?  = xxx or Hxxx       [  !? = even number of small cards

 

The mixed signal

The length signal and Quality signal are traditional signals - known and used for a long time. The Mixed signal, however, came about as a result of purely

theoretical speculation, and two questions have to be answered:

1) What is "mixed" information?

2) What is its practical use?

The mixed signal tells partner how many small cards you have:

even = Hxx or xxxx

odd  = xxx or Hxxx

As to the second question, some examples will demonstrate:

   

Problem 3–4:

 

 

 

 

      xxx

 

West leads a small card against a suit contract. East wins the trick with the ace and returns the suit, declarer winning with the king. When East gets in, he is faced with this problem:

1.  xxx

2.  Qxx

3.  Qxxx

 

AJxx

 

1.  KQx

2.  Kxx

3.  Kx

 

If West had Qxx there is a trick to take, but if  West had xxx or Qxxx he must look elsewhere for tricks. It is evident that neither length nor quality signals would help, as if West is known to hold three cards they may be three small cards, and if West is known to hold the queen it may be queen to four. Only the mixed signal is useful in this case.

 

Now for an example of problem 2–3:

 

      xxxx

 

Defending against a suit contract, West led a small card, East played the queen and South won with the ace. Now when East gets in, say with the ace of trumps, he is in trouble:

 1.  xx

 2.  xxx

 3.  Jxx

 

KQx

 

1.  AJ10x

2.  AJ10

3.  A10x

 

If West had xx - he can get a ruff; if West had Jxx - there are two tricks to take; if West had xxx - even cashing the king could be dangerous, as it sets up dummy's long card. Once again, only the mixed signal is of use, as East knows one of the following:

"I have xx or Jxx" (even number of small cards)

"I have xxx"       (odd number of small cards)

It would be best if this information were transmitted by the opening lead.

 

Now an example of problem 4 - 5:

 

      Qx

 

West underlead his ace against no-trumps, and South played the queen from dummy which held the trick. When West gets in again, he should lead:

A10xx

 

1. Jxxxx

2. Jxxx

3. xxxx

 

1.  Kx

2.  Kxx

3.  KJx

 

– the ace in case 1;

– look for an entry to partner's hand in case 2;

– lead any card in case 3.

West will only be able to make the right decision when East's small card is a mixed signal:

even number of small cards = Jxxxx or xxxx

odd number of small cards  = Jxxx

Finally, when the quality of any of declarer's suit is known, the mixed signal in that suit becomes a simple length signal; similarly, when declarer's length in any suit is known, the mixed signal in that suit becomes a simple quality signal.

 

Evaluating signals

Let us evaluate the three of signals by looking at the amount of information they transmit. Once again, here are the 4 basic possibilities:

 

Each signal means that partner has two alternatives, and the chance of distinguishing between them is obviously greater when the difference between them is greater. So we have to examine the difference in each of the 6 signals (L0  L1  Q0  Q1  M0  M1) and arrange them in order of ease of distinguishing between the two alternatives.

In this way we will be able to evaluate each of these signals.

The Quality signal (Q) gives these alternatives:

Q0 = xxxx or xxx

Q1 = Hxxx or Hxx

Both cases are the same, ie a small card disappears (or is added) which means that for the purpose of ease of distinguishing both signals are the same, so:    Q0 = Q1

 

The Length signal (L) gives partner these alternatives:

L0 = Hxxx or xxxx

L1 = Hxx or xxx

Both cases are the same, ie an honour changes to a small card, or vice versa, which means that:     L0 = L1

 

The mixed signal (M) gives partner these alternatives:

M0 = Hxx or xxxx

M1 = Hxxx or xxx

In these cases:    M0 = an honour changes to two small cards

                    M1 = an honour vanishes

 

So we have 4 types of difference:

Q  = a small card vanishes

L  = an honour changes to a small card

M0 = an honour changes to two small cards

M1 = an honour vanishes

To describe these differences numerically, we must establish the relative values of an honour and a small card.

Let us assume that:   1.5 small cards ≤  honour ≤ 2 small cards  which is borne out by practical experience, backed by calculations, in that trump support of xxxx is more or less equivalent to HHx or Hxx. So we have:

value of a small card = 1

value of an honour    = 1 + e     where 0.5 ≤ e ≤ 1

which means that the differences are as follows:

Q  = 1 (a small card vanishes)

L  = e (an honour changes to a small card)

M0 = 1–e (2 small cards change to an honour)

M1 = 1+e (an honour vanishes)

The greater the difference between two alternatives, the greater the information value of the signal. As 1-e < e < 1 < 1+e,  the order of value of the signals is M0 < L < Q < M1

From this we can see that, in spite of popular opinion, length signals are by no means better than quality signals.

Here is a graphical representation:

 

Let us try to convert this to percentages. Given that:

1) e = 0.75 (the most likely value)

2) The worst signal (M0) has a value of 50%

3) The best signal (M1) has a value of 68%

4) The value of a signal is proportional to the value of the difference between the alternatives

We get:     

 

M0 = 50%    Q = 56%    L = 59%     M1 = 68%

 

 

sources

SOURCES OF INFORMATION

 

As we now know approximately what information we are going to impart,   we have to discuss the method of imparting this information.

 

The first two tricks

Information will be transmitted only on the first two tricks in a suit, as by the third trick the position will usually be clear. These two tricks  need not be consecutive; they may be separated by one or more tricks in the remaining suits.

 

The key to signalling with small cards

This cannot be based on attributing specific information to a specific small card, for example we cannot say that the lead of the 3 means Hxx or the lead of the 7 is xxx or xxxx, etc., for the simple reason that the required small card may not be held. The only sensible method is to arrange the small cards in order of rank. Having three small cards, irrespective of their rank, there will always be a highest one, a middle one and a lowest one. So we could agree that from xxx we will lead the highest, from Hxxx the middle, from Hxxx the middle, from Hxxxx the lowest, etc.

 

A notation for the key

To simplify the description of the method we assumed that small cards are written in order of rank from left to right, ie:

 

Hxx

is

H73   or  H62   or   H85  etc

 

 

xx

is

64    or  52    or   43  etc

 

Hxxx

is

H642  or  H853  or   H954  etc

The field of the small card leading to the first trick is colored in light green – and the small card leading to the second trick is in dark green.

For example:

 

H

x

x

means that we first play the lowest one, followed by the highest one.

 

x

x

means that we first play the highest one, followed by the loweest one.

 

H

x

x

x

means that we first play the lowest one, followed by the middle one.

 

A formal definition of a small-card system

So we see that, from a formal point of view, a small-card system is the agreement of a specific method of playing all possible combinations.

For example:

x

x

 

 

 

 

x

x

x

 

 

 

x

x

x

x

 

 

x

x

x

x

x

 

x

x

x

x

x

x

 

 

 

 

 

 

 

H

x

x

 

 

 

H

x

x

x

 

 

H

x

x

x

x

 

H

x

x

x

x

x

The field of the small card leading to the first trick is colored in light green – and the small card leading to the second trick is in dark green.

 

 

Alternative plays

All the signals in the preceding chapter gave information about one of three alternatives:

        Length  = even or odd number of cards

        Quality = good or bad suit

        Mixed   = even or odd number of small cards

Let us now consider how we can signal to distinguish between any of these alternatives. If we wish to do this on the first round of a suit, then the best method is to play either the highest available small card or the lowest available small card. However, if we decide to transmit this information on the first two rounds of the suit, it is best to do so by playing small cards in either ascending or descending order, where ascending order means that the small card played to the second trick is higher in rank than the small card played to the first trick, and descending order the reverse.

 

A notation for alternatives

In the description of alternatives we shall use these symbols:

H  = playing the highest available small card

L  = playing the lowest available small card

A  = playing small cards in ascending order

D  = playing small cards in descending order

Examples:

Playing H in relation to the holding Q75  means playing the 7

Playing L in relation to the holding 982  means playing the 2

Playing D in relation to the holding Hxxx means playing either

 

H

x

x

x

H

x

x

x

H

x

x

x

 

Playing A in relation to the holding A8542 means playing either     

 

A

x

x

x

x

A

x

x

x

x

A

x

x

x

x

 

 

A

x

x

x

x

A

x

x

x

x

A

x

x

x

x

 

 

Sources of information about alternatives

Information about alternatives (either .....or) can come from one of the following sources:

        Source F  = First small card

                    either the highest (H) or the lowest (L) = H or L

        Source O  = Order of small cards

                    either ascending (A) or descending (D) = A or D

        Source S  = Second small card

                    either the highest (H) or the lowest (L) = H or L

It is vital to remember that the small card itself is unimportant, and that its rank is the transmitter of information.

 

Assessment of sources

Let us assess briefly the three existing sources of information:

Source F (first small card)

Information is quick but unreliable, as it is often not clear  whether partner has played L (Low) or H (High)

Source O (order of small cards)

Information is reliable byt slow (not clear until the second trick). However, it may be possible to work out whether the first small card is of the type L (Low) or H (High)

Source S (second small card)

Apparently the worst, as this information is both unreliable and slow. The situation is not as bad as it might be in that by the second trick a substantial number of small cards will have been played.

As you can see, none of the sources is ideal:

F  is quick but unreliable

O  is reliable but slow

S  is both unreliable and slow

 

transmitting

TRANSMITTING SIGNALS

 

Every signal (L, Q or M) transmits information about one of two mutually exclusive occurrences – X and Y, for example:

X =  even number of cards

Y =  odd number of cards

Every source (F, O or S) transmit information by one of two different ways:

H (Highest) or L (Lowest) – for sources F and S

A (Ascending) or D (Descending) – for source O

In order to transmit a signal via a source, we must of course decide which of the two occurrences will correspond to which signal. For instance, we could say that:

 

for source F or S

 

for source O

 

 

H means X

 

A means X

 

 

L means Y

 

D means Y

 

Equally, we could say the opposite, and theoretically it would make no difference which method we decided to use. So it follows that every signal (irrespective of the source used) can be used in two ways, one of which we will call the normal signal, the other the reverse signal.

 

Normal signals (classical)

Normal Length Signal ( L= Length )

L or A = odd number of cards

H or D = even number of cards

Normal Quality Signal ( Q = Quality )

L or A = good quality (an honour)

H or D = bad quality (small cards only)

Normal Mixed Signal ( M = Mixed )

L or A =  even number of small cards

H or D =  odd number of small cards

All of the above methods of signalling will be referred to as normal or classical and denoted by the symbols L Q M. The methods L and Q have been known and used for a long time, which justifies calling them "classical".

Method M, being a new one, has no long history, but one has to start somewhere.

 

Reverse signals

These differ from normal signals because their meanings are reversed:

Reversed Length Signal ( L* = Length* )

L or A = even number of cards

H or D = odd number of cards

Reversed Quality Signal ( Q* = Quality* )

L or A = bad quality (an honour)

H or D = good quality (small cards only)

Reversed Mixed Signal ( M* =Mixed* )

L or A =  odd number of small cards

H or D =  even number of small cards

Reverse signals will be denoted by a dot after the symbol for the signal:   L* Q* M*

 

 Normal or reverse

Theoretically, both variations of the same signal should be equivalent. In practice, however, the normal signal is better; this is because when

you hold, for example, Hxx, the highest small card is often a working small card, so it is best not to have play it to the first trick in order to signal.

 

 Which quality signal is classical?

The normal quality signal ( L or A  = good quality) is applied as follows:

 

x

x

 

 

that is:

lowest small card = good suit

highest small card = bad suit

 

x

x

x

 

H

x

x

 

x

x

x

x

 

H

x

x

x

It may come as a surprise that this method is called "normal" by the author, as popular nomenclature would call it "reverse". However, it is

worth noting that it is equivalent to Culbertson's small–card system, where you lead:

from small cards – your highest small card

from an honour – your lowest small card

Thus the method "L or A = good quality" is more classical than the opposite (“H or D = good quality”) and deserves to be called "normal". That this is not the case is the result of a misunderstanding.

 

The reason for the misunderstanding – encouragement

To understand why "H = good quality" has been given the name "normal", let us assume that you are defending in classical Culbertson style and partner has led the king (showing AK or KQ) against a suit contract. Which small card should you play from the following holdings:

 

x

x

Q

x

x

x

x

x

 

Since you are leading in classical style, it would be most convenient to use the same signals when following suit as when leading, ie low small card = good quality, which means you would play thus:

 

x

x

Q

x

x

x

x

x

 

However, this is not a good situation to be in as partner will not be able to distinguish between two and three small cards. So perhaps it is  better to use the normal length signal (low small card = odd number of cards):

 

x

x

Q

x

x

x

x

x

 

This is also bad, because now partner will not be able to differentiate between three small and queen to three. So in this type of situation neither normal quality signals nor normal length signals are useful (the same applies to reverse signals). To avoid this, encouragement was introduced; a small card played to a trick which partner has led to means:

continue the suit (encouragement)  or  switch (discouragement)

As it is instinctive to assume that “high = positive” and “low = negative”, the following system of encouraging was used:

High = encouraging

Low = discouraging

which, after a long period of use, came to be called normal (classical). In turn, because it is more common to encourage with an honour than with small cards only, "High" came to mean "good quality". This was termed normal in spite of the fact that the classical small–cars system is based on  "High = bad quality". One final point: "Low  = encouraging" is better, as when you possess an honour, a high small card in that suit is more often a working small card in a weak suit. So this method should be termed normal.

definition

DEFINITION OF A SMALL–CARD SYSTEM

 

There are two ways of defining a small–card system:

– either strictly formal

– or structural

 

Formal definition

Operates by agreeing which order small cards should be played (at both  the first and second tricks) for every particular holding:

 

x

x

 

 

 

 

x

x

x

 

 

 

x

x

x

x

 

 

x

x

x

x

x

 

x

x

x

x

x

x

 

 

 

 

 

 

 

H

x

x

 

 

 

H

x

x

x

 

 

H

x

x

x

x

 

H

x

x

x

x

x

Reminder:

1) We assumed that small cards are in descending order (from left to right),

     ie in the natural order you place them in the hand.

 

2) The field of the small card leading to the first trick is colored in light green

    – and the small card leading to the second trick is in dark green.

For example:

 

x

x

x

x

x

H

x

x

x

x

x

x

H

x

x

x

x

x

x

x

x

... etc

 

 

 

 

 

 

 

 

This definition enables you to describe any random small–card system, ie it is universal. However, it is uninteresting in that it does not delve into the meaning of the cards played, limiting itself to a formal description.

 

Structural definition

This describes a small–card system as the interaction of three sources of information, each of which transmits a given signal. We have three

 sources at our disposal: F O S, and six possible signals:

 

 

Normal

Reverse

 

 

 

Length

L

L*

 

 

 

Quality

Q

Q*

 

 

 

Mixed

M

M*

 

 

To define a small–card system we have to establish which signal will be transmitted through which source. This means that a small–card system can be described as tripartite:  SF  SO  SS   where:          

 

SF  = signal transmitted by F

For example:

LQM    QQL*    ML*Q   Q*MM   QL*M   M*M*

 

SO  = signal transmitted by O

 

SS  = signal transmitted by S

 

Interaction of sources

At first sight it might appear that each source could transmit any signal, irrespective of which signal the others transmit. However, it would be pointless to have the same ambiguous information transmitted by two sources, when one uses a normal signal and the other a reverse signal. If they transmit the same information, they should do so using the same version of the signal. The amount of information will not become less, and the signalling method will be more concise.

Conclusion:

Only one of two opposite signals should be used.

Equally, there is no point in transmitting the same information through sources O and S, as both source give the information at the same time (the second trick), and source O gives accurate information, so  SOSS. On the other hand, sources F and O can be used to transmit the same information. This ensures both speed (F) and accuracy (O). Also, this interaction of F and O means that it is often easier to determine whether the card played to the first trick was of the type H (Highest) or L (Lowest).

 

Nomenclature of classifiable systems

The symbolic name of a system which can be described using the structural definition (ie classifiable) is  SFSOSS 

Systems in which the lowest possible card is played at trick two will be denoted by SFSOLow or simply SFSO

Each source will be assigned a specific signal, with the proviso that the signal corresponding to S will be either in brackets or omitted.

Some examples:

QML = quality–mixed (length)

QL*Low  = quality–reverse length

L*M  = reverse length–mixed

In systems where SFSO, the first two signals will be the same,  for example:

QQM = quality (mixed)

MMLow  = mixed

L*L*Q   = reverse length (quality)

 

Reconstruction of a system using its name

Taking the small–card system QM:

Source F:  transmits signal Q, ie

Low  = good suit (an honour)

High = bad suit (no honour)

Source O : transmits signal M, ie

A  = even number of small cards

D  = odd number of small cards

Source S : Low    (as there is no third symbol in the name)

 

Thus, using the system  QM, you would play:

 

x

x

 

 

 

 

x

x

x

 

 

 

x

x

x

x

 

 

x

x

x

x

x

 

x

x

x

x

x

x

 

 

 

 

 

 

 

H

x

x

 

 

 

H

x

x

x

 

 

H

x

x

x

x

 

H

x

x

x

x

x

Note that L is not always the smallest card. To be more exact, it is  the smallest card consistent with the signal transmitted by source O. This also applies to H.

 

 

 

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