PART TWO |
SMALL–CARD SYSTEMS or signalling quality
and length |
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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:
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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: |
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Each small card is denoted by the symbol
"x", as its rank is totally irrelevant. |
But in this example:
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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: |
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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 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 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):
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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:
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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: |
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xxx |
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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 |
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AJxx |
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1. KQx 2. Kxx 3. Kx |
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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:
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xxxx |
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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 |
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KQx |
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1. AJ10x 2. AJ10 3. A10x |
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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:
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Qx |
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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 |
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1. Jxxxx 2. Jxxx 3. xxxx |
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1. Kx 2. Kxx 3. KJx |
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– 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:
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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 |
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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 |
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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 |
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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:
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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:
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M0 = 50% Q = 56% L = 59% M1 = 68% |
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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:
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Hxx |
is |
H73
or H62 or
H85 etc |
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xx |
is |
64
or 52 or
43 etc |
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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:
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means that we first play the lowest one,
followed by the highest one. |
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means that we first play the highest
one, followed by the loweest one. |
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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:
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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
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Playing A in
relation to the holding A8542 means playing either
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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 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:
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for source F
or S |
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for source O |
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H means X |
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A means X |
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L means Y |
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D means Y |
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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:
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that is: lowest small card = good suit highest small card = bad suit |
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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:
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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:
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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):
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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 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:
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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:
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... etc |
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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:
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Normal |
Reverse |
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Length |
L |
L* |
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Quality |
Q |
Q* |
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Mixed |
M |
M* |
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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:
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SF
= signal transmitted by F |
For example: LQM
QQL* ML*Q Q*MM
QL*M M*M* |
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SO
= signal transmitted by O |
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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 SO ≠ SS. 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 SF ≠ SO, 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:
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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. |