PERMUTATIONS OF SMALL CARDS
Some
readers may be of the opinion that the foregoing theory of smallcard systems
is not comprehensive enough, as it does not embrace such ideas as oddeven
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.
Oddeven 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 oddeven 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.
Oddeven
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 oddeven 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 smallcard 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 smallcard 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
smallcard 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
smallcard system with, amongst others, these plays:







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:







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:










... etc 
The
fact of equivalence (and especially reversal) saves us considerable effort when
researching into smallcard 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: 





Rev: 



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 smallcard 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 smallcard 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 counterexamples. 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 lengthquality problems.
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 
I_{0} 
I_{1} 
FS 
II_{0} 
II_{1} 
II_{2} 



543 
4 
L. 

45 
L. 

● 


542 
4 
L. 

45 
L. 

● 

532 
3 
3+ 
L 
35 
L 

● 

432 
3 
3+ 
L 
34 
3+ 
L 
● 


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 

● 


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:
Dopisać objaśnienia w tabeli z prawej 
Comments
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: I_{0} ≤ I_{1 }II_{0} ≤ II_{1}
≤ II_{2}
This fact enables us to simplify our symboling:
1) If there is a blank space in any column
(except I_{0} or II_{0}) it denotes the same
information as in the preceding column.
2) If the symbol ● appears in column
I_{0} 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 I_{1}, II_{2}), 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
I_{1}, 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 II_{0}
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 I_{0}
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 I_{0},
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:

I_{0} = 
7 
I_{1} = 
5 
II_{0} = 
2 
II_{1} = 
3 
II_{2} = 
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.
This answers the question:
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 M_{0}
56% for signal L
59% for signal Q
68% for signal M_{1}
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% M_{0}
= M_{1} = 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) M_{0} becomes less than 50%. To be
more exact, the value for M_{0}
is reduced by as much as it has been increased for M_{1}. This works
because the
ambiguity of M_{0} is always
linked with M_{1}(in the same
column). But M_{0} occurs
very rarely anyway.
3) M_{1 }(when not linked with M_{0}) remains at 67%. For every
three symbols relating to M_{1}(with
no dot in the middle and unreduced
by M_{0}) one success of the type *.
is added.
4) The role of extra
successes (for M_{1} 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 M_{1}over 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).