Never miss a game again? That’s easy –
simply bid a game on every board :) While Bob Hamman’s
note that “Bidding is only 3% of the game of bridge”
may be true, if you don’t bid your games you certainly
cannot catch up by making 13 tricks with brilliant play
at your 2
contract, while your notsobrilliant opponents make
only 10 tricks at their 4
contract. The experts know that, though – and
they bid “aggressive” games that “somehow,
magically” turn out to be cold. Experts use their
expert judgment which advanced and intermediate players
just don’t have yet. This article presents the
tool for the advanced and intermediate players to get
this expertlevel “aggressive” judgment
and never miss a game again – be it a “somehowmagical”
or just a “regular, plain” contract, and
to stop at a partscore when no game is in sight.
The ZarPoints theory is a result of exhaustive research
of hundreds and hundreds of “aggressive”
game contracts bid by worldclass experts like Hamman,
Wolff, Meckwell, Lauria, DeFalco, Zia, Helgemo,
Chagas, Sabine Auken,
Karen McCallum (I have great respect for the women experts)
and many others at various worldclass tournaments.
Following the 8020 rule, hand evaluation is 80% Initial
evaluation and 20% evaluation adjustment as the bidding
progresses. The initial evaluation (just as you pick
up your cards and have a look at what’s in there)
captures the three standard important aspects of every
hand: the shape, the controls, and the standard (Milton
Works 4321) HCP. The reevaluation covers the placement
of the honors and the suitlengths in the light of partner’s
and opponents’ bidding.
Here is the simple quick description of the initial
hand evaluation (Zar Points or Zars).
Calculating the Zar Points has 2 parts – calculating
the Highcard
Points (HP) and the Distribution
Points (DP).
For the highcard
points we use the 6421 scheme which adds the
sum of your controls (A=2, K=1) to your standard Milton
HCP, in the 4321 scheme (A=4, K=3, Q=2, J=1). You
will see WHY we have adopted this HP counting in
the second part of the article, but the short answer
is: NOT because “we feel that this is the best
way” :)
Calculating distribution
points is not news in Bridge – Charles Goren
introduced the Goren Points
more than halfacentury ago. It counts 3 points for
every void, 2 points for every singleton, and 1 point
for every doubleton. You understand, of course, that indirectly it also holds implicit valuation
for the long suits, since the sum of all the 4 lengths
is 13 – so, for example, the flat 4333 distribution
gives you 0 Goren distribution
points, while with 55 twosuiter
you get 3 Goren points (either
2+1 for a singleton and a doubleton or 3 for a void).
As we are going to see, there are only 39 different possible distributions in
a bridge hand. To get a feel of the enormous amount
of hands these 39 “types” of distribution
represent, just asks yourself how many deals are there,
in which YOU, sitting in the dealer’s position
(East, throughout this article) get a 13000
distribution. So – how many do you think?
The answer may surprise you –
337, 912, 392, 291, 465, 600
DIFFERENT deals in which you’ll have 13000
distribution  the LEAST probable distribution! How
about the MOST probable distribution of 4432? You
guessed it – “a bit more”:). You probably know by now that the number of all
possible deals in bridge is
53, 644, 737, 765, 488, 792, 839, 237, 440, 000
and the goal of the Distribution
Evaluation Methods is to put some order in this enormous
amount of “material”.
If we focus our attention on
a single hand, rather than all 4 hands constituting
a deal, the numbers certainly are many orders of magnitude
smaller. The total amount of hands you can have in bridge is only
635, 013, 559, 600
– a number you can handle much better, I
guess – at least in terms of pronunciation :)
From all the numbers above, the most important one is
probably the number 337, 912, 392, 291, 465, 600
–
the number of possible different deals in which you
have a 13000 distribution. Why, you may ask –
because it engenders the importance of reevaluation. Since there are so many
deals in which you hold the “stiffest” distribution,
you know that the number of deals for a FIXED “more
normal” distribution are orders of magnitude bigger
and you have to reevaluate your hand in the light of
the guidance given you by the line of bidding presented
at the table, thus adjusting your hand in this enormous space.
Now that we know what we are up against, let’s
continue with the way Zar Points are assigned to different
distributions. Let’s start with the initial evaluation
as you pick up your cards. Here is what you do. You
add:
The Highcard Zar points (HC)
you are already very familiar with (MiltonHCP + Controls or 6421)
The difference between the
lengths of the Longest and the Shortest suits (we call
it S2)
The sum of the lengths of the
Longest 2 suits (we call it L2);
That’s
all: HC + S2 + L2.
Why
the difference S2
between the longest and the shortest suit, though?
For simplicity, let’s denote your longest suit
with a, the second longest with b, the 3^{rd}
with c, and the shortest suit – with d. This means
that the following 3 hands have a 5332 distribution
with a=5, b=3, c=3, d=2:
A x xxx
K x x
K J x
x x
K x x
A x x
x x
K J x xx
Q x
x xx
A K x xx
J x x
Now,
the reality of Zar Points is that we add ALL
the 3 differences of your suits:
( a – b) + (b – c) + (c
– d).
But wait … look what
happens when you drop the parenthesis – both b
and c disappear and the expression becomes very
simple:
(a –
d)
So:The entire amount of the Distributional
Zar Points is:
(a
+ b) + (a – d)
It looks like the suit “c”
doesn’t participate in the Zar Points calculations,
but this is illusive, as you can see from the simple
algebraic manipulation that lead us to the (a + b) +
(a – d). If we continue a bit with the algebraic
manipulations, we get:
(a + b) + (a  d) = a + b + c + d 
c  d + a  d = 13 + a  c  2d = (13  2d) + (a  c)
If it is easier for you, you
may calculate the Distributional Zar Points from the
formula (13 – 2d) + (a – c).
Or make some other manipulation
that would better suit your memory. To me, (a+b) + (ad) is simple enough.
The flat 4333 distribution
has the minimum amount of Distributional Zar Points,
(4 + 3) + (4 – 3) = 8
points, while the 7600 has (7 + 6) + (7 –
0) = 20, for example. If you increase the length of
the longest suit, the valuation also increases, of course
– 9400 has (9 + 4) + (9 – 0) = 22, and
the wildest 13000 hand gets the max of (13 + 0) +
(13 – 0) = 26.
So
you have calculated the HP
portion first, and then have added the
DP portion for the Distributional Zars.
Now,
if the sum is 26
or better, you have an Opening
Hand. Here are some examples, to get your feet wet:
11+4+3+8=26
11 HCP
K J x xx
K x x
x xx
A x
10+4+4+9=27
10 HCP
x
K x xxx
K x xx
A x x
8+4+5+9=26
8 HCP
A x xx
A 10 x xx
x xxx
___
10+3+4+9=26
10 HCP
Q 10 x x
A x x
x
K J x xx
9+2+5+10=26
9 HCP
K Q x xx
K J x xx
x xx
___
7+3+6+11=27
7 HCP
K x xxxx
A x xxx
x x
___
If
Zar Points look a bit aggressive to you, let’s
have a look at several opening hands from the justpassed
First Open European Championship inMenton, France.
Qx
AKxxx
Jxxxx
x
Menton Bulletin 11:
"Chagas'
light distributional opening bid changed matters".
In fact the hand has 4 + 10 = 14 distributional
Zars, plus the 9 HCP (Qx) + 3 controls = 27 Zar Points,
well into the Opening Hand range. Nothing special
indeed. See the note about the implications
of having two 5card suits below.
Axxx
AJxxxx
J
xx
Menton Bulletin 9:
9 HCP, after you
discount the singleton J. Still Both Duboin and Ludewig opened the hand in the Open Teams. And indeed, the
distribution Zars are 5 + 10 = 15 Plus the
4 controls and the 9 HCP(singl. J) = 28 Zars! Well above
the opening minimum of 26.
Jxxx
x
KQxxx
KQx
Menton Bulletin 9:
11 HCP again, with only 2 controls ... but rich on distribution Zars: 4 + 9 = 13 points!
The total Zars are 11 + 2 + 13 = 26, an
opening hand. And indeed, Both BenitoGarozzo and
Andrew Robson
opened the hand in the Open Teams event.
Certainly, all “disabilitycombinations”
like KQ, QJ, singleton honor etc. discount the
standard way.
In the same time, you get 1 “upgrade
point” if all
your points are concentrated within 3 suits (if
you have a strong hand of 15+ HCP) or within 2
suits (if you have a “normal” opening
of 1114 HCP). In “light” opening
you never get this 1point upgrade. This upgrade
actually takes care of the value added by having
you honors “in combinations” rather than being
scattered around the 4 suits.
While we are on the wave of Menton, let’s
give you one final “touch” in the
Initial Hand Evaluation – it concerns holding
the Spade
suit – the so called “President’s
Suit”. In bordercases, when you have 25
Zar Points, you add 1 point for holding the Spade
suit. ONLY when you are at the border of opening,
holding the spade suit gives you the right to
add 1 Zar Point and get to the 26Zars opening.
If you think that holding the Spade suit is of no importance, let me tell
you – it may not be of any importance in
cricket, but in bridge it IS “:). Here
is an example of such an opening coming again
from Menton, with the ToBeEuropeanChampion Eric Rodwell
being in action:
AQxx
Jx
Axxx
xxx
Rodwellopened 1D and as
the commentator said "EW were talked
out of their game by Rodwell's light opening bid...".He actually has 11 HCP (depreciates the Jx but gets 1 pt back for 3suits concentration of points) plus 4 controls
for 15 points, plus the 2 + 8 = 10 DP for 25 Zars. When you upgrade the hand for holding the "president's
suit" of spades by 1 pt, you reach the 26.
And if you happen to actually open 1 S with 5 cards in spades, not only
you put the opponents on a defensivebidding
track, but you also cut the entire Levelone
bidding space.
So we see that a welldistributed hand with 89 HCP and 34 controls may
easily qualify for an opening. Let's ask the
more general question now: "WHY is it worth opening a "subopening"
hand, and WHEN?”
We already mentioned that the total amount of hands you can have in bridge
is 635,013,559,600. The more interesting thing
to note is that all the hands with 12 HCP or
more, all the way to 37 HCP, are
221,093,636,000
or 221 BILLION, while
the number of hands in the short811 HCP range is ... BIGGER(!) :
232,403,610,336
or 232 BILLION.
You see now that chances are better
for holding an 811
hand than to have ANY
"normalopening" hand. This "discovery"
should persuade you to consider "light
openings", even if you disregard the merits
coming from the very fact that you have entered
the bidding effectively putting
the opponents in a defensive bidding track.
Let's have a closer look at the Opening Hands with 811 HCP and determine
some General
Rules you need to follow in case you hold
an 811 hand, in the light of Zar
Points evaluation.
Hand with 8 HCP
4333 = 8needs10
controls = Pass
4432 = 10needs 8 controls
= Pass
5332 = 11needs 7 controls
= Pass
5422 = 12needs 6 controls
= Pass
5431 = 13needs 5 controls
= Pass
6322 = 13needs 5 controls
= Pass
5521 = 14needs 4 controls
= AA
5440 = 14needs 4 controls
= AA
6421 = 15needs 3 controls
= AA or AK
Hand with 9 HCP
4333 = 8needs 9
controls = Pass
4432 = 10needs 7 controls
= Pass
5332 = 11needs 6 controls
= Pass
5422 = 12needs 5 controls
= Pass
5431 = 13needs 4 controls
= AA only
6322 = 13needs 4 controls
= AA only
5521 = 14needs 3 controls
= AK or AA or KKK
5440 = 14needs 3 controls
= AK or AA or KKK
6421 = 15needs 2 controls
= A or KK
Hand with 10 HCP
4333 = 8needs 8
controls = Pass
4432 = 10needs 6 controls
= Pass
5332 = 11needs 5 controls
= Pass
5422 = 12needs 4 controls
= AA or AKK
5431 = 13needs 3 controls
= Any 3 controls
6322 = 13needs 3 controls
= Any 3 controls
5521 = 14needs 2 controls
= Any 2 controls
5440 = 14needs 2 controls
= Any 2 controls
6421 = 15needs 1 control
= K is enough
Hand with 11 HCP
4333 = 8needs 7
controls = Pass
4432 = 10 needs 5 controls
= AAK only
5332 = 11needs 4 controls
= AA or AKK
5422 = 12needs 3 controls
= AK or KKK
5431 = 13needs 2 controls
= Any 2 controls
6322 = 13needs 2 controls
= Any 2 controls
5521 = 14needs 1 control
= one K is enough
5440 = 14needs 1 control
= one K is enough
6421 = 15needs 0 control
= no need for any controls
In all PASS cases the decision is made on the fact that the point limitation
cannot accommodate the needed controls, e.g.
you cannot have 5 controls in 10 HCP since AAK
are already 11 Milton points.
This leads us to the following summary, which is worth remembering as a
general guideline, even if you are too lazy
to count Zar Points because “you are playing
for pleasure and fun” :)
SUMMARY:
1)
With 8
HCP  you need AT LEAST55, 64 or 5440 distribution with 2 Aces
2)
With 9
HCP  you need AT LEAST5431 distribution with 2
Aces
3)
With 10
HCP  you need AT LEAST54
distribution and corresponding controls
4)
With 11
HCP  you need EITHER a 5card suit OR 5 controls as a minimum
Simpleenough
guidelines, I hope.
How do you deal with “Normal” opening hands with
balanced distribution? And how do Zar
Points get involved after a balanced opening
of 1 NT for example (at the very end of these
discussions you’ll find some considerations
regarding different standard systems like “Two
over one”, “Standard American”,
“Strong Club” etc., which will give
you a general perspective about how Zar Points
fit into “your”
current system).
Let’s consider two boards
in which the opener EAST has the same hand with
balanced 15 HCP, but the responder WEST holds
completely different hands, although with the
same amount of 12 HCP:
1)
J
x x
J
x x
A
K x x
K
J x
K
Q x
K
Q x x
Q
J x
Q
x x
Would
East open the bidding to begin with? The answer
is yes, because he has more than 12 HCP and
it’s an opening hand by any system. If
the HCP power warrants an opening by itself,
you open the bidding the way you usually do
with the system you are using – most people
would open 1 NT with the East hand. NOTE, that
counting Zar Points with a balanced hand will
NOT help you – with these 15 HCP you collect
only 25 Zar Points which “formally”
means you should pass.
Zar
Points are geared towards aggressive
bidding with distributional
power rather than hands with brute HCP force
and balanced hands – every pair would
bid and make 3 NT on this first board with a
natural and simple sequence of 1NT – 3NT
(not even a Stayman
used :).
Now,
the second example:
2)
A
J x xx
A
x xxx
K
x x
___
K
Q x
K
Q x x
Q
J x
Q
x x
Here
the sequence is a bit different than 1NT –
3NT :). Normally WEST would transfer in one
of the majors and then rebid the second major,
allowing EAST to “upgrade” her K
Q holdings in both majors and proceed towards
the cold slam.
What kind of weight to put on the components you consider valuable is not
a matter of "expert judgment", but
a simple matter of solving a series of equations with unknown
coefficients  an obviously overdetermined
system of equations (you enter hundreds of equations
based on the hundreds of boards you feed in,
for finding the value of several of coefficients
– the weights you are interested in).
It is a wellknown fact that the standard 4321 valuation IS the one
that solves the system of equations when using
any of the standard distributionpoints systems
(Goren, Bergen, etc.)
and it is also known that the HCP + Controls
valuation (the 6421) is NOT a solution with
the standard distribution points.
How do you create the equations for a specific board in order to calculate
the “right” weights? We’ll
count HCP points and distribution points for
void, doubleton, and singleton (kind of Goren style).
because in the combined hands
you have2
Aces, 3 Kings, 2 Queens, 1 Jack, and 3 doubletons.
You make a collection of hundreds and hundreds of boards that have a game
(4 in major) and solve the overdetermined
system of equation to find the values of the
unknown coefficients. Simple.
For the board above (4 Spades), if we consider the plain 4321 Milton
Works points, assigning 4 for A, 3 for K etc.
and assign Goren distribution
points (3 for void, 2 for singleton, and 1 for
doubleton), we see that those ARE a solution
for our first (and only for the time being)
equation:
so you have to “collect”
25 points to get a game with Milton / Goren
points. The same way we have run the systems
for the Zar Points which have much more variables
to calculate.
While on the subject, you may come up with some “igneous idea”
that in bridge it’s only the Kings and
Jacks that count (because they are the “male
cards” :) and construct the corresponding
equations. And you WILL
find corresponding solutions for the coefficients
– so the natural question is “Why
not?” The answer is that such a solution
will have much
bigger deviation for the actual equations
(well, I’ll stop here :).
We will do an exhaustive comparison between Goren points (of Charles Goren), Bergen
Points (of Marty Bergen), Drabble
Points, and Zar
Points in the second half of this article so
you’ll get the picture.
You already might have guessed WHY the 4321 valuation solves the equation
system for the standard distributionpoints
systems while it does NOT solve the equations
for the Zar points (the 6421 is the one that
does).
The reason is the relative weight of the distributional points vs. the
weight of the honor points. As we have seen,
the distributional Zar points range goes to
26 (for the extreme case of 13000) while
the standard distributional points range goes
up to 13 at most (see below)  cannot compensate
the weight of the 6421.
Note also, that the experts know that the 4321 is a "twisted" solution, meaning that it undervalues
the A and K and overvalues the Q and J  that's
why they use fractions to "make it work"
(they count A for 4 1/2 while Q for 1 1/2, which
makes three Queens equal to one A, just like the 6421 valuation).
Zar points allow the natural 6421 honor count (which experts lean towards)
to be the solution of the overdetermined
system of equations.
Your judgment
Do you still need to apply your judgment and consider both hands of the
partnership in the bidding process? You bet.
Here is a simple example which covers both cases
 your own hand and the combination of the two
hands in the partnership.
Let's consider two different hands in opening position, with the same 26
Initial Zar Points, 10 HCP, 3 Controls, 5431
distribution. Which one do you like better?
NorthA
8 6 5 3 2
7 5 4 2
A
Q J
K
NorthB
A
J 10 8 4
K Q 10 9
10 9 6
4
I guess you have a preference here :). Certainly hand "A" will
be greatly downgraded from the initial 26 Zar
Points while the second hand "B" will
be upgraded for a number of reasons. BUT, let's
consider the Partner's hand in both cases and
how this dramatically changes the picture. In
both cases the partner has 27 Zar points  10
HCP, 4 controls, 5431 distribution.
SouthA
A 10 9 7 4
9
10
8 6 5
A
Q 6
SouthB
K
8
7 6
A
7 4 2
K
9 7 5 3
No comment needed  you'd reverse your "preferences" and you
would prefer the set "A". In bridge,
you always need your head on your shoulders,
at any stage of the game :)
This brings us to the next section where we consider
the adjustments to the partner’s and opponents
bidding.
So
your partner has already opened, the main consequence
being that YOU are in offensive bidding while
the opponents have
been already pushed in the defensive track.
How does this affect your hand evaluation?
You
first do the Initial hand evaluation that has
been covered in “The Opening” section
and THEN make certain adjustments  adjustments
to the partner’s suit and adjustments
to the opponents’ suit. The minimum pointcount
that allows you to talk is 16:

1 additional point for the trump honors (trump
10 counts for 1, trump A counts for 5) up to
MAX 2.

1 additional point for the Invitationalsecondsuit
honors (10 counts for 1, A counts for 5), MAX
2.
The total allowance here is two,
whether 2, 3, 4 or 5 are held (the rest goes
away as 'duplication values'). So how do you
judge the level you are ready to play at? Here
are the Game calculations:
 52 Zars for Game at level 4 ( two
opening hands make a game),
 57 Zars for level 5,
 62 Zars for a slam at level 6.
Plain and simple  5 points
per level.
These 5 points may come from an additional K
in the partner’s suit (3 points from the
HCP, 1 from the control, and the premium 1 from
the honor in the partner’s suit), from
an additional outside A (2 from the controls
plus 4 from the HCP) etc.
Let’s close this section
with four of the most
common situations in game bidding (the
longsuit
invitation has already been mentioned,
with the reevaluation of the responder’s
hand based on the 2 long suits of the opening
hand).
Q 10 x x
A x
x x
K x xxx
K J x xx
K x x
x xx
A x
East has 26count = 11 HCP
+ 8 Long (5+3) + 3 Short (52) + 4 Control
= 26. West has 21 = 9 HCP + 9 Long (5+4)
+ 3 Short (52) + 3 Control = 24. E
opens 1S, W corrects +1 for Q + 1 for 10 = 26 (K trump = 4, Q trump = 3, J trump = 2, 10 trump = 1).
1 S – 4 S.(2630)  52 deeded for level
4.
Add 10 Zars = AKx in and
you get the slam (5 Zars per level).
Q 10 x x
Q x
x x
K x xxx
K J x xx
K x x
x xx
A x
East has 26count = 11 HCP
+ 8 Long (5+3) + 3 Short (52) + 4 Control
= 26. West has 20 = 7 HCP + 9 Long (5+4)
+ 3 Short ( 52)
+ 1 Control = 20. E opens 1S, W corrects +1 for Q + 1 for 10 = 22.
1 S –
3 S.(2125)
Q 10 x x
x x
x x
K x xxx
K J x xx
K x x
x xx
A x
East has 26count = 11 HCP
+ 8 Long (5+3) + 3 Short (52) + 4 Control
= 26. West has 18 = 5 HCP + 9 Long (5+4)
+ 3 Short (52) + 1 Control = 18. E
opens 1S, W
corrects +1 for Q and 1 for 10 = 20.
1 S –
2 S.(1620)
A Q x x
A x
x x
K x xxx
K J x xx
K x x
x xx
A x
East has 26count = 11 HCP
+ 8 Long (5+3) + 3 Short (
52) + 4 Control = 26. West has
30 = 13 HCP + 9 Long (5+4) + 3 Short
( 52) + 5
Control = 30. East opens 1S, West corrects
+1 for Q + 1 for A = 32.
1 S – 2 NT.(32+) Long raise version of Jacoby 2NT. If E
has only Axx
in , this will bring 6 additional points and slam (62 min for slam)
It is worth noting that initially we had the responding
level at 18 rather than 16, but after some
additional experiments the responding level
was adjusted to 16, which fits perfectly with
the “5 points per level” calculations.
So, let’s say the opener has 26 and
opens 1
, while the responder has 16 and bids 2
. Now, if you put additional 5 points in each hand, this will bring
the point count to the gamelevel of 52.
During the initial hand valuation when you pick up your cards, it's a standard
procedure to depreciate shortsuit honors
by a point. While the bidding progresses,
you do a reevaluation of the hand, accounting
for the suits bid by your partner and your
opponents.
Just remember that the three important rules for evaluating the HCPportion
of your hand match the wellknown three important
rules for evaluating real estate  location,
location, location:
1) Location of your honors in partner's suits  add a point for each honor
(10 including) to a maximum of 2 (if you have
KQ10 add only 2, rather than 3).
2) Location of your honors in opponents' suits – subtract/add a point
for the honors in the suits bid by the opponents
depending on location of the opponent (chances
are you don't have many of these, so no limit
here): an AQ or Kx
behind (offside) the bidder can be upgraded
while QJx –
downgraded respectively. The same AQ or Kx
should be downgraded if you are in front of
(onside) the bidding opponent.
3) Location of your "depreciated" honors in short suits  add
the honors bonus points for the partner's
suits while further discount the honors in
short suits bid by your opponents. Doubleton
QJ in the opponents' suit can be dropped to
zero while in the partners suit it gets to
4 points, since the 1point discount for 'blank
honors' stays due to the inflexibility it
presents in playing the suit by blocking the
communications.
As the bidding progresses, you continue reevaluating the hand in the light
of both Partner's bids and the opponents bids.
A suit of AQx can be upgraded to AKx if the
suit is bid in front of you, while a KJx
can be dropped to 1 pt if the suit is bid
behind you. You use your head constantly.
So for the time being we are quite happy with the 52count for game – it works in a vast majority of hands. The
fact that we have a 6card trump suit, for
example (a good surprise to the partner who
has raised the suit expecting only 5 cards)
is already factored in via the Distributional Zar
Points, calculated on the basis of our 6card
suit. Or is it?
The fact that you have a 6card suit IS indeed factored in – what
is NOT factored in is the fact that partner
has raised that suit, believing that we have
a 5card
suit, which means that we have an
extended fit.
Besides the "real estate", your distribution points also get
adjusted depending on the way the two hands
of the partnership match. Your initial Zar
Points of 13 for a 5431 distribution are worth13 in the beginning when NO information from the
bidding is available, and may stay there if
the partner has 1345 and also gets 13 points
for distribution.
We have to be able to “calculate” the fact that we have extra
lengths in “our” suits and the
numbers that fit the calculations are:
3 additional HC points for any trump over the promised length,
i.e. 3 additional for 5 trumps, 6 for 6 trumps.
3 additional HC point for any Invitational Second suit card over
the length of 4 (secondary fit).
You know already that the calculations that led us
to these pointassignments are based on the
overdetermined system
of equations with “X_superfit”
additional variable, so let’s only notice
that these reevaluations are in line with
the “Law
of Total Tricks”, besides being
actually calculated as coefficients in series
of equations.
You may ask “How does it fit The Law when you say that 5 Zar Points
constitute 1 level and you assign only 3 points
for extra length?” And that would be
a reasonable question. The answer is that
2 points have already been factored in by
the fact that you have a 6card suit,1 from the (a + b) and 1 from the (a
– d). And you might say “But The
Law is only applicable when the HCP
power is relativelyequally divided between
the opponents” – and that’s
true. In aggressive gamebidding situations,
though, the HCP is also divided – it’s
the distribution and location that makes the
aggressive games, so there is no controversy
here either.
The second question often asked is “Since we add a lot of points
for superfits, don’t we have to change
the minimum limits of 52 for a Game and 62 for a Slam?”
The answer is “No”, you simply
will be able to arrive at more subtle (or
“aggressive” if you like this
word better) games and slams, while still
getting at the more normal “everyday”
ones.
The two good things that may happen when evaluating how well the hands
match together are:
1) you have one fit, but it is a superfit, i.e.
around 10+ cards in the suit.
2) you don't have a superfit, but you have double
fits, one 8+ cards, and another 7+ cards.
Here is how you reevaluate the hands in terms of additional Zar Points
for the main fit:
9th card  3 pt;
10th card  6
pt, i.e. a 10card fit brings you 3+3 = 6
points total just from the length.
11th card  9
pt, i.e. an 11card fit brings you 3x3 = 9
additional points from length.
For additional information on The Law
of Total Tricks see Larry Cohen's excellent books on that subject.
This is the “blind value”
of the additional trumps, though. Can
we do something better here? Does it matter
if these additional trumps can contribute
to additional tricks or if their power is
limited to “taking trumps out of defenders’
hands”.
Have a look at these two hands in which you have 5 trumps (spades) and
your partner has opened 1 spade.
Hand A has distribution 5044 while hand B has distribution 5332.
HandA
A 10 9 7 4
___
10
8 6 5
A
9 6 3
HandB
A
10 9 7 4
8
7 6
K
7
K
8 3
Granted, the difference is somewhat reflected in the (a + b) + (a –
d) calculation, but this
was before spades were named trumps!
Both hands have 25 Zar Points, 4 controls,
and 5 trumps, but are these 2 additional trumps worth the same
in both hands? You see the point.
I guess you would prefer hand A despite the fact that it has 8 HCP and
5 trumps, while hand B has 10 HCP and 5 trumps.
The reason is in the “Zar Ruffing
Power”.
Instead of assigning 3 points for every additional trump over the combined
length of 8 regardless of distribution, we
assign the entire amount of 3 points per trump
only “some times :) Here is how you
calculate the “Zar Ruffing
Power” of the additional trumps you
have.
You add (3d) points for every trump
over the combined length of 8, where d is
the usual shortest suit in your hand. If you
have a 4333 hand, your additional trumps
are worthless – you get (3d)
= 33 = 0, besides the fact that you have
already taken into account this length in
the initial Zar Points calculations.
If you have a singleton, you get (31) = 2 Zar Points for every additional
trump.
But wait! Can we find an easier form of the (3d) rule? Look what it actually boils down to:
Assign 3
points for every trump above the combined
length of 8 if your shortest suit is
void.
Assign 2
points for every trump above the combined
length of 8 if your shortest suit is
singleton.
Assign 1
point for every trump above the combined
length of 8 if your shortest suit is
doubleton.
Sounds
familiar?
Yes,
these are theGorenPoints
for distribution, but applied only for the
extratrumpholdings, and applied for everytrump above 8. So, the 2 additional trumps of hand A are worth
2x3 = 6 Zar Points, while the same 2 additional
trumps in Hand B are worth only 2x1 = 2 Zar
Points.
Here
is an important note on the Zar Ruffing
Power calculations. You can calculate and
add the points for a single additional trump even if you gave just the regular fit that
your bid shows (no
superfit), provided
there is a difference of at
least 2 between your trump length and your
shortest suit. So if you have 4 trumps
and a side singleton and you have raised your
partner’s 4card suit, you still can
count 2 points (that’s 31); same with
3 trumps if you have raised your partner’s
5card major – you count (31) = 2 points
for the singleton. However, with 3 trumps
and a doubleton you do not get the 1 point
since the difference in lengths is only 1.
We have tested these intensively and so could
you. This adjusted calculation takes care
of many competitive bidding situations where
you have to judge what to do.
You
realize that this brings new light to the
Law of Total Tricks – it allows
you to finetune your actions instead of blindly calculating the total trump
lengths.
For
additional developments of the The
Law of Total Tricks, see the last couple of
sections in the second part of this article,
called “The Finer Arts of Zar Points”
Having discussed the value of doublefit and superfit, let's pick the
following common hand with a 5card Major
suit:
Jxx
KQJxx
AQx
xx
You open 1H and partner raises to 2H, opponents pass carelessly.
What do you do? And how would your decision change if you hold a 6card
Heart suit? Pretty common question, you would
agree.
We hold 13 HCP with 3 controls for a total of 16 points, plus the 11 from
distribution (3+8) for a total of 27. An opening
hand, as we have already opened it, but nothing
more than that – so we PASS.
Let's get ONLY one nontrump card and move it to the hearts, making the
Hearts a 6card suit. How does that change
the situation?
Here is how. You guessed it  it depends :)
Depends on where you get this 6th card from. If you get from the doubleton, you make the hand 6331 and the distributional
Zars jump from 11 to 14, plus the 3 points
for a 6th trump (1 more than promised by
the bid) for a total of 33 points  enough
for Game Try since you
support the level 3 alone. So you bid
2S (invitation), asking partner for help in
this suit.
If you move the card from the Jxx, the distribution
would be 6322 for a jump from 11 to 13.
Plus you add 3 pt for the 6th suit and drop
a point for the resulting Jx
which gets your total from 27 to 31  still
most probably PASS, unless pushed in a competitive
bidding.
You see the meaning the Zar Ruffing Power calculations
now.
Same if you get the 6th card from the AQx 
you'll need to make 1 point deduction for
the AQ blank, adding 3 for the 6th suit, leaving
you again with a total of 31 – PASS,
unless in competition.
How easy and simple it is  if you can count to 32  Game try. If you can
count to 36 – Game. If you have only
a point of two extra – just let it go.
You can see how things change if you move
2 cards around and make the hand 6421 or
7321 and you would act accordingly.
Just one more hand on the 6card major suit theme:
Jxxxxx
xx
AQx
AQ
Again you open 1 S and partner bids 2S. The hand is from the exceptional
book of Jeff
Meckstroth"Win
the Bermuda Bowl with Me"
 this book should be your choice if you are
under the severe financial restriction to
buy only one bridge book :)
Jeff's view is that this hand is only worth a game try. Let's see what
"the calculator" would tell us.
We have 13 HCP and 4 controls, for a total
of 17 points, plus the 9 + 4 = 13 distributive
Zars (the 1 point for holding the Spades
suit only counts when you make a borderline
decision "to open or not to open")
for a total of 30.
If you are lazy and want to use the 3pointsperadditional trump rule,
you add the 3 points for the 6th suit you
reach a total of 33  enough for game try
(5 Zars are 1 level of bidding). To jump directly
to 4level you need 36+, as already discussed.
If you use the Zar
Ruffing Power
rule, than you add only 1 point because your
shortest suit is doubleton and get 31 points
– even lower that before.
So
do you have to be aggressive or conservative
in bridge? I hope you know the answer to that
question  it's both.
Karen McCallum said once “I’ve
never passed a hand with a void in my life”,
and when you think about it, a hand with
a void already has at least 14 Zars from the S2 and L2 components (as opposed to the
only 8 points that a flat 4333 hand would
give you – a minimum of6 points
difference). Put a couple of Aces for another
12 Zars (2 times 4 HCP plus 2 CT) and you
have a hand with 8 HCP but with 26 Zars!!!
Opening hand with 8 “standard”
points!
Here is an example of the “Two
opening hands make a game” rule in
the oldfashioned HCP style and the Zarstyle.
First  the common question “to game or not
to game” with 24 HCP, with standard
bidding (Std) and
Zar Points (Zar):
Q x xx
Q J x
Q J
K J x x
K J 10 x x
K x x
K x
Q x x
Std: East has 12count with a good 5card , West
has 4card support in and 12 HCP, both opening hands,
but not a chance for either 4 or 3 NT (the
defense will switch insooner or later).
Zar:
East has 12+3+3+8 = 26 points (bare
opening) while West has 12+1+2+8=23,
not an opening hand even with the
correction +1 for the Q of spades.
That is support to 3
(49 Zars, 52 needed for a game).
Now, the above mentioned “Karenstyle” approach to
the “to open or not to open”
question, again Std
vs. Zar:
K Q x xx
K J x xx
x xx
___
A x xx
A 10 x xx
x xxx
___
Std: East has 8 HCP and West has 9 HCP with notsowild
distribution  nobody has a suit
longer than 5cards! Admit it –
you would pass BOTH hands!
Zar:
East has 8+4+9+5 = 26 Zars. West
has 9+2+10+5=26 Zars. BOTH opening
hands! And two opening hands make
a game. The result  Cold 4
I am sure you have already
noticed that if you switch the and
in
EITHER hand (but not both : ) it’s
a GRAND! A GRAND SLAM that you
would simply have as an ALLPASS board!
Wouldn’t
that be a shame even for Bob Hamman
(arguably the most experienced bridge
player on Earth) with his 3% on bidding
 if only he’d have an ALLPASS
board here, though…
I am sure he wouldn’t and that’s
simply because his 3% are NOT your 3%!
Now you have the tool to come closer
to Bob’s 3%.
Looks strange, but …
only if you are still judging and evaluating
the hands based on HCP and “vague
feel” about things like shape,
controls, distribution, offensive power,
suitsupport, etc. – all of which
come into account with the Zar Points
evaluation system.
You only have to be able to
count to 26 and confidently open the bidding.
To finish the "aggressive
opening" subject, we just have
to show what REAL AGGRESSIVE
actually is. Some readers have already
noticed the hand in the beginning of
the article suggesting opening with
7 HCP. This sounds kind of crazy, I
hear you mumbling.
Let's
explore this avenue a bit, though 
is this the limit? Here are several
hands that will provide the answer to
that:
7+3+6+11=27
7 HCP
K
x xxxx
A
x xxx
x
x
___
6+2+6+12=26
6 HCP
K
x xxxx
K
x xxxx
x
___
5+1+7+13=26
5 HCP
K x xxxxx
Q x xxxx
___
___
4+2+7+13=26
4 HCP
10 x xxxxx
A x xxxx
___
___
Wow
... an opening hand with 4 HCP ????
Well
... you probably like "calmer"
hands, like 4432 with 8 HCP that
anyone in the world (me including :)
will pass:
Many
Bridge Federations, though, impose restrictions
on your bidding, some of them to a point
where you don't know what kind of a
game you are playing anymore ... (hey,
it's all 'games' :)
I
don't want to get into discussions about
the restrictions ACBL is imposing, but
would rather consider a more moderate
"average" set of restrictions
like the ones the French Bridge Federation
 "FédérationFrançaisede Bridge"
imposes on the bridge events under its
rules:
"An opening of one of a suit can be made ONLY under
the rule of 18 (HCP+a+b),
16 in third seat".
That's
like Marty Bergen's Rule of 20, only
kept to 18 in 1st and 2nd seat, and
lowered to 16 in 3rd seat.
Let's
see what happens with our "sublight"
openings with 7, 6, 5, 4 HCP respectively.
The 7 HCP hand > 7 HCP + 6 + 5 =
18
The
6 HCP hand  > 6 HCP + 6 + 6 = 18
The
5 HCP hand  > 5 HCP + 7 + 6 = 18
The
4 HCP hand  > 4 HCP + 7 + 6 = 17
So
 all these openings are just fine
and dandy, according to the “average”
National Federation restrictions, much
to the surprise of some people.
I
hope you are not one of them anymore.
Let’s
have a look at the
WBF definition of an opening hand.
The WBF doesn’t define an opening
hand in terms of Goren
Points, or Bergen Points, or (notabene) even in terms of
Milton Work HCP!
The
rule actually says that an opening hand
is a hand “better
than the average hand with a Queen worth!
This
means that basically regardless of how
you measure the hands, an opening hand
is a Queen better than the average.
So,
let’s measure the average Zar
Points hand. For comparison, and average
HCPmeasured hand is a hand with 10
HCP points. Hence, an opening hand in
Milton Work sense is a hand with 10
+ 2 = 12 HCP.
The
average hand in terms of 6421 Honor
Points contains 6+4+2+1 = 13
points. Now, for the Distributive
Portion, we have to go to the each distribution
and multiply the Zar Points it brings
with the probability this distribution
to occur. Then, when we addup all the
results, we’ll come up with the
average Zar Points distributional hand.
So,
let’s just do it.
Hand Distributions with their Probabilities and Zar Points assigned
for a total of1176, which after dividing by 100 to convert
the %, comes to approximately 11 Points.
So, the average Zar Points hand contains 13 + 11 = 24 Zar Points.
This in turn means that a Zar Points opening hand of 26 is a Queenworth (2 points) better than the
average
hand which has 24 Zar Points.
Hence,
according to the World Bridge Federation,
AND the "average" National
Bridge Federation standards, AND
the European Bridge Federation
Standard Zar Points OPENING HAND definition
of having 26 Zar Points is perfectly
OK.
It
is interesting to note that all the
3 methods  Goren, Bergen, and Zar 
fit EXACTLY the minimal conditions
imposed by the World Bridge federation.
Since the "average" hand has
5332 distribution with 10 HCP, here
are the three calculations:
1)
Goren gets 10 for the HCP plus 1
for the doubleton = 11 Goren Points
for the average hand  a Queenworth
from the opening bar of 13 Goren Points.
2)
Bergen gets 10 for the HCP plus
8 for the 2 longest suits = 18  again
exactly a Queenworth from the opening
bar of 20 Bergen Points.
3)
Zar gets 10 + 3 = 13 for the
HCP + Controls, plus (5+3) + (52) =
11 for the distribution for a total
of 24 Zar Points  a Queenworth from
the opening bar of 26 Zar Points.
Before leaving this section,
let's prove that Zar Points provide
a Monotonic Linear Evaluation with a
step equal to "OneJackWorth".
1) We will prove that Zar Points
are Monotonic
and Linear regarding the length changes throughout
the 4 suits a, b, c, and d:
First, let's prove the linearity
in the case of moving a card upwards
the suite, from d through a.
Moving a card from d to c:
(a) + (b) + [a  (d1)]= (a+b) + (ad) +
1
Moving a card from c to b:
(a)+ (b+1) + (a  d)= (a+b) + (ad) +
1
Moving a card from b to a:
(a+1)
+ (b1) + (a+1  d) = (a+b)
+ (ad) + 1
Now, moving cards downwords, from a through d.
Moving a card from a to b:
(a1)
+ (b+1) + (a1  d) = (a+b)
+ (ad)  1
Moving a card from b to c:
(a)+ (b1) + (a  d)= (a+b) + (ad) 
1
Moving a card from c to d:
(a)+ (b)+ [a  (d+1)] = (a+b)
+ (ad)  1
So the difference is 1 point
REGARDLESS of where you take a card
from and move it to the next suit in
the a, b, c, and d suits of the bridge
hand.
2) Every movement in distribution
by 1 card from one of the suits ordered
by length to the next is worth one Jack.
This stems from the fact that
the 1pointdifference proven above
is equal to the 1 point that Zar Points
assign to a Jack.
So Zar Points provide a Monotonic Linear Evaluation with a step
equal to
"OneJackWorth".
So
the aggression can go as low as 4 HCP
in the extreme case of 7600.
Let's
see how far we can push the things in
the opposite direction. This is to say,
what is the HIGHEST amount of HCP which
formally
does not allow you to open (in terms
of Zar Points), despite the already
stressed fact that with 12+ HCP you
will
open from “brute force power”
considerations. The fact that Zar Points
formally do not give you the green light
to open, presents you with the Zar Points
“orientation
correction”, explained in
this section.
You certainly understand that we are talking about
balanced hands here, rather than wild
twosuiters.
And we have already touched upon the
issue of balanced hands bidding in the
very first section of this article,
so you know that with balanced hands
basically what matters is brute force.
Is that the whole picture, though?
To set up the stage, let us consider the following
board. You are West
and your partner opens 1 NT. You are
playing a "special" 1 NT opening
(for the sake of the discussion, obviously)
ONLY with exactly 14 HCP, and
44 in Majors, and balanced 4432 distribution, and the
doubleton being in(hey, I am good at creating
bidding systems :)
Admit
it
 I am giving you more information
than you would have hoped for. Here
is your hand:
7 HCP
K
x xxx
A
x xxx
x x
x
So
... what do you want to play? Any
"scientific" ideas here?
:)
Face
it – you’ll scratch your
head and shoot in the dark.
To
show you what kind of a fog of uncertainty
you are manipulating in, here are 2
normal possible 1 NT hands:
14 HCP, balanced 4432 hand
A x xx
K x xx
A K x
x x
14 HCP, balanced 4432 hand
Q J x x
K Q J x
Q J x
Q x
So
 slam or partscore,
this is the question :) It’s
3 Levels difference we are talking about
here in this innocent example  slam
with the first and partscore with the
second …
Aggression
or Antiaggression, hitting the brakes
:)
You
see the difference between having 6
Controls vs. having 1 Control. The first
hand has 5 more controls than
the second one, meaning it has 5 more
Zar Points, everything else equal!
This brings the valuation for the first
balanced hand to 14 + 6 + 8 + 2 = 30
points, while the second one has 14
+ 1 + 8 + 2 = 25. So the first hand is at the border of having one additional level
“in excess” while the second
one doesn’t even have a formal
opening in terms of Zar Points –
you will open it just from “brute
force” consideration.
Now
we are ready to get back to our original
question of how far we can push the
"nonopening" hand, the quotation
marks being there to stress the fact
that you will open these hands because
of the brute HCP power of 12+ HCP, but
open
them "carefully, with caution”,
meaning that:
1)you would pass at the first
chance presented, and second,
2)you would push the hand toward
NT (see the “orientation”
remarks below).
Here
are the hands, this time in ascending
order of HCP:
12+2+8+3=25
12 HCP
Q J x xx
K x
Q J x
K x x
13+1+8+3=25
13 HCP
K Q x xx
Q J x
Q x
Q J x
14+1+8+2=25
14 HCP
Q J x x
Q x
Q J x
K Q J x
15+2+7+1=25
15 HCP
K J x x
K Q x
Q J x
Q J x
The
important distinction to notice and
remember is presented in the following
table:
Balanced Distribution
Maximum "nonopening"
HCP
4333
15
4432
14
5332
13
Now
you know what kind of focus
to have when dealing with balanced hands
against suitoriented hands. Zar Points
divide the balanced hands in the lowHCPlevel
(1215 HCP) in two groups.
The
first group is “max
3 controls”. The second group
is “min 4 controls”. This presents you with another evaluation parameter
called "orientation"
of the lowHCPlevel balanced hand –
NT or trump. It serves as your “halogen
lights” in the fog of uncertainly
that balanced hands present.
Now
that we know that Level1 "normal"
opening can happen with as low as 4
HCP, you probably think that preempts
can go to as low as a 12 HCP !
Sorry
to disappoint you  you are not even
close ...
The
"general" rules of the "standard"
bidding you know about, basically
states that for "normal" opening
you have to have about 12 HCP while
for a weak2 opening you need 8 HCP
and a relatively good 6card suit.
So,
for a weak2 preempt you need 2/3 of
the minimum for normal Level1 bid.
Things
with Zar Points
bidding are more conservative  you
need between 22 and 25 Zar Points
and a descent 6card suit.
The
main message you communicate there is
"I don't have the 26 points for
a normal opening, but I have a decent
6card suit and between 22 and 25 Zar
Points."
Here is a typical hand you would open 2 with, basically
in any system:
Kxx
KQJxxx
xxx
x
Let's see what happens in Zar Points. You get 9 + 2 = 11 for the HCP and
Controls, and 9 + 5 for the 6331
distribution for a total of 25 Zar Points
 not enough for opening at Level1.
How about preempts at level 3 and 4? Again the main message is "I
don't have 26 points, but I do have
a decent 7 or 8card suit respectively,
so you evaluate your hand respectively".
But
why does the limit of a "normal"
opening go as low as 4 HCP, while preempts
virtually do not drop below 7 HCP? The
answer is simple  playing power
and limitations of the hand.
With
preempts you virtually declare unisuit
with not much potential for variations
and reevaluation of the hand  you
basically say "That's
all I have".
We will assess the accuracy of four different methods of bridge distribution
evaluation via some standard common
mathematical approaches.
The first one is the already mentioned CharlesGoren’s system, known as the “321” system, named after
the points assigned for shortsuits
holdings.
The second method is the MartyBergen’s “Rule of 20” method
from his famous bookseries “Points
Schmoints”.
The approach of Bergen is to assign points equal
to the sum of the lengths of the 2 longest
suits of a hand, i.e. (a+b), using our notation.
We will also compare with the newest method from the
late 90ies, the Drabble
rule of “adding the 2 longest
suits, divide by 3, and subtract the
length of the shortest suit, rounding
downwards. Since Drabble’s
scale starts with 1 for the 4333,
we have adjusted it by shifting the
entire table with (+1) to eliminate
the negative numbers.
In all cases we consider the initial base points, before
the “fine tuning” in one
way or another.
The fourth method is the Zar distribution Points method you are already familiar with  assigning
the value of (a+b)
+ (ad), i.e. the sum of your 2 longest
suits, plus the difference between your
longest and your shortest suit (effectively
representing the SUM of all the 3 suitdifferences
of the hand).
As
we mentioned, there are 39 different
suitdistributions in a bridge hand.
The
table below covers them, along with
the probability of their occurrence:
Hand Distributions with their Probabilities
4333= 10.5%
4432= 21.5%
4441=3.0%
5332= 15.5%
5422= 10.5%
5431= 13.0%
5440=1.3%
5521=3.2%
5530=0.9%
6322=5.6%
6331=3.5%
6421= 4.7%
6430= 1.3%
6511= 0.7%
6520= 0.6%
6610= 0.1%
7222= 0.51%
7321= 1.88%
7330= 0.26%
7411= 0.39%
7420= 0.36%
7510= 0.10%
7600= ~0
8221= 0.19%
8311= 0.12%
8320= 0.10%
8410= ~0
8500= ~0
9211= 0.02%
9220= 0.01%
9310= 0.01%
9400= ~0
10111=
~0
10210=
~0
10300=
~0
11110=
~0
11200=
~0
12100=
~0
13000=
~0
The numbers marked as ~0 are
numbers less than 0.01%. It is worth
noticing that the 4333 distribution
is not among the top 3 most probable
distributions and that by far the most
probable one is 4432 – 6% above
the secondmostprobable 5332.
The distributive part of the
Zar Points varies from 8
for flat hand to 26
for the “wildest” hand with
3 voids. This means that it classifies
the hands in 17 categories. Here they go:
Zar Distribution Points for ALL distributions
4333=8
4432= 10
4441= 11
5332= 11
5422= 12
5431= 13
5440= 14
5521= 14
5530= 15
6322= 13
6331= 14
6421= 15
6430= 16
6511= 16
6520= 17
6610= 18
7222= 14
7321= 16
7330= 17
7411= 17
7420= 18
7510= 19
7600= 20
8221= 17
8311= 18
8320= 19
8410= 20
8500= 21
9211= 19
9220= 20
9310= 21
9400= 22
10111=
20
10210=
22
10300=
23
11110=
23
11200=
24
12100=
25
13000=
26
We are going to compare the 4 methods by 3 criteria:
1)span of base, given by the number of the groups the method classifies the hands in;
2)separation power, given by the maximum number of distributions which can fall in a single
group;
3)standard deviation, which is explained below
in the article.
To prepare for this exercise, we will present the following
table with the points assigned by all
four evaluation methods:
The
table is ordered by the amount of Zar
Points assigned, in ascending order.
As might be expected, ALL methods basically
follow the same ascending line, giving
the least amount of points for the balanced
distributions and the biggest amount
of points for the “wildest”
distributions. Since for everyone the
4333 case is the “base” to which everybody assigns
the minimum points we are going to consider
only the rest of the groups in the evaluation
methods (taking 4333 distribution as
base).
In
the table below, the columns of the
table are the displacements from the
“base”, (e.g. +1 means the
first group after the base of 4333)
while the actual number in the body
of the table represent the number of distributions the corresponding group.
Marty
Bergen’s Points classifies the hands in 6 groups, the 321in 9, Drabble
in
5, and Zar Points in 17.This means by the criteria of
span
of base (number of classification
groups) Zar points are between 2 to
3.4 times better than the rest of the
methods.
The
separation power of the methods is given
by the max number of distributions in
a group. In Zar Points this number is
4, while Bergen has 9, Goren – 8, and
Drabble – 13. Again  between
2 and 3.2 times better results.
When
we take into account the number of elements
(hands) in each group, we can now find
the Standard Deviation for each method and
see the difference there. Here is what
is meant by that.
The
rootmeansquare (RMS) of a variant
x, sometimes called the quadratic mean,
is the square root of the mean squared
value of x.
Scientists
often use the term rootmeansquare
as a synonym for standard deviation
when they refer to the square root of
the mean squared deviation of a signal
from a given baseline or fit.
Applying
the standard deviation from the basis
(the x coordinate) measure to the three
handevaluation methods (using the number
of hands in each group) yields the following:
Why would you care to convert Zar Points to Goren Points or Bergen Points when we JUST showed in three
different ways that Zar Points are three
times better that any other method?
“Why should I convert something ‘good’
to something ‘bad’?”
– I hear you already asking …
“I’d simply use the ‘bad’
directly rather than calculating the
‘good’ first and then scaling
it down to the ‘bad’.”
First, every method has its own ‘base’
of people who use it for one or another
reason – familiarity, convenience and habit
are some of the reasons that jump
rightoff your mind. Some people find
it easier to calculate 3*v + 2*s + d
(where v is the number of voids, s is
the number of singletons, and d is the
number of doubletons in the hand) than
calculating (a+b)
+ (ad) or calculating 2*a + (bd) for
example.
Why could that be – ask them :)
Second, when offered a new method that uses new ranges
and spans over new numbers of evaluation
metrics, you tend to “subconsciously” try to convert
or “squeeze” this new range
into the range you are comfortable with.
Such a conversion would enable you to
‘operate’ in a familiar
context and get better answers without
leaving the comfort of the familiar
‘dimensions’.
We already mentioned such a
‘convenient conversion’
when discussing that the experts use
the 4 ½  3 – 1 ½  ½ HCP scheme
instead of the regular Milton Works
HCP pointcount of 4321 to make the
‘good’ 6421 solution
look like the ‘bad’ 4321
HCP solution in terms of ‘dimensions’
and ranges (this was in theWhich HCP count is "mathematically correct”section of the this article).
So, how would that translate in terms of Zar Points?
On the HCP side it naturally translates to the abovementioned
HCP valuation of 4 ½  3 – 1 ½
 ½ that the experts use in their hand
evaluation methods.
On the DP side, let’s consider the two most common
‘conversions’  to Goren
Points and to Bergen Points.
Having a look at the comparison tables in the previous
section, you realize that to ‘convert’
Zar Points toGoren Points
you have to:
1)Subtract 8 from the calculated Zar Points count – this “equalizes”
or “aligns” the lower ends
of the scales while also aligns it with
0, thus preparing it for the second
scaling which follows below.
2)Divide
the result by
2  this “aligns” the
higher end of the scale (scale it back
to 9).
Thus for the 5521 distribution you get 3 Goren Points – just as many as for the 7330 distribution.
In Zar Points you get 14 for the 5521
and 17 for the 7330. To ‘see’
the 14 points in “Goren
Terms” we do (14 – 8) /
2 = 3. This means that in “Goren Terms” the 5521 valuation are the same in Zar Points and Goren Points..
For the 7330 we “scale down” the 17 Zar
Points by (17  8) / 2 = 4 ½. This is
a significant
difference – 4 ½ vs. 3 for
the same hand.
Now, let’s turn to the Bergen Points – again have a look at the comparison table. Here
we have to:
1)Subtract 8 to align it with 0 in order to prepare the appropriate division (conversion
of the higher end);
2)Divide the result by 2 – this
aligns the higher end of the scale;
3)Add 7 to
realign the lower ends (that brings
the lower end to 7 and the higher end
to 13)
Thus for the 5521 distribution you get 10 Bergen Points
– just as many as for the 7330
distribution. In Zar Points you get
14 for the 5521 and 17 for the 7330.
To ‘see’ the 14 points in
“Bergen Terms” we do [
(14 – 8) / 2 ] + 7 = 10.
This means that in “Bergen Terms”
the 5521 valuation are
the same in Zar Points and Bergen
Points..
For the 7330 we “scale down” the 17 Zar
Points by [ (17
 8) / 2 ] + 7 = 11½ .This is a significant difference – 11 ½ vs. 10 for the same hand.
Just one more example  the 6511 and 8320 distributions
to both of which Goren
assigns 4 points and Bergen 11 points,
but Zar Points vary from 16 to 19. Convert
and see the corresponding results.
You can do this ‘conversion’ for any other
distribution of course, and see the
distinction in you familiar setting
– be it Goren
or Bergen.
To finish this section, let’s examine a couple
of hands and evaluate them at the point
of picking up your cards. We’ll
evaluate them in Goren
Points, Bergen Points, Zar Points, Zar
Points “converted” to Goren,
and Zar Points “converted”
to Bergen.
The first hand has a 5521 distribution and relatively
low amount of controls, the second one
has a 7330 distribution and relatively
high amount of controls. Both hands
with the same HCP count of 15:
HandA
K
Q 8 3 2
K
Q J 9 2
A
4
7
Hand B
A K 10 8 7 5 4
A 10 9
A
9 6
__
Both
hands are decent hands with playing
strength, yet looking somewhat different
…
1) Goren Points Count
Both hands
in Goren Points
are worth 18 points  15 + 2 + 1 for Hand A and 15 + 3 for Hand B.
2) Bergen Points Count
Both hands
in Bergen Points are worth 25
points  15 + 5 + 5 for Hand A and
15 + 7 + 3 for Hand B.
3) Zar Points Count
Hand Ain Zar HCP Points is worth 15 + 4 = 19 (4 pts for the
4 controls). For the 5521 we get 10
+ 4 = 14 Zar Points for a total of 33
Zar Points. We add 1 point for HCP concentrated
in 3 suits = 34 pt.
Hand B in Zar
HCP Points is worth 15 + 7 = 22 (7 pts
for the 7 controls). For the 7330 we
get 10 + 7 = 17 Zar Points for a total
of 39 Zar Points. We add 1 point for
HCP concentrated in 3 suits = 40
pt.
A
difference of 6 Zar Points between the
2 hands. Note that this is one
level difference!
4) Zar Points “converted” to Goren
Hand Ain “Converted” Zar HCP Points is worth 4
½ + 6 + 3 + 1/2 = 14 (1 pt less than
the 15 if using the standard Milton
Works 4321 HCP count).
Hand B in “Converted”
Zar HCP Points is worth 3 x 4 ½ + 3
= 16 ½ (1 ½ pts more than the 15 if
using the standard Milton Works 4321
HCP count).
Hand A in “GorenConverted”
Zar Points is worth (as calculated in
the beginning of the section) 3 points
– the same amount as in the actual
Goren Points.
Hand B in “GorenConverted”
Zar Points is worth (as calculated in
the beginning of the section) 4 ½ points
– 1 ½ pts more than the actual
Goren Points.
So, Hand A is
worth 14 + 3 = 17 “Gorenconverted”
Zar Points – that’s
1 point less than the Goren
valuation itself (17 vs. 18).
Hand B is worth
16 ½ + 4 ½ = 21 “Gorenconverted”
Zar Points – that’s
3 points more than the Goren
valuation itself (21 vs. 18).
In other words, with the Hand A
Zar Points are 17/18 = 94% more conservative than Goren (in “Goren Terms”)
while with the Hand B Zar Points are
21/18 = 117 % more aggressive than Goren (in “Goren
Terms” again).
5) Zar Points “converted” to Bergen
Hand Ain “Converted” Zar HCP Points is worth 4
½ + 6 + 3 + 1/2 = 14 (1 pt less than
the 15 if using the standard Milton
Works 4321 HCP count).
Hand B in “Converted”
Zar HCP Points is worth 3 x 4 ½ + 3
= 16 ½ (1 ½ pts more than the 15 if
using the standard Milton Works 4321
HCP count).
Hand A in “BergenConverted”
Zar Points is worth (as calculated in
the beginning of the section) 10 points
– the same amount as in the actual
Bergen Points.
Hand B in “BergenConverted”
Zar Points is worth (as calculated in
the beginning of the section) 11 ½ points
– 1 ½ pts more than the actual
Bergen Points.
So, Hand A is
worth 14 + 10 = 24 “Bergenconverted”
Zar Points – that’s 1 point less than the Bergen valuation itself (24 vs. 25).
Hand B is worth
16 ½ + 11 ½ = 28 “Bergenconverted”
Zar Points – that’s 3 points more than the Bergen valuation itself (28 vs. 25).
In other words, with the Hand A
Zar Points are 24/25 = 96% more conservative than Bergen (in “Bergen Terms”) while with the Hand
B Zar Points are 28/25 = 112
% more aggressive than Bergen (in “Bergen Terms” again).
Zar Points enable you to essentially do two things:

Stop at partscore
with 24HCP when no game is in sight
but the crowd bids a game “from
general considerations”;

Bid 17HCP
games or slams when the crowd has
an “allpass” board, from
the same “general considerations”.
Here is how:
Opener:
1)Add your HCP and your Controls
in the hand;
2)Add the sum of the two longest suitsto the difference
between the longest andshortest;
3)If you can count to 26 or more – you can comfortably
open the bidding;
Responder:
4)Make the openinghand calculation
mentioned above;
5)Add +1 pt for every honor in the partner suit (up to +2) and
+3 pt for extra
length;
6)If you can count to 16, comfortably raise to level 2; if you
count to 26
– it’s a game.
Zar
Points demonstrate unsurpassed precision
in evaluation of the distribution power
of a hand – about 3 times better
that any other method in the wonderful
game of bridge.
I have heard people sometimes complaining that Zar
Points are a bit toocomplex. But the
term “complex” is a relative
thing. What do you expect to beat these
experts out there with – with
bare hands? They’ll call the police
:)
If you only see their systems written down, you’d
be stunned to see 100, 200, 300 pages!
No joke – I have copies of systems
with these exact “mileages”.
And these are systems of world champions,
who know that “nothing for nothing”
is not a good deal… You have to
make an effort – complex or otherwise.
By the time you sit around the bridge table, it’s
already too complex :)
So, are Zar Points too complex for you?
Think again – and good luck at
the table: