Calculating the Zar Points
in a hand is straightforward, as you already know:

(HCP + Controls)

+

(a + b) + (a – d)

wherea, b, c, and d
are the lengths in descending order of the 4 suits of
the hand (ranging from 13 to 0).

To open, you need 26 points.
To go to a game – double the opening amount, or
52. You simply count and bid.

NOTE, that you can do just fine without this
second part of the article, which gets into somewhat
deeper stuff.

Pace yourself comfortably.

How does this fit in the bidding
space, though? And what is the bidding space to begin
with? How does the fit and misfit affect the bidding
and what is a fit to begin with? How often do you have
a fit? What are the bid-pips and the foot-prints? What
is The Theorem and how you can use it?

Questions like these will be
answered in the discussion below.

Let’s now have a look at the entire board and
see what the global evaluations would be. Now we will
consider the suits in the combined hands and the evaluation
formula will be based on the shapes of both hands.

The considerations below have been inspired by a board
given to me by Mike Lawrence as a challenge for an initial
version of the article - thank you, Mike. Here are the
2 hands of the board:

K J x x x

x

A x x x

K x x

x

K J x x x

K x x x

A x x

The best contract is 2
and the question is “Can you stop there?”I will allow myself, instead of this board to
consider 2 “almost identical boards” –
this will make it easier for me to unveil the point.
Here they come:

1)

A x x x

K x x

K J x x x

x

K x x x

A x x

x

K J x x x

2)

x

K J x x x

K x x

A x x x

K J x x x

x

A x x

K x x x

These hands in the 2 boards are “almost completely”
identical. They have:

-the same shape;

-the same HCP;

-the same Controls;

-the same top honors;

-the same Zar Points;

-the same offensive power;

-the same suit-support;

-the same level of best contract (level 2).

Still … which of these 2 boards do you like better?I’ll make it a bit harder – FORGET
that with the first board the best contract is 2
and you will score 110 while in the second one it’s
2
and you’ll score “only” 90. Let’s
pretend for a moment that each of the four suits brings
30 pt, i.e. both boards would bring you 110. NOW which
one do you like better? If any – after all they
are “almost completely” identical…

I personally STILL like the first one better. FOUR
TIMES better! Why? And why four times?

Because it gives me FOUR TIMES bigger bidding space than the second one!

Do you see that? Four times is a lot!

Let’s introduce the term bid-pips (pips is a term we use in backgammon to describe the steps
in the backgammon space). A bid-pip is any bid in the
bidding space, so there are 5 bid-pips per level, and
35 bid-pips in the entire bidding space (as opposed
to only 24 pips in backgammon – that’s why
backgammon is a simpler game :-) - and why I love it
so much). So, there are 2 bid-pips between the bids
1
and 1
, 4 bid-pips between 1
and 2
etc.

Now you probably see why board 1) has 4 TIMES bigger
bidding space than board 2)… In board 1) East
opening bid is 1
and there are 8 bid-pips to the ‘best contract’
of 2
, while in board 2) East opens 1
and there are only 2 bid-pips to the best contract of
2
! Don’t tell me that you’ll stop at the
best contract of 2
here – I’ll call the Director :-)

Things like bid-pips, bid-space, and what I call the
inherent HCP-inertia
(the fact that it is almost impossible to stop at a
contract like 2
if you have 25 HCP in the combined hands simply because
you need room to express the “additional”
and “undisclosed” power of the hands) are
all things that you HAVE to keep in mind during the
bidding process and in the same time things that can
be grasped neither by Zar Points, nor by “Czar”
points if they exist :-) As Kozma Prutkov said nearly
a century ago, “Nobody can grasp the ungraspable”
– that’s where the beauty of the game of
bridge comes from. Bidding sequences like 1
- 2
have to catch your attention and to alert you that you
have already “eaten-up” 8 bid-pips without
communicating that much of information, and to consequently
make you more conservative for this board. Let alone
the opponents’ interventions and even worse –
their pre-emptive bidding. It’s a jungle out there
:-)

To make things more clear, let’s consider the
following scenario. NOTE how important this hypothetical
scenario is in order to recognize that the bidding space
is HUGE, contrary to your beliefs, probably.

So … we are going to consider a “slightly”
different game of bridge – a game in which ALL
the 4 people are “partners” in the sense
that they cooperate towards a common goal – the
goal to REVEAL the holdings of EVERY player, ALL the
13 cards of ALL the 4 players. And you can go as high
as possible – like bid 9 NT, 13 SP, 21 CL etc.
as needed. BUT – at the end you can write down
the positions of ALL 52 cards at the table. Interesting
…

Remember that the number of all possible deals is pretty
large –

53, 644, 737, 765, 488, 792, 839, 237,
440, 000

last time I counted.

This hypothetical question has been answered by an
old friend and former partner of mine – Manol
Iliev. And the answer is pretty surprising. It turns
out (mathematically proved, of course) that everything
will be clear by the level of 6 Clubs! That’s
all!

At level 6 you’d know EVERY card of the 52 cards
on the table!

How big is the bidding space indeed, or how many different
bidding sequences are possible in the regular game of
bridge? The answer will surprise you more than the answer
about the number of all possible deals – it is

2, 400, 000, 000, 000, 000, 000 TIMES bigger

than the above-mentioned number
of deals !!!

The total number of bidding sequences including doubles,
redoubles, and passes is a bit more than

128 E+45, that is -128 times 10 to the power of 45

Not enough room to fit-in the actual number :-)

So … there is room at the table – you just
have to use it wisely.

Sounds a bit ambitious, so … let’s get
right down to the proof.

Applying the Dirichlet'sPrinciple from
mathematics, we see that THE WORST-CASE scenario when
talking about fit and misfit is that you either have
at least two 7-card fits (the so called “Italian”
fits) or one 8+ card fit.

We have 13 + 13 = 26 Dirichlet’s Balls (the cards
in both hands) and 4 Dirichlet’s Drawers (the
suits). You can easily see that (13 + 13) – 4
x 6 = 2(the
6 comes from filling-in all 4 “boxes” to
6 each) and these 2 “loose” cards will have
to fall in the one or two of the Dirichlet'sPrinciple Drawers (suits in our case),
making the “fit” of at least 6+1 = 7.

This means that you virtually always have a fit or fits somewhere.

The best-case scenario is of course a board with two
13-card fits:

A Q x x x x

K J x x x x x

___

___

K J x x x x x

A Q x x x x

___

___

Do you like this board? I don’t – it would
be a wash on any tournament …Almost everybody
will bid a GRAND. Unless someone decides to fish for
a top and bids 7 NT, hoping for a favorable lead :-)

The Theorem has deep
implications on the bidding process simply because
you do know that your goal is to findthe pre-existing fit(s)
rather than approaching the bidding trying to find out whether or not you have fits. Think about it!

The best example is the balancing. You have noticed that part of the “aggressiveness”
of the experts is that these guys will almost never
let you play at low levels, provided that you wish to stop there. They will try to push you up or to get the contract
in “their” suit.

Do they know they “have” a suit? At least
they “hope”, that’s for sure :-) The
Theorem gives you the confidence to shoot for finding
your best spot, simply because you know that it exists.

If you get a little greedy and ask the question “How
often do I get into the worst-case scenario of 2 7-card
fits”, I have good news for you. For that to happen,
you have to have special cases of only four possible combinations: 4333 vs. 4333, 4333 vs. 4432, 5332
vs. 4432, and 5332 vs. 5332 distributions. Which special
cases? The following ones: 4333 vs. 3433, 4333 vs. 2344,
4432 vs. 3244, 5332 vs. 2335, and 5332 vs. 2434 (and
slight variations of these - the unbalanced combinations
like 6160 vs. 1606, 7060 vs. 0706 etc. have a negligible
probability ~0%).

So when you run the probabilities, you reach the following
form of The Theorem:

In bridge you always have a fit:

–about 85% of the time at least
one 8-card fit

–about 15% of the time at least
two 7-card fits

If you are a careful reader, you probably have noticed
that there is a chance for you to have 3 7-card fits
(the 4432 vs. 3343 case). This chance is included in
the 15% chance for having only 7-card fits. These results
were also re-checked with several generations of deals
via DealMaker, DealPump, and Deals programs (chunks
of 1,000,000 boards each time).

One last word on this subject, stemming from the fact
that if you have only 7-card fits (neglecting the cases
with ~0% probability like 7060 vs. 0706), both hands are balanced.In this case the only thing that matters is brute
HCP power. If you have it - generally play in NT.
If they have it - let them suffer, because if you show
your head ‘above the water’, they’ll
make a salad out of you.

Think twice before balancing with 4333 despite The
Theorem. Better yet – think twice and then
pass :-)

The only exception to that rule would be when in Matchpoints
you have to push your opponents out of a non-vulnerable
1 NT - arguably the point in which you have to be most
aggressive. But think once before doing it :-)

Let’s now get back to the other bridge theory
addressing the issue of levels of play – The Law,
and how it can be extended in made more explicit. This
is the subject of the following “Superfits Theorem”.

Is there any relationship between The Theorem and The
Law of total tricks?

As mentioned above, for a comprehensive study of The
Law and how it influences your bidding-decision-making
process, your best source is the set of Larry Cohen’s
books on the subject. He also has his own website which
you can visit at your convenience.

To see the correlation between The Theorem and The
Law, let us apply the same
Dirichlet'sPrinciple from mathematics (as we did
in the proof of The Theorem) to the case when the opponents
have one superfit (defined as a 10-11 cards-suit) first,
and than to the case when the opponents have a double
superfit (at least 20 cards in 2 suits).

In the first case, when the opponents have a single
superfit, this leaves you side with 24 cards in 3 suits
which means that according to Dirichlet your side has
at least three 8-cards fits instead of the guarantied minimum of 2 7-card suits
(in the worst-case scenario).

In the second case of double superfit on the opponents
side, you will be left with max 6 cards in these 2 suits
and 20 cards in the other 2 suits – again applying
the Dirichlet Principle you would be guaranteed at least
two 10-cards fits yourself in the worst-case scenario!

To summarize:

1)If the opponents have a single superfit, The Theorem guarantees
you that your side has (in the worst-case scenario)
a triple 8-cards fits.

2)If the opponents have a double
superfit, The Theorem guarantees you that your side
(in the worst-case scenario) also has a
double superfit.

Easy-to-remember
guidelines, I believe.

To
make it even simpler, the finding can be summarized
by the principle “The more they have, the more we have”
:-) And vice-versa, of course – the moment you
realize you have a double-superfit you know that the
opponents also have a double-superfit, whether they
know it or not. This may result in a tactical decision
to either:

1)hide your second fit while
still account for it, or

2)make a psychic bid in one of the
opponents’ suits, knowing that partner is “not
rich in that suit” anyway and chances that you
are going to mislead him are slimmer.

Is
the second option perfectly legal and the correct thing
to do? I don’t want to open this kind of discussions,
but the short answer is “oh, yeah …”.
As long as your partner get’s the same information
(or mis-information
for that matter) and acts accordingly, you are fine
and dandy. The fact that you know
that chances are 95% that you will mislead the opponents
and only 5% that you will mislead your partner only
speaks about your smarts, and if you don’t approve
of that statement you probably disapprove the bodycheck
in hokey and the fact that in boxing the guys hit each
other in the face and the referee pretends he doesn’t
notice :-)

Can
we push this simple superfits discovery a bit further,
though …

Let’s
try – it may be worth the effort.

Let’s
assume that the opponents have a double-fit in 2 of
the suits and the total amount of cards in both of these
suits is N.

Watch
what kind of simple calculations we are going to do
now.

The
total amount of cards in these 2 suits (we’ll
call them “their” suits) is 26.

This
leaves us with the amount of (26-N)
cards in their suits.

But
we have 26 cards in our combined hands between me and
my partner, so for the other 2 suits (I’ll call
them “our”
suits) we are left with

26
– (26 - N) = 26 – 26 + N =
N cards in our suits.

This
leads us to The
Superfits Theorem:

IF:Opponents have N cards in 2 suits

THEN:We have the SAME
amount ofN cards

in THE OTHER2suits.

If
they have 16 cards in the minor suits, we have 16 cards
in the major suits.

If
they have 16 cards in the major suits, we have 16 cards
in the minor suits.

I
am sure you see already where you can utilize The Superfits
Theorem. Yes – after Michaels cue-bids and after
unusual 2 NT. Such bids should not panic you, but rather
sharpen your senses and watch for the other two suit,
because most probably they are the ones you can find
a fit in.

The
easiest way to remember The Superfit Theorem is by “Whatever they have, we have”.

I
am sure you have already noticed the appearance of this magic number of 26
again …

Do
you remember it from somewhere?

Correct
– it’s the
opening amount of Zar Points that you need.

Now that we know we virtually always have a fit somewhere,
and that “Whatever they have, we have”,
let’s get back to the question of misfit.

In Zar Points we don’t deal with misfits –
we deal with “footprints”
and controls. The “footprint” of a suit
is the shorter side of the suit in both hands:

K x x x xx xhas a footprint of 2

A x x x xhas a footprint of 1

A x x xx
x x has a footprint of 3

You see now how easy it becomes to evaluate the immediate
losers using footprints (FP) and controls (CT):

If FP < CT, you have 0 immediate losers.

If FP = CT, you have min( 1, FP) losers

(1 in most cases; 0 with void against xx(xxx)
).

If FP > CT, you have min( 2, FP) losers

(2 in most cases; 1 with x against xx(xxx)
).

I already hear you screaming “That’s too
complex, man … That’s for a computer…”I am not going to get deeper in the manipulation
of the footprints and controls, immediate losers, and
their cooperation with the suits where your fit(s) are
– it goes beyond the scope of the article, but
you get the picture, I hope. Here is a simple example which illustrates the point.

The
same fits “13 – 7 – 6 – 0”,with same
10 HCP but with different
footprints:

1)

A Q x x x x

x x x x x x x

___

___

K J x x x x x

___

x x x x x x

___

This
board has 0 losers in any of the 4 suits. The contract
- 7
(the defenders
wouldn’t find the killing trump lead :-).

The
footprints of all the 3 non-trump suits are 0. Now,
the second example:

2)

A Q x x x x

x x x x

x x x

___

K J x x x x x

x x x

x x x

___

This
board has 3 losers in both and
suits, since the
footprints are 3 and controls are 0. The
contract - 1.A significant difference by any standard, I
would say :-).

As you certainly know by now,
Zar Points is a hand-evaluation system rather than bidding
system by its own. You can continue using your own conventions
and systems, while still constantly evaluating and re-evaluating
your hand as the bidding progresses and act accordingly.

Having said that, there are
systems and … then there are systems :-) Which
ones are suitable for direct Zar Points
involvement and which ones are not? Which bidding principles fit the concepts of Zar Points and
which don’t? This is an important question, which
I would like to shed some light on at the end of these
discussions.

To answer this question let’s
state the most important basis on “revealing”
the Zar Points in the two hands – the basis is
in the suits and fits in both hands, which means that
the bidding itself is concentrated around showing the
primary and secondary suits of the hands.

Talking about “Standard
Systems”, the first than comes to mind is “Standard American” - no surprises
here :-) Is Standard American (and the popular SAYC
– Standard American Yellow Card) suitable? The
answer is … regrettably not. And the reason for
that is in the weakest part of that system – the
fact that 1 NT is NOT forcing after the opening 1
/1
by partner. Having 1NT available as a “pass-through”
bid allows the partner to reveal his hand and gives
YOU the opportunity to re-count your Zar Points and
act accordingly.

SAYC’s two over one bid
is fine, though, as it is with all the other systems.
Even if two over one is NOT a Game Forcing (in case
the suit is rebid), the important thing is that it is
still a round forcing and allows for a variety of hands
to be bid through a two over one bid.

The other 2 major “standard”
bidding systems are “Two over One” and the
Strong Club systems (no room here for “twisted”
systems like “Strong Pass” etc. which have
their own merits).

Both “Two over One” and “Strong Club” (this “bag”
includes “Precision Club”, “Polish
Club”, Blue Club”
etc.) have the sanity of using 1NT as a forcing bid.
And the two-over-one is forcing anyway. That’s
what makes them suitable for Zar Points
valuation in practice.

How about popular conventions
like Jacoby 2NT
and Bergen raises over 1
/1
opening?

Both of them fit perfectly.
There are several modifications on the Bergen raises
part (rather than having the original limits of 7-9
and 10-12 in HCP for the 3
/3
bids)
which you may or may not wish to make, but this goes
beyond the scope of these discussions anyway.

How does a “standard bidding” system like
2/1 cope with Zar Points, though? The answer to this
question is “In a similar, forcing-based way”.

This means, that the Zar Points “reserve”
that you eventually have, enables you to make a forcing-bid, baring the responsibility
for ensuring the next Level on your shoulders. So, if
you have 31 Zar Points and

your partner has bid 2
over your opening bid of 1
, you have the “insurance” of one additional
level in your pocket (the 5 additional Zar Points that
it) to make an invitation via your secondary suit and
enable the partner to re-evaluate his hand accordingly.

You are not supposed to have additional power to make
a bid, BUT
you ARE supposed to have additional power to make a
forcing bid.

Since you are already familiar with the minimum requirements
for Level 4 (52 Zar Points) and Level 6 (62 Zar Points,
it will be easy to grasp the entire bidding scale in
terms of Zar Points.

The LEVELS

Keeping
in mind the Levels will allow Zar
Points utilization beyond the initial “prima-vista”
Hand Evaluation, enabling you to extend the Zar Points
communication throughout the bidding process, as both
partner and opponents keep climbing the bidding ladder.

Here
are the levels and the linear Zar Points scale that
you are partially familiar with already:

Level
7 – Grand Slam- 67 (responder has 41+ Zar Points if
the Opener has the minimum of 26)

Level
6 – Slam- 62 (responder has 36+ Zar Points if the Opener
has the minimum of 26)

Level
5 – Slam Try- 57 (responder has 31+ Zar Points
if the Opener has the minimum of 26)

Level
4 – Game- 52 (responder has 26+ Zar Points if the Opener
has the minimum of 26)

Level
3 – Game Try- 47 (responder has 21+ Zar Points
if the Opener has the minimum of 26)

Level
2 – Part Score- 42 (responder has 16+ Zar Points
if the Opener has the minimum of 26)

Level
1 – Opening- 37 (responder has 11+ Zar Points, as low as
it may sound)

Note,
that the amount of Zar Points reflects the
combined Zar Points power of the partnership (at
the specific moment of bidding). The names of the Levels
are the “de-facto-standard” labels you use
to describe the Levels of Contracts in bridge.

Having
the Levels cleared, let’s summarize the most
important dilemmas that you actually can encounter
and try to resolve with the help of Zar Points. I call
them Critical
Decision Points. Note the interesting part that
all of them are at Levels 1, 3, and 5 – the Opening
Level, The Game Invitation Level, and the Slam Invitation
level.

At
the Opening Level:

1)To open or pass – this includes postponing
your intervention for a later stage;

2)To respond or pass – this includes
a “trap-pass” and cases like passing you
partner’s 1NT opening, “inviting”
the opponents to “show their heads up”.

At
the Game Invitation Level:

3)To invite for a game or pass – this
includes competitive-bidding decisions;

4)To accept the invitation or pass.

At
the Slam Invitation Level:

5)To invite for a slam or pass – this
includes competitive-bidding decisions;

6)To accept the invitation or pass.

As
mentioned above, the main principal in the Zar Points
bidding, in respect to applying it to your current system, is that every forcing
bid by any partner promises 5 more points, thus warranting
the safety of the next level of bidding.

Let’s
consider just one simple sequence, using Zar Points
with standard “2 over 1” bidding system.
It goes to the Game Try Level, presenting the Responder
with a chance to re-evaluate his/her holdings in the
light of the secondary suit of the Opener.

Bid - Opener

Meaning

Bid - Responder

Meaning

1

26+,
5+ cards in

2

16+,
3+ cards in

3

4+
cards in
, Level 3 guaranteed (I have 31+ and our total
is at least 47, so you can bid). Forcing –
please re-evaluate your hand based on my secondary
suit.

3

No
additional upgrade points to my initial 2
bid. You can go to game only if you make the Game-total
of 52 based on my initial 16 Zar Points.

Pass

OK,
I only have 31, so we stop at the part-score level.

This
example just comes to illustrate the way you incorporate
the Zar Points “background thinking” in
your current bidding attitude, without requiring drastic
changes in your everyday bidding practices. For a complete
utilization of all the implications that Zar Points
introduce, you would need deeper changes of both system and attitude.

And fundamentally, as you know
already I hope, a Bidding System is not a bunch of conventions
“collected” from here and there, but a philosophy
shared by the two partners in the partnership. Calculations
in bridge are important (especially the ability to count
to 13 :-), but common philosophy, along with synchronization
and imagination make this game “human” -
otherwise Zia wouldn’t be able to beat The Computer
:-)

Talking about computers, I am also finishing a program
playing Zar Points so you would be able to play around
with it and see how things work. The program is kept
in Java script only so you can run it directly from
you browser with no server-side operations (it will
come free, of course). You’d be able to run it
from your own hard drive, distribute it free to anyone,
publish it as an HTML page wherever you like, make modifications,
additions, improvements (hey, nobody’s perfect
:-).

I
hope all these discussions gave you a fresh
look at the wonderful game of bridge!