Counting Distribution Tutorial
------------------------------
In this tutorial, I will discuss techniques for determining the
entire distribution of all 4 hands. This often lets you play as if
you cheated by peeking into your opponents' hands! However, the
techniques I will discuss are all perfectly legal, and are
commonly employed by many experienced bridge players.

You may think that this topic is much too advanced for you. Please
keep an open mind. Hopefully, after you read the discussion and
do the examples, you will see that the techniques I discuss are
just common sense. For those of you who like to read mystery novels
and see if you can solve the crime before the author reveals the
guilty party, you should find this topic particularly fun.

There are two basic techniques that you can use to count distribution --
positive inferences and negative inferences.

Positive inferences are straightforward. You can determine many
facts about the other hands by the bidding and card play. Here are
some examples:

1) Your partner opens 1H/1S. Therefore, he or she must have at least
five spades. (I told you it was easy!)

2) Similarly, your opponent has pre-empted. If the opening bid was
a weak 2, the opener has a six card suit; if the opening bid was
at the 3 level, the opener has a seven card suit (you already know
about more than half the cards in his/her hand!).

3) Declarer opens 1NT. The hand cannot contain more than 1 doubleton
(and therefore no 6 card suit), and no singletons or voids.

4) Declarer draws 2 rounds of trump. You have 3 trumps, and your partner
shows up with 1. By looking at dummy, you can determine how many
trumps declarer has.

5) Your partner leads the 2 of a suit against a no trump contract.
He/she has exactly four. By looking at dummy, you can determine how
many cards in that suit declarer has.

Negative inferences are similar to the famous mystery story in which
Sherlock Holmes solves the crime because the dog did not bark.
You can often determine what someone has by what they did not do.
For example,

1) Partner opens 1C/1D. It is almost certain that he/she does not have
a five card major.

2) You open 1H and partner bids 1NT. Partner has neither 3 hearts
(would have raised to 2H) nor 4 spades (would have bid 1S).

3) Partner opens 1C, you bid 1H, he/she bids 1NT. Since the rebid
was neither 1S nor 2H, partner has at most 3S and 3H. Partner
also has at least 4 clubs  -- if he/she only had 3C, the hand must
contain 4 diamonds, and therefore 1D would have been the opening bid.
Look at all the information that can be gleaned from such a 
common auction!

4) Partner leads the 2 of a suit against no-trump, showing 4.
Partner does not have another 5 card suit (unless it was bid
by the opponents).

5) You bid a suit and partner does not lead your suit -- the opening
lead is a low card in another suit. Partner better have a void!

As defender, you can usually determine declarer's distribution within
a card or two by the time trumps are drawn. If you lead a side suit
and more than one round is played, you often can determine the
distribution of that suit. Similarly, once trumps are drawn, you know
the trump distribution. Finally, declarer has bid, so you have 
additional information about the hand. I recommend that you
develop the habit of counting declarer's hand on every deal;
it should become automatic after a while.

One technique that helped me develop my counting skills was to try
and count the hand when I was dummy. This can be difficult,
because you can only see 13 cards, not 26. However, as dummy you
have little else to do anyway, and it makes your counting skills
that much sharper.

As promised, let's look at the bonus problem from the NT Declarer
Play tutorial to see how we can make use of these techniques
Remember, this was an actual hand, so this is not a purely
academic exercise.

Your left hand opponent opens 1NT followed by 3 passes. Your partner
leads the 2 of hearts. You observe the following:

    (Dummy)
    xxx
    10xxx
    Q10xxx
    A   
                (You)
                Qxxx
                xxx
                KJx
                Kxx
                
You should be able to determine declarer's exact distribution.
What is it?

The most clear cut positive inference is the heart distribution.
Partner has 4, and you can see 7 hearts between you and dummy --
therefore declarer has 2.

The key to this hand is the club suit. There are only 4 between
you and dummy so there are 9 clubs between partner and declarer.
You can therefore determine the club suit distribution by 
making use of both a negative inference (partner cannot 
have 5 clubs, because they would have been led) and a positive
inference (declarer opened 1NT and therefore cannot have a six
card suit). Therefore, declarer must have 5 clubs 
and partner must have 4.

Since you have already determined that declarer has 2 hearts and
cannot have more than 1 doubleton, you know declarer's entire
distribution -- 3 spades, 2 hearts, 3 diamonds, and 5 clubs.

Quiz
--------
1) Sitting West, you hear the following auction:

S   W   N   E
--  --  --  --
1H  P   2C  P
2D  P   2H  P
3C  P   4H  P
P   P

What is South's most likely distribution?

This type of auction comes up reasonably often, so you 
might want to remember your answer for future reference.
In fact, when there is a lot of bidding, you can often determine
declarer's distribution, particularly once you see the opening lead and dummy.

South's most likely distribution is 5 hearts, 4 diamonds, 3 clubs, and
one spade. He could also be 0-5-4-4 or 0-5-5-3, but these distributions
are statistically less likely. The important point is that South has
at most one spade (so a trump lead may be best).

2) You are sitting South. The auction proceeds as follows:

S   W   N   E
--  --  --  --
    2H  D   4H
4S  P   P   P

Here is your hand and dummy's hand:

               North
               -----             
             S AQxx
             H x
             D xxxx
             C AQ9X


               South
               -----             
             S KJxxx
             H xx
             D Kxx
             C K10X

(I'm sure all of you have taken my bidding lessons to heart,
and everyone would confidently bid 4S with the South hand, right?)

West leads the heart K and then shifts to a low diamond.
East wins the A and plays the Q of diamonds, which West trumps.

West now plays the heart A which you trump. 
You play 2 rounds of spades East follows to both rounds,
and West plays a spade and then a heart.

Of course, all of you planned the play at trick 1, and realize you
must take 4 club tricks in order to make the contract. Because you
have determined the distribution of both defenders, you know
if you should play for the 3-3 break or finesse someone for the J.
What should you do?

You can count West's distribution. He has shown up with 2 spades
and 1 diamond. Since he opened 2H, he has six hearts. Therfore, he
has 4 clubs. Therefore, you should play the A then K of clubs. 
If the J does not drop, you should finesse West for the J.
One other point - you should play the 10 of clubs under the A
so you do not block the club suit.

3) You are sitting West; your partner is the dealer.

S   W   N   E
--  --  --  --
            1D
1S  P   2S  P
4S  P   P   P

You lead a low diamond. Here is your hand and dummy:

               North
               -----
             S KQx
             H 9xx
             D xx
             C Jxxxx

West
----
S xx
H Q108xx
D xxx
C Qxx

Your partner takes the A and K of diamonds, declarer following with
2 low diamonds. Your partner shifts to the heart J, declarer winning
with the A as you signal with the 8. Declarer leads
a low spade to the K. Your partner wins the A and returns a low
heart. Declarer wins the K. Declarer plays a spade to the Q, partner
following with the J. Declarer plays another spade, your partner
following as you discard a diamond. Declarer plays 2 more spades, 
and you discard 2 hearts while partner discards 2 low clubs.

The moment of truth has arrived. Declarer plays the Q of diamonds,
leaving you in this position:

               North
               -----             
             S -
             H 9
             D -
             C Jxx

West
----
S -
H Q
D -
C Qxx

You should be confident you will beat the contract, because you
have determined declarer's distribution. What is it, and what should
you discard?

You can determine partner's distribution. He has shown up with 5 diamonds and
3 spades. Since he played high low in hearts, he has 2 hearts, and, therefore, 3 clubs.
You now know South's distribution (5 spades, 3 hearts, 3 diamonds, 2 clubs).
So you should throw a low club.

4) As South, you are in 6 spades. 

               North
               -----             
             S AQ9x
             H xx
             D Axxx
             C AJX

               South
               -----             
             S KJ10xx
             H Axxx
             D x
             C K10X
             
The opening lead is the K of hearts.

The key to the hand is determining who is more likely to have
the Q of clubs. You may be able to accomplish this by using 
what is known as a "discovery" play -- discovering the distribution
of each opponent. In particular, you should play the opponent
who has more clubs for the Q. 

Therefore, you must delay playing clubs for as long
as possible. Perform all your red suit ruffs and draw trumps
before playing clubs. Discuss how you would play the hand using
this technique, and how you would determine which way to finesse
the club Q. 

To make it easier for yourself, duck the heart A at
trick 1. I will stipulate that East has at least 2 hearts.
Assume West plays a second heart at trick 2. For the purposes
of slightly simplifying the play of the hand, I
will also stipulate that trumps are 2-2.

After ruffing 3 diamonds and 2 hearts, you will be able to determine
the exact distribution of the red suits. Since you also know how many
spades each opponent has, you are able to determine the exact club
distribution.

For example, suppose one opponent has 4 hearts and 4 diamonds. Since
he started with 2 spades, he has 3 clubs. Therefore, his partner has 4
clubs, and you should finesse his partner for the Q of clubs.

As another example, suppose one opponent has 3 diamonds and 4 hearts (and
2 spades). He has four clubs, so you should finesse him for the club Q.
