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.