Inferences

By Marshall Miles

The key to successful play and defense is to visualize the unseen hands. The first and most valuable technique is to determine the distribution. In the following lesson I explain not only how to count the hands but also what you can do with that information.

The bidding is:

West leads the ten of spades.

Because of the duplication, each hand having the same distribution, you are going to have to guess who has the queen of hearts. In a newspaper column you usually find out that one hand has a singleton so that you can cash a high honor and then have a cinch finesse against the other hand. In real life, you seldom discover that someone has a singleton, but you follow the same general procedure to determine the opponents' distribution. You cash all your winners outside the crucial suit, hoping that someone will show out so as to give you at least a partial count. In this case at least one opponent must show out when you cash four rounds of diamonds.

Both opponents follow to three rounds of clubs. West follows to three rounds of diamonds while East has only two. When you play spades you obtain an important clue. East shows out on the third round. Now you know that West had 5-2-3-3 distribution or possibly 5-1-3-4. (You won't know who has the 13th club unless someone discards a club.) You now have enough information to play the hearts correctly. You cash the ace and finesse the jack on the next round. The odds are at least five to two in your favor since East has five (or possibly six) hearts and West has, at most, two.

You were somewhat lucky that East had a doubleton spade, which helped you immensely in getting an accurate count. However, you were bound to get some sort of clue. Suppose both defenders follow suit to three rounds of both black suits, and the diamonds split 3-2.

The defenders have to make a total of three discards on the diamonds. If someone discards a club, for example, and both opponents follow to three rounds of clubs, at least you will know that the player who discarded a club started with four. On the third round of spades, if West doesn't play the nine, he must have started with four since he would hardly lead the ten without the nine. An expert would play the nine of spades on the third round whether he had to or not, but most defenders would not think of that falsecard, and that would give you a clue. Somehow or another, you will almost always know, after ten tricks, which defender has the greater length in hearts. It makes you happier when hearts are 5-2 (unless you happen to lose to the doubleton queen), but even when they are 4-3, you have 4 to 3 odds in your favor if you play the longer hand for the queen.

So you don't consider 4 to 3 odds impressive? Suppose you and an opponent were allowed to draw cards from a hat. If you got to draw four cards and he got to draw three, isn't it obvious that you would have a better chance to draw the crucial card?

gainst expert and devious opponents, the defender with four small hearts may go out of his way to show you he has four hearts, and the hand with the queen may discard down to the doubleton queen. But most opponents will keep the queen of hearts amply guarded, and it will certainly pay, in the long run, to get a count on the hand. Another small point: Suppose dummy had Al0x of hearts and A10x of clubs. Your hand is KJx of hearts and KQx of clubs. It would be a good idea to play your four rounds of diamonds first. The defender who doesn't have the missing queen of hearts won't know whether clubs or hearts is the crucial suit. If he knew, he would keep cards in the crucial suit so as not to give away his partner's holding, but since he doesn't know, he may give the show away.

The contract is 3NT. West leads the jack of clubs. You duck in dummy, but East wins the king and ace, then leads the third round. West, who started with CJl087, wins the third and fourth round of clubs and exits with a heart. (You discarded a spade and a small heart from the dummy on the third and fourth rounds of clubs, leaving the blank ace of hearts in dummy.) How can you take the rest of the tricks?

It will be easy if the spades break 3-2, but when you cash the ace and queen of spades, East discards a heart on the second round. Since the spades split 4-1, you are going to need three diamond tricks. You follow the normal procedure by cashing your sure winners in the other suits, and both opponents follow to three rounds of hearts. What do you know about the opponents' distribution?

It is easier to count West's hand than East's. West has shown four clubs and four spades and at least three hearts--consequently no more than two diamonds. Again the percentage play is to finesse through East with 5 to 2 odds in your favor.

There is another clue which, this time, you don't need: East's discards on the fourth round of clubs and the second round of spades. A top expert might try to trip you up, but 99% of your opponents would discard first from their five-card suit. They know their fifth card won't be a winner, nor will discarding it create a winner for you (directly). But it could cost them a trick to discard from a four-card suit since you might have four of that suit. Even when there is no real danger (because you can't have a side four-card suit), players tend to discard their fifth card of a suit as a reflex action, since USUALLY it won't cost. Only when it helps you to count the hand.

At notrump it almost always pays to cash your winners in the suits where there are no options in order to get a count of the opponents' distribution. When your problem suit is trumps, the way to get the most accurate count might be to cash winners until someone ruffs one of your winners. That won't be a good idea when the ruff is the setting trick (and seldom a good idea in any event), so you have to act on less than full information.

In a 3-card or longer suit, count the queen as a winner as long as there is at least one other honour in the suit. If not, count the queen as only half a winner.

West leads the king of spades, which you win with dummy's ace. You have one sure loser in every suit but hearts, so you need to avoid a trump loser (by ruffing or otherwise). West presumably has six spades to East's one. That means West has seven "non-spades" to his partner's 12 "non-spades." West has seven chances of holding the queen of hearts to East's 12 chances. With odds like that, the correct play is to cash the king of hearts and, if the opponents follow suit, finesse the jack or ten on the next round.

West leads the jack of spades, overtaken by East's queen and your ace. What now?

East has seven or eight spades (seven is more probable) to West's one or two. You cash the ace of diamonds and both defenders follow suit. East has (probably) five cards in hearts and clubs to West's ten. East might or might not hold the king of hearts for his 3S bid, but with 2 to l odds, based on the distribution, it seems best to take a straight finesse in hearts (through West) rather than a ruffing finesse (through East) to get rid of your spade loser.

You are the declarer in 4S (after opening lNT and a transfer response by partner). West leads the ten of diamonds. East wins with the ace and returns the five of diamonds, which West ruffs. West cashes the ace of spades and leads another spade, both defenders following suit to the second spade. How would you play from this point?

You know that West has three spades and one diamond (to East's two spades and five diamonds). That leaves West with nine cards in hearts and clubs to East's six cards in hearts and clubs. So the odds are 9 to 6, or 3 to 2, that any finesse through West will win and any finesse through East will lose. So your best play is to lead a low heart to the ten spot rather then take the club finesse.

Incidentally, although the major theme of these hands is counting, it doesn't hurt to consider other clues. East could have returned the deuce of diamonds to ask for a club lead (but he didn't), and West didn't return a club. That is another indication that the club finesse won't work. Of course, you could finesse against either defender in hearts, but playing West for the queen is the percentage play.

Once you know the percentage play in a suit, it doesn't pay to stop counting.

You open lH. LHO bids 4S and partner bids 5H, ending the auction. West leads the ace of spades and another spade. East ruffs the second spade and returns the jack of hearts, which you win, while West follows suit. Since West has nine major cards to East's three, the odds are pretty good that the club finesse will work. So the careless play is to lead a club to the king and finesse the jack of clubs next round. Horrors! You lose to the doubleton queen. Have the cards no respect for the odds? Yes, they do. They punish you when you take a 67% play rather than a 100% play. What you should have done was to cash the high diamonds (discarding a club), followed by a diamond ruff. If West had followed suit to the third diamond, a second-round club finesse would be bound to win. When he showed out on the third diamond, you would know he had 8-1-2-2 distribution. So you cash all your trumps (saving dummy's remaining diamond). With three cards left, dummy has a diamond and CK7; you have CAJ9. You lead a club to the king and (if the diamond isn't good) lead a club to the ace, paying no attention to which club East plays. If East had the CQ, he will have to play it on this trick, since his other card is a high diamond. If West has the CQ, he will have to play it on this trick, since he started with a doubleton club. You should reject a club finesse, even though it will usually work, because you have a 100% play by adopting a "show-up squeeze."

It was easy to count the distribution of the previous hands. Nothing helps you count a suit better than noticing that someone shows out. But many times it isn't practical to play your suits until someone shows out. There may be entry problems or dangers of a ruff, for example. In such cases you have to draw inferences from the bidding and the way the defenders play (in particular, their opening lead and play to tricks one and two). The following hand is from a matchpoint game with both sides vulnerable.

The opening lead is the D9. East wins the queen and ace of diamonds, then leads the third round of diamonds, giving West a ruff. West then leads the jack of spades to East's ace, and East returns a spade to your king. Since the opponents have already taken four tricks and you are still missing the ace of hearts, even Eddie Kantar would fail to make this hand. While -100 may be a decent result (if the opponents could make 110 in spades), -200 would be a disaster. The only thing you "know" (and could bet your life on) is that West had two diamonds and East had five, but you can draw certain inferences. Since West bid lS, he should have at least four spades (but no more than four since East raised). (Many serious players now play support doubles and redoubles, in which cases East would redouble instead of raising, but this hand was from several years ago.) Anyway, West must have four spades and two diamonds. How many hearts?

No more than three since with 4-4 in the majors, he would respond lH. Therefore, West has four trumps. West also has the HA, which you know from the bidding. You can ruff one spade, and you need to lead hearts twice toward dummy to get a discard for your fourth spade. You will also need to finesse West for the queen of clubs, (preferably after cashing the CK). So the rest of the play will go: Heart to the king; club to the king; heart. West wins and plays, say, a spade. You ruff in dummy, ruff a heart, take the club finesse and pull trumps.

The following hand is not quite so clear-cut as the previous hand, since it depends on an inference which you may or may not be able to rely upon. It depends upon the bidding skill of the opponents.

West leads the king of hearts and shifts to the eight of spades. East plays the queen and you win with the ace. What can you figure out about the distribution?

In one respect this hand is similar to the previous hand. East must have four spades and three hearts since he didn't respond lH. What is West's spade holding? It looks like KJ8, which leaves East with SQ532. East can't be very anxious to play spades opposite three-card support, so wouldn't he bid 2D if he had DQxx or DQJx? (With Qxxx of spades and no heart honor, East needs at least the queen of diamonds for his response.) There is a very good chance (from East's point of view) that West has a five-card diamond suit, and if East bids 2D and West passes, 2D is almost sure to be a better contract than 2S. Therefore, East probably has fewer than three diamonds.

Have a little courage and trust your judgment! Cash a high club. Lead a heart to the jack and return a club, finessing the nine if East fails to split his honors. East had: S Q532 H l042 D QJ C Jl062.

If West had continued hearts at trick two, forcing you to use one of dummy's entries prematurely, you would have to take a first-round double finesse in clubs! West's singleton could have been the jack, ten or any of three smaller cards. If you could be sure of the distribution, the odds would favor the first-round finesse by 3 to 2. Even to me, that first round finesse looks rather scary.

Sometimes, when you know the distribution, you can make some unusual plays in a suit.

You are vulnerable against not. You open lS. West bids 2NT. Partner passes and East bids 3D. You bid 3S and partner, figuring all of his high cards are working, raises to 4S. West leads the king of diamonds. You win and play the ace of spades and a low spade to the queen, both opponents following suit. What do you do now? You can be pretty sure of the opponents' distribution. West's most likely distribution (since he followed suit to two rounds of spades) is 2-1-5-5. With 6-5 or 5-6 in the minors, he might have bid more with favorable vulnerability. And if he has a void in hearts, you can't make the hand unless East carelessly keeps nothing but hearts and allows himself to be endplayed.

If West has a singleton heart, how should you play the hearts? Obviously, if his singleton is the king, you should lead low to the ace, but if his singleton is the jack, ten, or nine, you can make your contract by leading the queen. With three chances instead one, it must be right to lead the queen.

Marshall's book on inferences is available at your local bridge supplier or book store, and the full selection of email courses is available at Bridge Today University (www.bridgetoday.comwww.bridgetoday.com), where you can also get an annual membership which includes daily quizzes, back issues of Bridge Today, and other extras.