Match 2 - Board 9

Board 9
Opponents vulnerable

♠ 9 5 4 ♥ 8 3 ♦ 7 5 2 ♣ A K 9 8 6

Partner opens one heart, and RHO overcalls two diamonds. With three diamonds and two defensive tricks, I'm quite happy to pass. If partner has diamond shortness, I will hear from him.  If he doesn't, defending two diamonds may be our best chance for a plus score.

LHO raises to three diamonds, partner bids three hearts, and RHO bids three spades. I could bid four hearts, but four clubs leaves partner better prepared to act over four spades or five diamonds. This bid should show heart tolerance. If I couldn't bid clubs on the previous round, I wouldn't be bidding them now with a misfit.

LHO doubles and partner passes. I assume he would run to four hearts with a singleton club. RHO passes, and I bid four hearts. LHO bids five diamonds, passed around to me. Obviously I'm not bidding five hearts. Should I double? If the opponents think they can make five diamonds and partner agrees with them, I don't see any reason to suspect they're wrong. Probably only one of my clubs is cashing. Whether they can make this or not is apt to depend on how good partner's spades are, which is something only he knows. If I were confident we were making four hearts, I might double in an attempt to protect our score.  But I have no particular reason to think we were making it. Accordingly, I pass.

My first inclination is to lead a trump, hoping that partner has good spades and that we can stop spade ruffs in dummy. One opening lead I would never consider would be a high club. Dummy has good clubs, declarer has no place to pitch whatever club losers he has in his hand, and I may need the club entry to lead a second trump. A heart lead might be right, since declarer may be able to pitch dummy's hearts on his spade suit. But if his spades are that good, we probably can't beat this. So I go with my first instinct and lead the five of diamonds.

	NORTH ♠ 10 2 ♥ 10 5 ♦ K Q 9 8 ♣ Q J 7 3 2
WEST ♠ 9 5 4 ♥ 8 3 ♦ 7 5 2 ♣ A K 9 8 6

West	North	East	South
		1 ♥	2 ♦
Pass	3 ♦	3 ♥	3 ♠
4 ♣	Double	Pass	Pass
4 ♥	5 ♦	(All pass)

Dummy is no surprise. Declarer plays the eight--four--jack. If declarer has only five diamonds, he presumably has only four spades. If we assume partner would not have passed four clubs doubled with a singleton club, that gives declarer a 4-3-5-1 pattern. Declarer plays a second diamond to dummy's nine, and partner pitches the four of hearts. Declarer plays a heart from dummy. Partner hops with the ace, and declarer plays the seven. A poor choice. He should play the deuce, the card he's known to hold. (Partner would not have discarded his second lowest heart on the previous trick.) I now know that declarer begain with king-seven-deuce of hearts, so it appears I was right about his shape. I play the heart eight.

Partner returns the queen of hearts to declarer's king. Declarer plays the five of clubs. I wouldn't hop without both honors, so there is no point in false-carding. I play the club king, and partner plays the four.

Declarer has three major suit cards that he might potentially want to ruff, so I play my last trump. Declarer wins in dummy, and partner throws the club ten. Declarer plays a spade to the queen, cashes the ace, and ruffs a spade. He now has one heart and one spade left in his hand. He leads the club queen, pitching his spade loser. I win and must return a club to dummy, allowing him to pitch his heart loser. Down one.

	NORTH ♠ 10 2 ♥ 10 5 ♦ K Q 9 8 ♣ Q J 7 3 2
WEST ♠ 9 5 4 ♥ 8 3 ♦ 7 5 2 ♣ A K 9 8 6		EAST ♠ K J 8 6 ♥ A Q J 9 6 4 ♦ 4 ♣ 10 4
	SOUTH ♠ A Q 7 3 ♥ K 7 2 ♦ A J 10 6 3 ♣ 5

A high club lead would have allowed declarer to make this for the very reason I suspected: I need the entry to play a second trump. A heart lead, though, would have beaten it, provided partner wins the ace of hearts at trick one and shifts to a trump, a play that should not be difficult for him to find. (Although, as it happens, partner would not have found it. I replayed the board to see. After a heart lead, he wins with the ace and returns one.)

We get 11 matchpoints. Only one other pair beat five diamonds. Of the five pairs who allowed it to score, four defended it doubled. So, strangely, most of our matchpoints came not from my opening lead but from the decision not to double. We would have still won nine matchpoints had we allowed them to score this contract.

Score on Board 9: +100 (11 MP)
Total: 82 MP (75.9%)

Current rank: 1st

A note about comments

I've changed the settings on this blog to require my approval of each comment before it is posted.  This is not meant to discourage anyone from commenting.  Nor it is an attempt to exercise editorial control.  I did this because I have been bombarded with spam comments containing suspicious-looking hyperlinks.  I delete them whenever I see them, but that doesn't seem to discourage the perpetrators from trying again.

If you try to post a comment and don't see it right away, this is why.  I will approve all non-spam comments, but it may take 24 hours before you see it.

Match 2 - Board 8

Board 8
Neither vulnerable

♠ 10 7 4 ♥ K Q 5 3 ♦ Q ♣ Q 9 6 5 3

Partner opens one diamond in second seat. Pass to me. I respond one heart, and partner rebids one spade. I bid one notrump, which ends the auction. West leads the deuce of clubs.

	NORTH ♠ Q J 9 6 ♥ 7 6 ♦ A K 9 8 6 ♣ K 7
	SOUTH ♠ 10 7 4 ♥ K Q 5 3 ♦ Q ♣ Q 9 6 5 3

West	North	East	South
Pass	1 ♦	Pass	1 ♥
Pass	1 ♠	Pass	1 NT
(All pass)

If East has the ace, I might do well to duck this in dummy. But if West has it, rising with king will create more problems for the defense. To take two club tricks, the next club play will have to come from East. He may not have enough entries to play both clubs and hearts (assuming playing hearts is a good idea). In addition, if he has jack doubleton and fails to unblock under the king, taking two club tricks becomes even more problematic.

I rise with the king, and East plays the four. If I play the three, West will know East played his lowest club. So I follow with the five.

My basic plan is to take one club, three diamonds, two spades, and a heart. It could be difficult to do this, since the opponents can hold up in spades to deprive me of a second spade trick. But, if they do, perhaps I can manage an extra trick in hearts or clubs.

There are two ways I can start.  I can unblock diamonds, then play a spade; or I can use this opportunity to play a heart toward my hand.  If East has the heart ace, this might be a good idea, because I may need two heart tricks.  But if West has the heart ace, it will be a very bad idea.

I don't know at this point whether I need two heart tricks or not.  So which line is better? Actually, this decision is not hard if you think about the hand from your opponents' point of view, as you always should.  East knows his partner has only four clubs and you have five (unless West has led from ace third), so, if you play a diamond to the queen and a spade, what is East going to do when he wins the trick?  Looking at that dummy, he will surely play a heart.  So even if it's right for you to play a heart now, you probably don't need to. East will do it for you.  I play a diamond to the queen. East plays the seven; West, the four.

I lead the spade seven (getting the card out of the way just in case I need to finesse against the eight later on). West plays the three, I play the queen from dummy, and East wins with the ace.

As expected, East shifts to the deuce of hearts. This confirms that clubs are four-two. If West had led the deuce of clubs from ace third, East, with three clubs himself, would not know this and would probably continue the suit. His spurning a club return suggests that he has a doubleton and knows I have five. And, since West didn't lead the jack, East's remaining club is probably the jack or ten.

If I play the heart king and West has the ace, he will suspect I have the queen. I don't know if he will have enough confidence in this inference to duck, but I'm not taking any chances. I want to know who has the heart ace. It is much harder to duck the queen than the king, so that's the card I choose. West wins with the ace and continues with the heart ten. East follows with the four.

Unless West is being tricky, East has the heart jack. So, if I win this, East will have an entry to put a club through. When the defense wins the spade king, they will be able to cash two more hearts and two clubs, bringing their total to seven tricks. It's possible they can't cash all these tricks if East failed to unblock the jack of clubs. But, even then, I have no clear route to making this contract if I win this trick. All the defense has to do is duck the next spade to give me serious problems.

What happens if I duck the heart ten? Suppose, first, that West plays ace and a spade to strand me in dummy. I can then cash one high diamond and the long spade, coming down to:

	NORTH ♠ -- ♥ -- ♦ K 9 8 ♣ 7
	SOUTH ♠ -- ♥ K ♦ -- ♣ Q 9 6

I need two more tricks. There are lots of way I might take them, depending on what cards the opponents have come down to. If West keeps a diamond, a heart, and two clubs, for example, I can cash the king of diamonds and play a club, covering East's honor, and score my king of hearts at trick thirteen. Or, if West keeps one diamond and three clubs, I can cash the diamond king, pitching my king of hearts, and score a club at the end. If he keeps one heart and three clubs, my hand is squeezed on the lead of the king of diamonds. In that case, I will need East's remaining club to be the jack, giving me a winkle. My exact play will depend on the other three cards in East's hand. If East has two diamonds and a heart, I can cash the diamond king (pitching a club), lead a club, and duck. If, instead, East has jack-ten third of diamonds, I can lead a club immediately and duck, scoring two diamond tricks in dummy.

West might choose to lead a low spade instead of ace and another to deprive me of a long spade trick. But that doesn't seem to help. Without going through all the possibilities, that play seems to make life easier for me by giving me an extra way to toss him back in the lead. In short, ducking the heart looks as if it gives me lots of chances, some, but not all of which, require East to have a singleton jack of clubs. Winning the heart requires clubs to be blocked. And I'm not necessarily home even then. So I duck.

West takes yet another approach to the defense. He cashes the club ace, stripping dummy of its exit. East follows with the jack.

West shifts to the five of diamonds. I win with dummy's ace. East plays the deuce as I discard a heart. I cash the king of diamonds, pitching a club as West discards the club eight. I need to find three tricks in this position:

	NORTH ♠ J 9 6 ♥ -- ♦ 9 8 ♣ --
	SOUTH ♠ 10 4 ♥ K ♦ -- ♣ Q 9

I lead the nine of spades from dummy. If East has the spade king, I'm down. If West has it, he must duck, else I have an entry to my hand. I can then continue with the jack of spades, smothering my ten. If West has the king or king-eight of spades remaining (having foolishly failed to unblock the eight on the previous trick), he must give me the last two tricks in my hand.

As the cards lie, West can't even stop the overtrick now. He takes the spade king and plays a heart, My hand is high. Making two:

	NORTH ♠ Q J 9 6 ♥ 7 6 ♦ A K 9 8 6 ♣ K 7
WEST ♠ K 8 3 ♥ A 10 9 8 ♦ 5 4 ♣ A 10 8 2		EAST ♠ A 5 2 ♥ J 4 2 ♦ J 10 7 3 2 ♣ J 4
	SOUTH ♠ 10 7 4 ♥ K Q 5 3 ♦ Q ♣ Q 9 6 5 3

What if West, after cashing the club ace, had exited with king and a spade instead of a diamond? I would cash the long spade, and East would presumably pitch the jack of hearts. I would then cash a diamond, reaching this position, needing two more tricks.

	NORTH ♠ -- ♥ -- ♦ K 9 8 ♣ --
	SOUTH ♠ -- ♥ K ♦ -- ♣ Q 9

The opponents have three diamonds left, so a low diamond from dummy, pitching a club, guarantees my two tricks. If East has all three diamonds, he is endplayed. If not, whoever wins the diamond must put me in one hand or the other.

They couldn't beat me once East won the spade ace. Had he ducked the first round of spades, I don't see any way I can make it.

This result is worth a full 12 matchpoints, and that would have been true even without the overtrick.  One North-South pair defended one notrump making. The others declared either one notrump or two spades, down one in every case.

Score on Board 8: +120 (12 MP)
Total: 71 MP (74.0%)

Current Rank: 1st

Match 2 - Board 7

Board 7
Both sides vulnerable

♠ K 9 8 7 5 ♥ 8 6 4 3 ♦ 8 4 ♣ 10 9

I pass in first seat, and LHO opens with one notrump (12-14), which is passed around to me.

If both opponents have maximums for their actions, partner will have 13 high-card points. He probably doesn't have much more than that, since he didn't double. The fact that I don't have much in the way of high cards myself doesn't dissuade from balancing, but the fact that I don't have a singleton does. With a balanced hand, it's unlikely we can do better in our own fit than we can defending notrump. I pass, and partner leads the three of diamonds.

	NORTH ♠ 10 4 ♥ J 10 7 ♦ Q 10 9 ♣ A Q J 8 3
		EAST ♠ K 9 8 7 5 ♥ 8 6 4 3 ♦ 8 4 ♣ 10 9

West	North	East	South
		Pass	1 NT
(All pass)

Declarer plays the nine of diamonds from dummy. If partner has the king of diamonds, he would like to know my diamond count, so that he can continue diamonds if declarer began with a doubleton ace. If partner has the diamond ace, however, he will want an attitude signal, since he will be primarily interested in whether or not I have the diamond king. (I can't afford to play the king from king third in this layout.)

In general, if either of two signals might be useful, attitude takes precedence. Defending notrump, I can give count at trick one only if partner can infer I don't have an honor either from the auction or from my failure to play it. That's not the case here. If I play the eight, trying to give count, partner may well continue diamonds when a spade shift would be more productive. So I play the diamond four.

Declarer overtakes dummy's nine with the ace, presumably from ace-king third. He should have overtaken with the king, leaving open the possibility that he has king-jack third. The ace is a revealing card.

Declarer leads the four of clubs to the deuce and jack. I drop the ten. Declarer plays the seven of hearts, I cover with the eight, and declarer wins with the ace as partner plays the five. Again, the ace is a revealing card. Declarer would hardly play a heart to the ace unless he has the king as well. Had he played a heart to the king, neither I nor partner could be sure where the ace was.

I can now place declarer with 14 high-card points, so partner must have the rest. Declarer has five club tricks, three diamonds, and two hearts for ten tricks. My job now is to persuade declarer that I have the heart queen. I would certainly not pitch a heart if I had queen fourth, so I must clutch all my hearts as long as possible.

Declarer plays the club five--king--ace--nine. He cashes the ace queen of clubs. I drop the five of spades. Partner, like me, can place all the high cards now, so there is no need for attitude signals. This card should show count. Partner pitches the deuce of diamonds. On the next two clubs, I pitch the seven and eight of spades. Partner pitches the three of spades and the six of diamonds. Declarer, on the fifth club, pitches the deuce of spades.

Declarer cashes the queen of diamonds. The reason I never pitched my diamond was I that I wanted to be able to follow to this trick. It doesn't gain anything to pitch the diamond, and, as a general rule, you want to avoid showing out when possible to make it harder for declarer to count the hand. In this particular case, declarer has an accurate count in diamonds anyway from partner's carding. And, frankly, I want him to have a count in diamonds so he knows I have heart length. But, as Aristotle pointed out, virtue is all about instilling good habits

Declarer now falls for the trap. He leads the ten of hearts and passes it, losing to partner's queen. Partner plays the jack of spades. I overtake with the king and return a spade. Partner cashes the ace and queen and concedes a diamond to declarer's ace at the end. We've held declarer to nine tricks.

	NORTH ♠ 10 4 ♥ J 10 7 ♦ Q 10 9 ♣ A Q J 8 3
WEST ♠ A Q J 3 ♥ Q 5 ♦ J 7 6 3 2 ♣ K 2		EAST ♠ K 9 8 7 5 ♥ 8 6 4 3 ♦ 8 4 ♣ 10 9
	SOUTH ♠ 6 2 ♥ A K 9 2 ♦ A K 5 ♣ 7 6 5 4

This is worth eight matchpoints. Every other pair who played notrump made twelve tricks. Two were in game, and two were not. We lost only to the two tables where North-South played club partscores.

Score on Board 7: -150 (8 MP)
Total: 59 MP (70.2%)
Current Rank: 1st

Match 2 - Board 6

Board 6
Opponents vulnerable

♠ Q 10 6 5 ♥ K 10 5 2 ♦ Q 4 2 ♣ Q 4

Partner opens two notrump (20-22) in fourth seat. Since we have at least 29 high-card points between us, we can probably take as many tricks in notrump as we can in a four-four major-suit fit, so I simply raise to three notrump. West leads the jack of clubs.

	NORTH ♠ Q 10 6 5 ♥ K 10 5 2 ♦ Q 4 2 ♣ Q 4
	SOUTH ♠ A K 2 ♥ A 7 6 ♦ A J 9 8 7 ♣ A 7

West	North	East	South
Pass	Pass	Pass	2 NT
Pass	3 NT	(All pass)

My decision to raise to three notrump didn't effect the final contract, but it may give me an advantage in the play, since the opponents don't know declarer doesn't have a four-card major. According to the opponents' card, the club jack says nothing about the king, so I play the queen. East covers.

At IMPs, I would duck in the vain hope that clubs are eight-one or that they are seven-two and I can keep West off play. But at matchpoints, the strategic advantage of retaining a club exit is too great to surrender lightly. It's unlikely West began with seven or eight clubs, and it would be annoying to develop some kind of end position and then notice I didn't have a small club to throw the opponents in with. So I win the first trick.

I can't afford to lose the lead, so I need to the king of diamonds to be onside to come to eight tricks. My ninth will have to come either from the long spade or from picking up the diamonds.

What is my best play to pick up the diamonds? If the suit splits three-two, there are three possible ways to run the suit:

(A) Lead low to the jack, picking up king doubleton onside.
(B) Lead the queen. If it's covered, win with the ace and cash the jack, picking up king empty third onside.
(C) Lead the queen. If it's covered, win with the ace, return to dummy, and finesse again, picking up king-ten third onside.

Note that, provided the suit splits three-two, all three plays are equally likely to succeed . There are three small diamonds, so each play wins in exactly three layouts. (C) frequently offers the advantage of allowing you to pick up king-ten fourth onside. But that's not the case here, since I don't have enough dummy entries.  (A) might enable me to pick up a singleton king onside. If I lead low from dummy and the king appears, I can win and float the nine. Normally I wouldn't even consider that play against a good opponent, since the falsecard with king-ten doubleton is too well known. But, in this case, it's worth considering, since East might be afraid to falsecard for fear I have ace-empty sixth of diamonds. In any event, I decide to postpone my decision until I get some kind of count in the side suits.

I lead the spade king (more likely to elicit an honest count card than the ace). West plays the seven; East the nine. On the spade ace, West plays the three; East, the four. I suspect both opponents are giving honest count. As we've discussed previously, the proper way to card is for the hand with the jack to give false count while the other hand gives honest count. This combination makes it impossible for declarer to read the position. I continue with a third spade. West pitches the three of clubs.

The three of clubs? Surely he would keep enough clubs to beat me if he gained the lead, so he must have started with at least six. That means he started with at least eight black cards while East started with at most seven, so I should play East for diamond length. I lead the diamond queen--king--ace--six.

I can't handle a four-one diamond break. So, a priori, it's a complete toss-up whether to return to dummy for a finesse against the ten or whether to cash the jack. But West might have pitched a diamond from two small in preference to a club. And, other things being equal, I'd just as soon retain the heart entry to dummy. So I cash the diamond jack. West drops the ten. So far so good.

I cash the third diamond, and West discards the four of hearts. On the next diamond, he discards the jack of hearts. So he started with jack doubleton? If so, then East began with a 4-4-3-2 pattern. I pitch a club from dummy, and East pitches the three of hearts. On the last diamond, I pitch a heart from dummy; both opponents pitch clubs. There are only three hearts outstanding. Either my hearts are good or East has queen third of hearts and the high spade. I play a heart to dummy's king. West shows out, so I exit with the spade ten, forcing East to lead away from his queen of hearts. Making six.

	NORTH ♠ Q 10 6 5 ♥ K 10 5 2 ♦ Q 4 2 ♣ Q 4
WEST ♠ 7 3 ♥ J 4 ♦ 10 6 ♣ J 10 9 8 6 5 3		EAST ♠ J 9 8 4 ♥ Q 9 8 3 ♦ K 5 3 ♣ K 2
	SOUTH ♠ A K 2 ♥ A 7 6 ♦ A J 9 8 7 ♣ A 7

Was it necessary to win the first trick to preserve the throw-in? I suppose not. If I win the second trick and play the same way, I can finish with a simple squeeze in the majors against East. Since it didn't gain, perhaps it was wrong to win the first trick. Switch the ten and five of diamonds, for example, and I would be going down on my line, whereas I would make four had I ducked.

Of course, there are layouts where I would be happy I didn't duck. For example, give West:

♠ 7 3 ♥ Q J 4 ♦ 10 6 ♣ J 10 9 8 6 5

Now, assuming I play spades and diamonds as I did, retaining the club exit allows me to make six in the endgame. Actually, my "decision" to win the first trick was more of an instinctive reaction than a decision. As a rule, I tend not to cater to seven-two breaks if doing so requires me to give up other chances. But I'm not entirely sure I was right this time, since the chances I'm giving up are rather obscure. At a different vulnerability, my decision would be more defensible, since it is more likely we would have heard from West in the auction if he began with seven clubs.

This turns out to be a top. Two pairs made five notrump, two made four notrump, one made four spades, and one played three notrump, down two.

Score on Board 6: +490 (12 MP)
Total: 51 MP (70.8%)
Current rank: 1st

A Digression

This is a special post to discuss an issue that came up in the previous deal. Specifically, what is West's proper strategy in the following layout:

	NORTH ♠ x x
WEST ♠ Q J 10 9		EAST ♠ x x
	SOUTH ♠ A K 8 x x

That is, assuming West wants to keep declarer in the dark about how spades split, what sequence of plays should he choose in following to the first two tricks? My instinct told me that West should avoid sequences that he would be unlikely to choose from a doubleton (i.e., queen or jack followed by a non-touching card) and should choose his sequence at random from the remaining possibilities. It turns out my instinct was wrong. It is quite possible that this question doesn't interest you in the slightest. If you wish to stop reading right here, I won't be offended. In fact, I'll be none the wiser. If the question does interest you, I suspect you'll find the answer somewhat surprising.

To get a handle on how to approach this problem, let's start with a more familiar position:

	NORTH ♠ x x x x
WEST ♠ Q J		EAST ♠ x x
	SOUTH ♠ A K 10 x x

Here, as we all know, West should play the queen half the time and the jack half the time. But why, precisely, is that the right strategy? Most people, I suspect, would answer that West must randomize so that declarer cannot read him. If West tends to favor one card or the other and if declarer figures that out, declarer can gain an advantage. Actually that isn't the reason. In fact, it's not even true. Unless you play the same card almost all the time (more than 93.75% of the time to be precise), declarer can't gain an advantage.

Let's say, for example, that you play the queen from queen-jack doubleton 80% of the time and declarer knows that. How should declarer play when you drop the queen? If he finesses, he wins whenever you were dealt a stiff queen. If he plays for the drop, he wins 80% of the time that you were dealt queen-jack doubleton. Since those holdings are almost equally likely, the odds are roughly 1 to .8, or about 5 to 4, in favor of finessing. Your deviating from the correct strategy does not affect declarer's proper play.

Why, then, should you play the queen only half the time? If declarer plays correctly, it doesn't matter what you do. But what if declarer is a fool? What if declarer, perversely, decides to finesse if you play jack but to play for the drop if you play the queen? You still have an edge against such a declarer, but you don't have the edge you are entitled to. This declarer will pick up a singleton jack in your hand, and he will pick up 80% of your queen-jack doubletons - a total of 1.8 out of 3 cases (the three cases being singleton jack, singleton queen, and queen-jack doubleton). He's entitled to pick up 2 out of the 3 cases by finessing, so you do have an edge. But you would have an even greater edge if you played the queen only half the time. In that case, you would hold declarer to 1.5 wins in 3 cases. While you can hold declarer to even fewer wins than that by playing the queen even less often, such a strategy would not fully punish the opposite kind of fool: the declarer who finesses if you play the queen but plays for the drop if you play the jack. By playing the queen half the time, you hold both fools to 1.5 wins out of 3. The reason you must randomize, then, is to avoid having to guess what kind of fool declarer might be. It is to guarantee you the same edge against a declarer who misplays regardless of how he misplays.

How do we determine the optimum frequency with which to play each card? You must play each card with the same frequency with which you would play it if you actually did have a singleton. Since you hold a singleton queen and a singleton jack with equal frequency, you must choose the queen or jack from queen-jack doubleton with equal frequency. It doesn't always work out that neatly.

Let's look at the following slightly more complicated position:

	NORTH ♠ x x
WEST ♠ Q J 10		EAST ♠ 9 x x
	SOUTH ♠ A K x x x

Spades are trumps. You tap declarer. He cashes the ace and king of spades. He must now decide whether to play a third round, losing lots of tricks if trumps are four-two, or to play on side suits and allow you to ruff, losing a trick unnecessarily if trumps are three-three. As in the previous example, your correct stategy is to play your cards with the same frequency with which you would play them if you had a doubleton. There are three possible ways for you to have two honors doubleton: queen-jack, jack-ten, or queen-ten. In the first two cases, you would play up-the-line half the time and down-the-line half the time. In the third case, you would, in all likelihood, always play up the line. (To make this easier, let's assume declarer can't afford to cater to a five-one break, so there can be no gain in falsecarding with the queen from queen-ten doubleton.) If you have a doubleton, then, you will play as follows with the indicated frequencies:

Q J	1/6
Q 10	0
J Q	1/6
J 10	1/6
10 Q	1/3
10 J	1/6

So you should choose your play from queen-jack-ten with the same frequencies. I think most players intuitively know not to play queen-ten, since that effectively marks you with the jack. I suspect, however, few players are aware of the corrollary: that you should play ten-queen twice as likely as any other sequence.

Obviously, jack-ten-nine works similarly. Your plays should have the following frequencies:

J 10	1/6
J 9	0
10 J	1/6
10 9	1/6
9 J	1/3
9 10	1/6

What about queen-ten-nine? You aren't going to drop the queen at trick one or trick two, so you will play as follows:

10 9	1/2
9 10	1/2

And with queen-jack-nine? As before, let's assume there is no gain in representing a singleton. So playing an honor on the first round is pointess. In fact, it's worse than pointless, because, when you follow with the nine on the second round, you would render it unlikely that you have a doubleton. Under this assumption you would play as follows:

9 Q	1/2
9 J	1/2

Now we are prepared to solve the original problem: How should you card with queen-jack-ten-nine to offer declarer no inference as to whether we began with two, three, or four trumps? As before, the answer is: you must mimic the frequency with which you would choose each sequence if you actually began with a doubleton or trebleton.

To choose one example, how often should you play queen-jack?  You would play queen-jack from queen-jack doubleton half the time, and you would play queen-jack from from queen-jack-ten one sixth of the time. There are six relevant doubletons and four relevant trebletons - a total of ten cases. So the frequency with which you would play queen-jack from the relevant two- and three-card holdings is 1/2 times 1/10 plus 1/6 times 1/10, or 1/15 of the time. 1/15, then, is the frequency with which you should choose queen-jack from queen-jack-ten-nine. If you work it all out, you wind up with the following frequencies:

Q J	.07
Q 10	0
Q 9	0
J Q	.07
J 10	.08
J 9	0
10 Q	.13
10 J	.08
10 9	.12
9 Q	.15
9 J	.18
9 10	.12

To generalize, you should usually play either nine-jack or nine-queen. Somewhat less frequently, you should play ten-queen or the ten and nine (in either order). Less frequently still, you should play touching honors in either order.

This, of course, is of theoretical interest only. No one is ever going to bother with this strategy. But I find it interesting that my intuition was so wrong.