Match 2 - Board 43

Board 43
Neither vulnerable

♠ A K 9 4 ♥ A K 3 ♦ 8 4 ♣ A 10 6 4

I open one spade in first seat; partner bids four notrump. Back on Board 108 of the previous match, I took Blackwood off our convention card when I discovered I was unable to raise opener's two notrump rebid to four without hearing how many aces he had. So, whatever this bid means, it's not Blackwood. I suppose it shows about 17 or 18 high-card points, probably with a 3-4-3-3 pattern, since he would surely make a two-over-one response if he had a biddable suit. In any event, it seems clear to raise to six notrump. Everyone passes, and West leads the three of diamonds.

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

West	North	East	South
	1 ♠	Pass	4 NT
Pass	6 NT	(All pass)

Four notrump? This hand arguably isn't even worth three notrump. Is this partner's way of insisting I put Blackwood back on the card? I don't have a clue what partner was thinking about.

RHO plays the diamond king. How are the diamonds divided? Jack tends to lead high from worthless holdings against notrump. I can't imagine he's leading from a jack on this auction, especially when he has a perfectly safe heart lead. So he must be leading from the ten. Even leading from a ten isn't that attractive, so I suspect he's short in hearts. He might prefer a diamond lead from ten third or ten fourth to a heart lead from two small. (I don't agree with this preference, by the way. I know some people shy away from small doubletons against six notrump for fear they will pick off partner's queen or his jack fourth. But I think that fear is misguided. If partner has length in a suit, declarer may well get a count and guess to finesse against partner anyway. Picking off partner's doubleton queen or jack, on the other hand, could be a disaster. That's a finesse declarer was apt to get wrong if left to his own devices.)

I have ten cashing tricks. If I can manage four spades trick, I have eleven. To come to twelve, I'm going to need a second club trick. Ostensibly, I need to hope West has both club honors. But, in practice, the double finesse may work even when the club honors are split. If West has the queen of clubs without the nine, he may not cover the jack for fear I'm fishing with king-jack-nine.

I take the diamond ace and play the jack of clubs. West plays the queen, I duck, and East plays the three. West now plays a card I'm surprised to see: the jack of diamonds. Surely he has the diamond ten as well. He really led low from jack-ten fourth against this auction? I'm glad he told me. I never would have figured it out. Now that I know he guards diamonds, I rate to be able to make this provided the club king is onside. If spades are three-three, I have twelve cashing tricks. If West has four spades, I can squeeze him in spades and diamonds. And if East has four spades, West probably has club length. So I can squeeze West in clubs and diamonds.

East plays the five of diamonds. I win with the queen and play the six of spades--deuce--ace--three I cash the heart ace--four--deuce--six, then the heart king--five--ten--seven. I'm virtually certain from West's weird opening lead that he doesn't have another heart. That means the opponents aren't giving honest count, and I don't blame them. I play the four of spades--eight--queen--seven. I'm down to this position

	NORTH ♠ K 9 ♥ 3 ♦ -- ♣ A 10 6
	SOUTH ♠ 5 ♥ Q J ♦ 9 2 ♣ 7

The moment of truth has arrived. If spades are three-three, I can cash my tricks in any order. If not, I must retain the entry in the suit West is squeezed in. If he has

♠ J 10 7 2 ♥ 7 6 ♦ J 10 x x ♣ K Q x,

I must play a club to the ten, cash the club ace, then run hearts to squeeze him. If he has

♠ 7 2 ♥ 7 6 ♦ J 10 x x ♣ K Q x x x,

I must play a spade to the king and run hearts. Is there any indication which hand West is more likely to hold?

While I find a diamond lead strange with either hand, I can almost think of a reason to produce it with the first hand. With stoppers in three suits, West is going to be in serious trouble on the run of the hearts. Perhaps it makes sense to attack declarer's communications in an attempt to break up a squeeze. Of course, if I were to lead a diamond for that reason, I would lead the jack, not a low one. The last thing I would want to do is to force partner to part with a diamond honor.

I decide to go for the spade-diamond squeeze. I play a club--deuce--ten--king. The opponents cash their two diamond tricks, and I'm down three.

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

At least I was right about West's pattern.

It occurs to me that, if West did have the hand I was playing him for, it would be a nice play to drop the jack or ten of spades under my queen. In theory, I wouldn't even need the squeeze now. I could finesse the spade and take four spade tricks. But I would never do that, since I would look pretty foolish if spades were three-three all along. So, in practice, the play would simply convince me that West didn't have four spades, and I would play for the wrong squeeze.

We are, of course, the only pair to go minus. Three pairs made four notrump; three pairs made five. Anyway, partner gets his wish. If that's what he thinks a natural four notrump bid looks like, I'm putting Blackwood back on the card.

Score on Board 43: -150 (0 MP)
Total: 334 MP (64.7%)

Current Rank: 1st

Post Script

In playing around with this deal afterwards, I discovered I could have made it. If, at trick two, I lead a heart to dummy and play a low club toward my jack doubleton, East hops with the king. I'm now cold on a spade-diamond squeeze against West if I work it out.

I had fleetingly thought about playing clubs that way. If both club honors on onside, it's just as good as leading the jack, and it gives East a chance to make a mistake when the honors are split. But I've never seen anyone make that mistake outside of the novice game. So leading the jack appeared to have better swindle potential.

Why did East hop? The only reason I can think of is he actually thought his partner might hold the diamond queen. It would be right to hop if I held

♠ Q J x ♥ Q J x x ♦ A J 10 x ♣ Q x.

I'm annoyed with myself. I should have found this line. I had my head in the sand, playing as if I were playing against a human. While a human might sometimes choose not to cover the jack of clubs, Jack would always cover. He assumes declarer is double-dummy, so he would not worry about taking away a guess. He would cover because it's right any time his partner has the king. Furthermore, while hopping with king fourth of clubs is a mistake a human would be unlikely to make, it is a mistake Jack could make quite easily. Jack doesn't draw inferences from his partner's plays, so he has no compunction about playing his partner for the diamond queen. I played correctly against a human but incorrectly against Jack.

Playing bridge is all about seeing the deal through your opponents' eyes. Doing this can be a little harder than normal when your opponent is a computer. But, after playing 171 boards against Jack, I understand how he "thinks" sufficiently that I should be more adept at it than I was on this deal. It would have been quite a coup to bring this contract home. The next time I'm faced with a problem like this, I'm getting it right.

Match 2 - Board 42

Board 42
Both sides vulnerable

♠ K Q 10 5 2 ♥ A 3 ♦ A 9 8 ♣ J 7 4

Most weak notrumpers would open one spade with this hand. But I prefer opening one notrump with all balanced minimums, even those with a five-card major. When I learned this style, I was told that you gave up something when you opened one notrump. But you got it back when you opened one of a major and partner knew you were either unbalanced or had extra high cards. My experience, however, has been quite the opposite. The negative inference when partner opens one of a major seldom seems to make any difference. But the one notrump openings themselves tend to work out surprisingly well. More often than you would expect, partner raises to three and they lead your suit. So here goes. One notrump.

Partner bids two clubs, I bid two spades, and partner bids three notrump. West leads the seven of diamonds.

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

West	North	East	South
		Pass	1 NT
Pass	2 ♣	Pass	2 ♠
Pass	3 NT	(All pass)

See what I mean? The opening bid worked out just fine. If they had led a club, I wouldn't be very happy. But they didn't. It looks as if four spades rates to make five. So if I can make five notrump, I ought to have quite a good result.

It appears West has led highest from a weak suit, so East should have the queen and ten of diamonds. I'll capture his ten with my ace, then drive the spade ace. If the defense still doesn't attack clubs, I can lead a diamond to the king, trying to drop queen-ten doubleton to make six. If the queen doesn't drop, I can drive it to make five.

What if the defense switches to a club after winning the spade ace? Since I expect plus 660 to be an excellent result, I'm less concerned about making six than I am about maximizing my chance to make five. So, if the club shift comes from West, I intend to duck. I'm then cold for five provided I guess who has the remaining club honor. If East has it, he's squeezed in the minors. If West has it, I can cash the diamond king and the club ace, then execute a double squeeze with hearts as the double threat.

What if the club shift comes from East? Now I can't isolate the club threat. My best play in that event is to win the club shift and cash the diamond king (retaining my chance of making six). If the queen doesn't drop, I run spades, coming down to this position:

	NORTH ♠ -- ♥ K Q 10 7 ♦ J ♣ --
	SOUTH ♠ -- ♥ A 3 ♦ 9 ♣ J 7

I now need to decide which defender has kept four hearts. If it's East, I cash the heart ace and toss him in with a diamond. If it's West, I cross my fingers and take a heart finesse. (Unless West started with the king-queen of clubs. If so, he has had to pitch one of his honors. If I work out that's what's happened, I can just set up my club jack and avoid the heart finesse.)

I suppose I didn't really need to carry the analysis quite so far. But I did want to have some idea of what the end position might look like before I played to trick one. It would be frustrating to realize all of a sudden that I needed a diamond entry to my hand and should have won the first trick with the king of diamonds.

I play low; East plays the queen. So all my plans have gone up in smoke. Did West lead second highest from ten fourth? Or did East falsecard? For a human expert, falsecarding with queen-ten doubleton would not be all that difficult, provided he thinks the seven is high (rather than low from nine-eight-seven). He can see I'm going to drop his queen. So why not play the queen now and give me a losing option? As usual, just play the card you're know to hold.

I know good and well Jack is incapable of this falsecard, however, which is too bad. This would be a more interesting problem if I had confidence in him. I play the five of spades--six--jack--ace. East returns the three of spades. I win with the king, and West plays the eight. I know I can just float the nine of diamonds. But I might as well pretend I'm worried about the falsecard just for practice. I cash another spade. East pitches the deuce of clubs. I'm guessing he's 2-4-2-5, which leaves West with 3-3-4-3.

On the next spade, West pitches the four of diamonds. What's this? Why would he ever pitch a diamond from four? If he did, dummy's diamonds are now good even without a finesse. The only time I need to finesse is if he began with ten-seven fifth. Is that possible?

I pitch a club from dummy. I expect to see another club pitch from East, since he knows I don't have four clubs. But he pitches the deuce of hearts. This suggests five hearts, which would make East 2-5-1-5 and West 3-2-5-3. It appears West did lead the seven of diamonds from ten-seven fifth. On the last spade, both opponents pitch clubs as I pitch a heart from dummy. I have enough information to float the nine of diamonds now even if I thought East was capable of the falsecard. I lead the nine of diamonds. West makes it easy for me by covering. Making six.

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

I suppose the seven of diamonds was a sensible lead. I would rarely lead high from a five-card suit. But with such a bad hand as well as a useful holding in the suit partner is most apt to want to shift to, I might make an exception. Add another high card to the hand, and I wouldn't be so pessimistic about establishing my own suit.

Everyone else, of course, was in four spades. One pair made five; everyone else made six. Plus 690 is a top. It's about time we got another one of those.

Score on Board 42: +690 (12 MP)
Total: 334 MP (66.3%)

Current rank: 1st

Match 2 - Board 41

Board 41
Opponents vulnerable

♠ 7 6 5 ♥ K 7 5 ♦ Q 5 3 ♣ Q 8 7 4

Partner opens one notrump (12-14), and RHO doubles. Our convention card says, unhelpfully, "Escape after double." I've figured out from previous deals that this means my pass is forcing, requiring partner either to run to a five-card suit or to redouble, after which we wing it. So I pass.

LHO bids two spades, passed around to me. With three small spades, I can't safely act. It would be nice to know that partner would have doubled for take-out with an appropriate hand. Then I could pass with some confidence. But Jack plays opener's double of two spades as a penalty double. It's hard to see how partner can even have a penalty double if I can't redouble one notrump. But these are the methods I'm stuck with. Instead of passing with confidence, I have to pass a bit nervously. Partner leads the four of diamonds (third and lowest).

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

West	North	East	South
1 NT	Double	Pass1	2 ♠
(All pass)
1Forcing

Declarer plays the jack. I win with the queen, and declarer plays the seven. Taking this finesse at trick one is a strange play. With three diamonds, it would be normal to go up with the ace or king. He might be able to pitch his third diamond on a club. And, if not, he can always take the finesse later. So my first inclination is to assume that declarer has a doubleton diamond and wants to take a quick pitch. But declarer's seven can't be from a doubleton. It would have to be from seven-deuce, giving partner ten-nine-eight-six-four. And from that holding, partner would have led the ten.

If declarer wanted to conceal his count from me, then playing the seven was an error. Let's continue our discussion of a few weeks ago and examine how declarer should know that. We start with a hypothetical layout. (In the actual deal, East, not South, has the three.)

	NORTH ♦ A K J
	SOUTH ♦ 10 7 3

West leads the four of diamonds, third best from four. Dummy plays the jack, and East wins with the queen. Suppose declarer wants East to worry that West has led from a five-card suit. (Perhaps he wants to entice him into embarking on an ill-advised cashout.) He could slap his forehead, feigning disgust. Or he could just follow Rule #1 of Rules for Scrambling the Opponents' Count Signals and play the three to "signal" an odd number in West's hand. The three is the only card that works. From East's point of view, if the four is West's lowest card, then declarer must have three-deuce doubleton. So if declarer doesn't play the three, East will know his partner doesn't have five.

But in the actual deal, declarer doesn't have the three; East has it. So declarer is looking at something like this:

	NORTH ♦ A K J
	SOUTH ♦ 10 7 6

Declarer no longer has a card lower than West's, so he can't follow Rule #1. He must fall back on a more basic rule: "Play the card you're known to hold," or, more accurately, "play the card you might be known to hold." If West has the nine, then declarer is marked with the ten, since West would have led the top of a sequence. So declarer must play the ten. It is possible he has ten-deuce doubleton, but, if West has the nine, declarer cannot hold seven-deuce or six-deuce. So the play of the seven or six risks marking declarer with at least three cards. (If East has the nine, declarer's play will not matter. Ten-deuce, seven-deuce, and six-deuce are all possible holdings.)

Why, you might wonder, doesn't the "card you're known to hold" rule apply in the first example? Actually, it does. In the first layout, if West has the nine, there is nothing declarer can do to conceal his length.  If the four is West's lowest card, then declarer must have the three and the deuce. If three-deuce doubleton would give West a sequence, then declarer simply can't have a doubleton. So declarer's only chance for concealment is to follow Rule #1 and hope that East has the nine.

Back to the problem at hand. I know from declarer's careless seven that he can't have a doubleton, so he has no quick pitches. What else can I conclude about his pattern? Since my pass was forcing, there is no particular reason South should have been in a hurry to run, especially to the highest-ranking suit. That means he is unlikely to have only four spades.

What about declarer's high cards? He has from four to six high-card points. Since I'm not playing with Lowenthal, I assume from partner's failure to lead a club that declarer has the ace or king of clubs. (You can't imagine how comforting it is to be able to draw inferences like that again.) That gives him at most a king in addition. So partner must have the heart ace and probably has the ace and queen. That makes a heart shift by me perfectly safe. If partner could have queen-nine third or fourth of hearts, a heart shift would not be safe.

While I know there is no hurry to cash hearts, it may make the defense easier for partner if I do so. If I can find something else more important to do, however, I shouldn't hesitate. If I don't shift to a heart, partner should suspect I have a heart honor. I would strain to shift to a heart from a worthless holding (possibly getting partner off an endplay). So failure to shift to a heart should be a red flag (both to partner and to declarer) that I have a heart honor.

Is there anything better to do? Could I give partner a club ruff for example? If partner has a doubleton club honor and a trump entry, then it might be right to shift to a club. Suppose partner has

♠ K Q x ♥ A x x x ♦ x x x x ♣ K x

Then a club shift beats two spades. Should I play for this? Not only is this a very specific hand, but it also runs counter to my assumption that declarer has five spades. If I had complete confidence in partner, I might shift to a club anyway rather than trust declarer's bidding. But I'm afraid drawing the inference that I have a heart honor is beyond Jack's capabilities. If I shift to club and partner doesn't find an immediate heart shift from a hand such as

♠ K x ♥ A Q x x ♦ x x x x ♣ K x x

then declarer will be able to set up a club for a heart pitch. The layout where a club shift is right is sufficiently obscure that I don't want to risk it.

Accordingly, I shift to the five of hearts. Declarer plays the six, and partner wins with the queen. Partner then shifts to the ace of clubs. What's this? Maybe partner does have a doubleton club after all. He's hoping I have the king and can give him a ruff. I discourage with the four, and declarer plays the five. Partner cashes the heart ace--ten--seven--deuce, then plays the three of clubs.

Partner is determined to get that club ruff, isn't he? I play low, expecting dummy to win the trick, but declarer wins with the king. What's going on? Why would partner play ace and a club with ace third? I certainly hope declarer is 5-2-4-2. If he has a third heart, he can now establish a club trick in dummy and pitch it.

Declarer cashes the king of spades--nine--eight--six. He then plays the six of diamonds to dummy, as partner plays the deuce. Oops. Partner has four diamonds, so declarer is 5-3-3-2, and we've dropped a trick. Declarer plays the ten of clubs. There is nothing to be gained by covering. I play low, and declarer ruffs with the deuce of trumps. Eventually we score a heart trick, holding declarer to his contract.

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

How could ruffing the club ever be right? Obviously declarer thought West could have ace-queen-third of clubs. Since Jack has no way of drawing inferences from his opponents' plays, he has no basis for ruling out ace-queen-third as a possible holding. It's always a little surprising when you see this flaw in Jack's design manifest itself. But the real surprise should be how well Jack plays despite this handicap.

How about partner's club continuation? Can that ever be right? Yes, it can--assuming he attaches no significance to my discouraging four of clubs. If I held this hand:

♠ K x x ♥ x x x ♦ Q x x ♣ K x x x ,

then three rounds of clubs beats declarer two tricks. On any other defense, declarer could play a spade to the jack and pick up spades for one loser. But if he does that after three rounds of clubs, I can play a fourth round of clubs for partner to ruff with the trump queen.

If declarer had made an overtrick, it would have been my fault. I know I want partner to cash hearts, so I should have played the king of hearts at trick two instead of a low one. Why rely on signals when you can force partner to do the right thing?

Only one other pair held spades to eight tricks. Three pairs were minus 140, and two pairs were allowed to play one notrump, going down either one or two. So we receive seven matchpoints for this result.

Declarer could have made nine tricks without any help from us. He had to refuse the diamond finesse at trick one and start on clubs. If partner wins the club ace and continues diamonds, he must refuse the finesse again. It's hardly clear to do this. Playing that way is essentially guessing that West holds the spade queen rather than the diamond queen. And why should he?

I suspect the declarers who made nine tricks did so not by superior guessing but by playing from the North side (after a one notrump overcall and a transfer). So the weak notrump strikes again!

Score on Board 41: -110 (7 MP)
Total: 322 (65.4%)

Current rank: 1st

Match 2 - Board 40

Board 40
Neither vulnerable

♠ 10 ♥ K J 10 9 3 ♦ Q J 10 7 5 3 ♣ 8

RHO opens two clubs, strong and artificial, in third seat. I have two choices:

(A) I can bid slowly, trying to find the right spot for our side. This means starting with two diamonds (hoping to keep the auction low), then showing hearts later (possibly via an unusual notrump sequence). We don't play methods that allow me to show both suits simultaneously. Even if we did, I'm not sure I would use them. Diamonds may well play several tricks better than hearts when partner has equal length in the red suits.

(B) I can forget about investigating the right spot for our side and go for maximum obstruction. Four diamonds seems like the right call if that's the route I choose. If I bid four diamonds, I have to be content to bury the heart suit. It makes no sense to make the opponents guess, then take them off the hook by making the last guess myself.

I'm pretty much on the fence with this one. (B) could certainly work out well. Having an undisclosed major sometimes makes it hard for the opponents to find a sensible call, since it may deprive them of a take-out double. But we could also wind up under-competing if we have a big heart fit.

I decide to go with (A). I'm not sure how necessary it is to preempt over a two club opening. Two club auctions are difficult enough even when the opponents stay out of your way. So why preempt? If I bid two diamonds and partner can't raise, the auction rates to stay low (Two diamonds--pass--pass--two spades seems like a serious possibility.), so I will be able to introduce hearts easily on the next round.

I bid two diamonds. LHO bids two spades, partner passes, and RHO raises to four spades. I guess I was wrong about the auction's staying low. I've painted myself into a corner now. There wasn't much point in bidding only two diamonds unless I was intending to introduce hearts on the next round. I bid four notrump. LHO doubles. Two passes to me. I bid five diamonds. LHO doubles again--pass--pass back to me.

I wonder if Jack understands this auction. Probably not. It's quite possible we have a heart fit and partner doesn't realize it. I should have thought of that earlier. Now that I'm in this mess, I have to make the best of it and figure out what my percentage action is.

(A historical note: Cliff Bishop once insisted to me that an immediate five diamonds over four spades on an auction like this promised hearts. The way to show a one-suiter, he contended, was to bid notrump (ostensibly natural), then run when doubled. Either this was standard expert practice before the invention of unusual notrump or Cliff was making it all up to shift the blame for our disaster. I never figured out which.)

It seems unlikely partner has three-card diamond support. He could have raised diamonds over two spades, and he could have run to five diamonds when four notrump was doubled. On the other hand, if he doesn't understand this auction, he could easily have four or five hearts. I hate to make a unilateral decision. But I find it hard to believe that five diamonds doubled is a good spot for us. There probably isn't much downside to pulling to five hearts. And the potential gain is huge. I can't believe I'm bidding this way in print, but here goes. Five hearts. LHO doubles again, ending our inelegant auction. West leads the six of spades.

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

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

Well, I was right about one thing. I don't think my five heart bid cost us many matchpoints. Minus 800 is the best I can hope for in either contract, and I'm unlikely to do even that well. I play the four from dummy, and East wins with the king. The opponents play fourth best leads, so I'm guessing West has queen fifth of spades and East has ace-king fourth. East shifts to the seven of hearts. I play the ten. West wins with the queen and returns the deuce of hearts to his partner's ace. This is a pretty good start for me. They should be tapping me. Instead, they're wasting all their entries. East plays the spade ace. I pitch my club, and West follows with the deuce.

East plays the diamond ace--five--deuce--six, then shifts to the nine of spades. If trumps are three-three (as they rate to be given the opponents saw no urgency in tapping me), I can hold my losses to one more trick. I ruff and cash the heart king. Trumps split. I drive the king of diamonds and claim. Down four. A triumph! Minus a mere 800.

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

I don't understand jumping to four spades with only three trumps. I must have telegraphed the fact I was intending to bid again. Pretty crafty, that Jack.

Obviously the opponents can do at least one trick better. East could have given his partner a diamond ruff. Or he could have simply not cashed the diamond ace. If he hadn't given me that tempo, I would have to let him score the club ace in order to keep control.

Is it possible for the opponents to do better yet? Since East has only three spades and West has no side-suit entry, the way to do better is to use clubs as a tap suit.

West knows enough about the hand to see that. He knows his partner has long clubs. And he knows that his partner will have to lead clubs from his side of the table if declarer gets tapped out. Surely it makes sense to use his only entry to put a club through.

Say, after winning the heart queen, that West shifts to the queen of clubs. I cover, and East wins with the ace. East continues with a club to the ten and jack as I pitch a diamond. If I play a heart off dummy, East hops with the ace and plays a low club while his partner still has a trump. I'm tapped out, and East's hand is high except for the four of diamonds. I score my trump tricks and eventually capture East's four of diamonds for down six. Minus 1400.

Should the defense really find this line? Yes, it should. In general, when you are playing a forcing game, you should imagine yourself as declarer and ask yourself how you would play the hand. If you imagine East as declarer and West as dummy, this line is not hard to find. What could be more natural than setting up "declarer's" hand? But, because we are conditioned to think along different lines as defenders, it's hard even to think of this line. Leading dummy's long suit (except perhaps as a communication-killing maneuver) is just not a tactic we give much thought to most of the time.

Of course, since it's matchpoints, the opponents had plenty of leeway. They could have dropped another trick and still wound up with a top. Most pairs made an overtrick in four spades. One managed a second overtrick (I presume after a club lead), and one pair played three notrump, making five. Three notrump, eh? That's a very nice result. I feel just terrible taking their well-deserved top away.

Assuming partner would have led a diamond had I passed four spades, my four notrump bid cost us eight matchpoints. I don't mind taking a flier so much if we were destined to score average or worse. But I certainly mind it when we were destined for a good result. Part of the reason I bid was it seemed likely we were poorly placed. I could have preempted and didn't, and the opponents appeared to be in their best spot. How could I possibly imagine I was ahead of the field?

Score on Board 40: -500 (0 MP)
Total: 315 MP (65.6%)

Current rank: 1st

Match 2 - Board 39

Board 39
Both sides vulnerable

♠ K 8 6 5 ♥ 8 7 4 3 2 ♦ K ♣ A 6 2

I pass, LHO passes, and partner opens one notrump (12-14) in third seat. I bid two clubs, and partner bids two hearts. I was intending to pass two spades. But, with a ninth trump, my hand is worth a raise. I bid three hearts, and partner goes on to four. RHO leads the queen of clubs.

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

West	North	East	South
	Pass	Pass	1 NT
Pass	2 ♣	Pass	2 ♥
Pass	3 ♥	Pass	4 ♥
(All pass)

Not your classic weak notrump opening. But it worked out OK.

I have four potential losers. I need the heart king to be onside, or I need to be able to establish a second spade trick for a club discard. I may be able to do that if East has ace doubleton or tripleton of spades. I will need some luck for this to work, like the opponents' unaccountably not continuing clubs or perhaps five-two clubs with the diamond ace favorably located. But it doesn't hurt to try. I can always fall back on the heart finesse if necessary.

I win with the club ace. East plays the eight, and I play the four. The eight presumably shows the ten. I play the five of spades--four--queen--ace. So much for developing a second spade trick. I expect West to continue clubs, since East's signal has told him it is safe to do so. But he plays the jack of spades--king--seven--deuce. Jack tends to play up the line after his initial count signal. So it looks as if East has nine-seven-four of spades and West has ace--jack--ten--three.

I could take a heart finesse now. But I don't see any reason not to exit with the king of diamonds first. If I put the opponents on play, maybe they will do something stupid. Perhaps West will win with the ace and try to cash the ten of spades, crashing his partner's nine. Even if they don't do anything stupid, it's always a good idea to gather information if you can.

I play the king of diamonds--ace--five--ten. East plays the club ten--king--three--deuce. I ruff the three of diamonds to dummy, intending to take a trump finesse. To my surprise, West contributes the queen; East plays the deuce. Can West really have queen-ten doubleton of diamonds? It's possible, but I doubt it. That would mean East passed twice with ace-jack seventh. Even if he chose not to pre-empt in second seat, he could have overcalled with two diamonds on the next round. So I suspect what is going on is West has queen-jack-ten of diamonds. If so, he surely has only three. With longer diamonds, he would have preferred a diamond lead to a club lead from queen-jack-nine.

The fact that West has at most three diamonds means it is likely he has five clubs, which opens up an interesting possibility. If East is out of clubs, there is no need for me to take a heart finesse (unless hearts are three-zero). I can strip the hand and cash the heart ace. If the king drops, great. If it doesn't, I play another heart, endplaying East. Can I determine for sure whether East is out of clubs?

I know West has two or three diamonds. If he has two hearts, either I'm down or East has a singleton king and I have nothing to worry about. So I might as well assume he has a singleton or void in hearts. I also know he has no six-card suit, since East has already followed twice to each black suit. Thus there are only four relevant patterns West can hold: 4-1-3-5, 5-1-3-4,  5-0-3-5, or 5-1-2-5.

The latter two patterns, while possible, are unlikely, since West probably would have bid over one notrump with five-five in the black suits and an opening bid. 5-1-2-5 is particularly unlikely, as I said earlier, since it requires East to have passed twice with seven diamonds. So I'm going to focus on handling the first two patterns. If I can handle the latter two as well, fine. If not, I'm not going to worry about it. I should start by ruffing a spade to my hand to get a count in that suit.

(A) If East follows (as I expect him to, given his count signal), then I know West is 4-1-3-5 (or he has two hearts, in which case my play doesn't matter). I cash the heart ace. If the king doesn't drop, I ruff my last diamond and play a heart, hopefully endplaying East.

(B) If East shows out, he probably has another club. So there is no endplay, and I need the heart king to be onside. I ruff a diamond to dummy and play a heart to the queen. If West follows low, I'm home. If he shows out, I don't have an entry to repeat the finesse. But I don't need one. If East has three hearts, he is out of clubs. I cash the heart ace and play a third heart to endplay him.

What happens if West does have a doubleton diamond?  When I try to ruff the third round of diamonds to dummy, he may ruff in front of dummy with the nine or king. If so, I can pitch dummy's club loser. But if East is out of clubs, West can lead a club for an overruff. That means if West is 5-1-2-5 with a singleton king or nine or hearts, I'm going down.

I don't mind going down if he has a singleton king. My alternative line of taking an immediate trump finesse would not have worked either. But if he ruffs in with a singleton nine, I've gone down on a hand where the more straightforward line would have worked. As I said, I'm not going to worry about that. I think West is very unlikely to be 5-1-2-5.

So my discovery play of the king of diamonds paid off. I'm now going to make this on some layouts where West has a singleton king of hearts. Obviously, I didn't envision this when I chose to exit with the king of diamonds. When you embark on a journey of discovery, you never know where it might lead you.

I play a spade from dummy. It turns out, East did give false count in spades. He shows out, but, instead of pitching, he ruffs with the six of hearts. That makes it easy. I overruff and cash the heart ace, dropping West's king. Making four.

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

So East did pass twice with ace-jack seventh of diamonds! And West passed over one notrump with five-five in the black suits and an opening bid! What got into Jack? Not only that, East handed me the contract by ruffing the spade. I was expecting to go down on this particular layout. Giving East an opportunity to misdefend was a bonus for my line that I hadn't even considered. If I ever use this problem in an article, I'll switch the eight and nine of hearts, so I needn't rely on misdefense.

Every table played four hearts. Half the declarers made it; half did not. Strangely, everyone who went down failed by two tricks. Where does the extra loser come from? Somehow East must manage to ruff a winner. It took me awhile to figure it out, but I think I have it. Club queen to the ace. Spade to the queen and ace. At my table, West played the spade jack, perhaps hoping his partner had a singleton spade. What happens if, instead of a spade, West plays another club? I win and play a diamond to the king and ace. East then plays a heart. If I finesse, West wins, cashes his club, allowing East to pitch a spade, then gives East a spade ruff. Down two.

Should I finesse the heart in that position? I don't think I would. I would probably assume from East's failure to play a club that he doesn't have one. In which case, I would rise with the heart ace and, if the king doesn't drop, go for the strip and throw-in. It would take quite an imaginative East indeed, holding another club and the king of hearts, to envision that refusing to cash the club presents declarer with an attractive losing option.

Score on Board 39: +620 (9 MP)
Total: 315 MP (67.3 %)

Current rank: 1st

Match 2 - Board 38

Board 38
Opponents vulnerable

♠ J 6 2 ♥ A 8 7 ♦ Q 8 2 ♣ J 10 7 6

RHO opens one notrump (15-17) and buys it.

After one notrump--three notrump, it would be normal to lead a major, hoping to hit a five-card suit in partner's hand. Against one notrump--pass, however, the situation is different. For one thing, since partner didn't balance, he is more likely to have a long minor than a long major, particularly when playing Astro. For another, when you are looking for seven tricks rather then five, there is less urgency in finding partner's long suit right away. You rate to have more entries defending one notrump than defending three, so you may have an opportunity to correct your course later on, and there is less reason to speculate at trick one.

This is a long-winded way of saying I'm leading fourth from my longest and strongest: the six of clubs.

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

West	North	East	South
			1 NT
(All pass)

Declarer plays the eight from dummy, partner plays the queen, and declarer wins with the ace. Partner has eight to ten high-card points, and I've just seen two of them. So he has six to eight unaccounted for.

Declarer would usually duck the first trick with only the ace. So it is a fair inference that either (A) declarer has the club king as well or (B) there is a shift declarer is afraid of. If (A), as we've seen before, declarer would have done better to win with the king. Then neither partner nor I could draw this inference.

Declarer leads the heart queen. If partner has the king, I'd rather he win this trick so I can retain my entry to the long clubs. If declarer has the king, there is probably no hurry. This doesn't look like the kind of deal where he wants to sneak one trick through, then switch to a different suit. I play the seven--four--deuce. If the deuce is honest count, then declarer has king-queen-jack fourth or king-queen-nine fourth. If the latter, he is not taking his best play in hearts, so there is an inference he has some communication problems. With, say, queen fourth of spades, he might have used dummy's spade entries to lead up to his hearts twice.

Declarer plays the spade three. I play the six (routinely giving false count with the jack)--king--seven. As I said, I don't think partner has two small spades. I suspect partner has either queen-seven, queen-third, or nine-eight-seven. If declarer has the queen, then, when he cashes it and sees my deuce, he will know that either I gave false count with jack third or partner gave false count with jack fourth. If he thinks the latter, then perhaps he has some sort of endplay available against partner, in which case we will have given him a losing option.

Declarer plays the heart ten--jack--king--ace. If declarer had king-queen-nine fourth, he would have led low from dummy in case partner had jack doubleton. So it appears declarer has five hearts.

I also know that partner has at least one high diamond honor. This conclusion comes from the inference I drew earlier: either (A) declarer has the club king (in which case both diamond honors would give him 19 high-card points) or (B) there is some shift declarer is afraid of, which could only be diamonds.

If declarer has the spade queen, we are in a cashout situation. Declarer has at least nine cashing tricks (four spades, four hearts, and the club ace) and might easily have a tenth (the club king or diamond ace). So we should try to cash as many tricks as we can before giving up the lead. Partner can still have as many as seven high-card points that I haven't seen yet. So there is room in partner's hand for the club king or for the ace and king of diamonds. Either minor could be running:

(A) ♠ Q x x ♥ K Q 9 x x ♦ A x (x) ♣ A x (x)

If we cash out our clubs, we hold declarer to nine tricks. If we don't, he makes ten.

(B) ♠ Q x x ♥ K Q 9 x x ♦ J x (x) ♣ A K (x)

If we cash our diamonds, we hold declarer to seven or eight tricks (depending upon whether partner has four or five diamonds). Again, declarer takes ten tricks if I don't find the right shift.

Rather than win this trick and guess which suit to shift to, I could duck again, so that partner can signal when I win the heart ace. Can I afford to do that? How many tricks might declarer take if he abandons hearts? At most, he has two hearts, four spades, the club ace, and one more minor-suit trick, bringing his total to eight tricks. The only time we can hold him to fewer than eight tricks is if partner has ace-king fifth of diamonds, so ducking, while not always best, at least works out better on balance than a blind switch.

Accordingly, I duck. Declarer continues with the six of hearts--ace--ten--three. So declarer had only four hearts after all! Maybe I was hasty in assuming declarer would lead low from dummy with king-queen-nine fourth. Can leading the ten ever be right?  I suppose it would be right if partner had jack-eight fourth. But few defenders would duck with an offside doubleton ace. If I were declarer, I would rate that layout unlikely.

Partner wasn't able to signal on the heart ace, but I did find out declarer has one trick fewer than I thought he did. That makes a club shift less attractive. If declarer has only nine tricks to cash, there is no hurry to cash our three club tricks.

Could it be right to lead a club for some reason other than to cash out, perhaps because breaking diamonds gives away a diamond trick? How about this layout:

♠ Q x x ♥ K Q 9 x ♦ A J ♣ A x x x

The diamond shift establishes a diamond trick for declarer. That wouldn't matter if the minors were three-three. But with this pattern it costs a trick because the clubs are blocked. Still, the fact that I had to work to construct this layout suggests that a diamond shift is the percentage choice.

The question now is which diamond? A low one gives partner an impossible problem if he has the very hand I am hoping for: ace-king fourth or fifth of diamonds. How is partner supposed to know whether I am leading from the jack or the queen? If I lead low, partner might reasonably play ace, king, and another, playing declarer for queen doubleton. Leading the queen would make things much easier for partner.

Of course, the queen could be a spectacular failure. Imagine, for example, that declarer has

♠ x x x ♥ K Q 9 x ♦ K J x x ♣ A K

But I'm not so sure I should worry about that. The queen is wrong only when leading diamonds is the wrong idea altogether. If I'm going to lead the wrong suit, how much worse is it to lead the wrong card as well? Better to make sure I maximize my advantage if I happen to have made the right decision.

I shift to the diamond queen. Partner takes his ace, and declarer drops the five. Partner returns the five of clubs, declarer wins with the king, and I play the seven. The five is the highest outstanding club, so declarer began with three or four clubs. His hand is probably some variation on:

♠ Q x x ♥ K Q 9 x ♦ K x x ♣ A K x

That's 17 high-card points, so there isn't room for the diamond jack. But he might have the diamond jack if he's missing the spade queen. Let's hope not.

Declarer cashes the nine of hearts. I pitch a diamond; dummy and partner pitch clubs. If I'm right about declarer's hand, he's down to

	NORTH ♠ A 10 5 ♥ -- ♦ 10 6 ♣ ---
	SOUTH ♠ Q x ♥ -- ♦ K x ♣ x

His spades are good, so he has four of the last five tricks. But he doesn't know that. If he thinks I have the diamond jack, his best play now is a spade to ace, then back to the queen, strip squeezing me. This gives him four tricks any time I have the diamond jack, regardless of the lie of the spade suit. As the cards lie, he will be disappointed to take only three tricks. My queen of diamonds may have been a serendipitous choice.

I don't know if declarer doesn't see this line or if he is simply unconvinced that I have the diamond jack. For whatever reason, he banks on spades coming home. He cashes the spade queen, then plays a spade to the ace. My jack drops, so dummy's spade is good. It looks as though we are going to score partner's jack of diamonds at trick thirteen. That is, unless declarer has the nine of diamonds and finesses. That's a scary thought. Wouldn't partner have ducked my queen if he didn't have the nine of diamonds? Who knows? Maybe he was afraid I had king-ten fourth of clubs. If declarer does have the nine, let's hope he thinks I've come down to a singleton jack of diamonds and a winning club for my last two cards.

No worries. Partner has the nine of diamonds. Making four.

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

Four pairs held one notrump to nine tricks. It looks to me as if it takes an initial diamond lead to do that. I shouldn't think that would have been such a popular choice.

One pair reached three notrump, so we wind up with three matchpoints.

Score on Board 38: -180 (3 MP)
Total: 306 MP (67.1 %)

Current rank: 1st

Match 2 - Board 37

Board 37
Our side vulnerable

♠ Q 9 2 ♥ A 7 3 ♦ K 9 7 ♣ 9 7 6 2

Two passes to me. I pass. LHO passes. Oh, well.

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

West	North	East	South
	Pass	Pass	Pass
Pass

The opponents can make three diamonds, but it's hard to get there. If RHO opens, as he would have done in the old days (he does have two and a half-plus honor tricks after all), he would probably wind up in two notrump going down. Every other table passes it out as well, so we wind up with six matchpoints.

Now that we have some time on our hands, let's go back to last week's deal. I glossed over the play of this suit, and it bears some further discussion:

	NORTH ♥ K Q 9 8 3
	SOUTH ♥ 7 2

In the post, I said simply that I led the deuce of hearts from my hand. What I didn't say was it would have been an error to lead the seven. To see why, note what happened. West took the ace, and East, holding ten-four and not wishing to waste his honor, played the four. West was now unable to read the lie of the heart suit. East might have ten-four or he might have ten-seven-four. Had I led the seven, West would have known the count. He might not know whether his partner held ten-four or four-deuce. But he would know his partner held at most two hearts. There is no three-card holding East would play the four from. This was no accident; I led the deuce precisely for this reason.

In this particular case, concealing the heart count from West made no difference. But it might matter on a different deal, so it's worth discussing how one knows that the deuce will be less revealing than the seven. One possibility is to tabulate all of East's possible holdings and figure out which play makes East's card unreadable in the greatest number of cases. Of course, this method is laborious and time-consuming. An easier way (and one less apt to generate late-play penalties) is to follow a few simple rules:

Rules for Scrambling the Opponents' Count Signals

In the ensuing discussion, I will use the name 'Fred' to refer to the defender from whom you wish to conceal information. I will use the name 'Ethel' to refer to his partner, that is, the defender whose holding you are trying to obscure. Sometimes you know ahead of time which defender is which; sometimes you don't. Sometimes Fred may be, for example, whichever defender holds the ace.

Rule 1 - If you wish to represent a particular holding for Ethel, signal as if you held that holding.

Most players know this rule when it comes to attitude signals. Declarer should play high to encourage and low to discourage (or the opposite if the opponents signal upside down). Many players do not know, however, that the rule works with count signals also.

	NORTH ♣ Q J 10 9
	SOUTH ♣ 7 6 2

Fred leads the king of clubs against a heart contract; Ethel plays the four. If Ethel has four-three doubleton, you want Fred to think that it is possible she has three clubs. So you signal as if you had three clubs (which, by sheer coincidence, you do). Since the opponents play standard count, you play low. ("Low" means any card lower than the card Ethel played. If you have choices, you should choose among them at random.)

Suppose, however, you want Fred to continue clubs. Perhaps you know--for whatever reason--that Ethel has three clubs (presumably eight-five-four), and you want Fred to waste a tempo cashing his club ace. You want to represent a doubleton in Ethel's hand, so you signal as if you had a doubleton. Since the opponents play standard count, you play high. Any card higher than Ethel's will do. By concealing the deuce, you leave open the possiblity that Ethel has four-deuce.

Rule 1 is fine if you have a specific objective in mind. But what if you don't? What if you just want to play the card that has the greatest chance of scrambling the opponents' signals? In that case, you need Rules 2 and 3.

Rule 2 - With three or more cards, play your lowest or second lowest card, choosing whichever is closest to middle in rank. If the opponents signal upside down, choose similarly between your highest or second highest spot card. 

For example,

	NORTH ♣ K Q J 10
	SOUTH ♣ 8 5 2

Which card should you lead toward dummy to scramble Ethel's count signal? Assuming standard signals, your choice is between the deuce and the five. The five, being a middle-ranking card, will work more often. Playing the deuce is right only if Ethel has four-three doubleton, since it leaves open the possibility that she has eight-five-four. Playing the five is right anytime Ethel has three clubs, since it leaves open the possibility she has a doubleton.

Note that if dummy had the nine of clubs instead of the ten, the five would work in even more cases. It would work against against any three-card holding and against any even holding that includes the ten (provided Ethel is unwilling to signal with the ten), since it leaves open the possibility she has ten-eight third.

	NORTH ♣ K Q J 10
	SOUTH ♣ 9 8 4

Now the choice is between the four and the eight. Again, the middle-ranking card, the four, is the percentage play. The eight will create an ambiguity only if Ethel has seven-six-five. The four will create an ambiguity any time Ethel has an even number.

Rule 3 - With a doubleton, give correct count with one exception: If Ethel is apt to have a card she doesn't want to part with, give false count. 

Why should you usually signal an even number with a doubleton? Because, in general, it is easier to scramble Ethel's odd signal than it is to scramble her even signal. To scramble her odd signal, all you need to do is to conceal one card lower than the card she plays (or higher if they signal upside down). To scramble her even signal, you must conceal two cards higher than the card she plays (or lower if they signal upside down).

	NORTH ♣ K Q J 10
	SOUTH ♣ 9 2

Signal as the opponents would. Play the nine if the opponents play standard signals; play the deuce if they play upside down. You hope that Ethel has an odd number of clubs, in which case playing the proper card will make it possible that she has a doubleton. It is impossible to create an ambiguity when Ethel has an even number of clubs.

	NORTH ♥ K Q 9 8 3
	SOUTH ♥ 7 2

In this layout, there are two cards outstanding that Ethel might well choose not to signal with: the jack and the ten. Ethel is apt to have one or the other of them, so the exception applies, and you should give false count.

Why does this situation create an exception? Because there are now many three-card holdings from which Ethel's play is automatically ambiguous:

♥ J 6 5

♥ J 6 4

♥ 10 6 5

♥ 10 6 4

Ethel will play low from any of these holdings, but she will also play low from:

♥ J 5

♥ J 4

♥ 10 5

♥ 10 4

(Actually, it doesn't matter whether she will play low from these doubletons or not. All that matters is that, from Fred's point of view, she might play low.)

Normally, against standard carders, you want to retain the deuce when you hold a doubleton. That way, if Ethel plays low from three, it will be possible she holds a doubleton. But if Ethel holds the jack or ten, you don't need to do that. It is already possible that she holds a doubleton. Her signal from three cards is already ambiguous without your having to do anything.

Therefore, you need to shift your attention to scrambling her signal when she holds an even number. And, as we have seen, the way to do that is signal an odd number. If Ethel plays her highest spot card from:

♥ J 6

♥ J 5

♥ J 4

♥ 10 6

♥ 10 5

♥ 10 4

♥ J 6 5 4

♥ 10 6 5 4

then retaining the seven leaves open the possibility that she has three. (If they signal upside down and she plays the lowest spot card, then retaining the deuce leaves open that possibility.)

One addendum: Declarer's choice of equals rarely matters. So, whatever card the above rules instruct you to play, you are free to play any other card of equal rank. Randomizing among equals will help to keep Fred from  exploiting these rules to divine your holding.

Am I being too cautious in saying "rarely" instead of "never"? No. I can't think of a case where declarer's choice among equals matters for an even-odd ambiguity. But it can certainly matter for a two-or-four ambiguity:

	NORTH ♣ K Q J 3
	SOUTH ♣ 9 8 2

You lead the king from dummy, and East plays the seven, presumably from seven doubleton. By Rule 1, if you want to represent three cards in East's hand, you should play low; if you want to represent four cards (so that you might have a singleton), you should play high. But, if you play high, you must play the eight. The nine will not work, since East would play the eight from eight-seven fourth. This falls under the heading of "playing the card you're known to hold."

So there you have it. A simple set of guidelines for scrambling the opponents' count signals. What's particularly nice about these rules is that the opportunity to apply them typically comes up several times per session. So, if you didn't know these rules already, your expected per session score has just gone up perhaps a couple of matchpoints.

Score on Board 37: 0 (6 MP)
Total: 303 MP (68.2 %)

Current rank: 1st