Sunday, October 29, 2017

How many errors can GIB make on one hand?

The other day, I was playing one of those 25 cent all-day tournaments on BBO (IMPs). On the last hand (none vulnerable), my GIB partner picked up the following hand: J875 96 AK8763. RHO dealt and passed, as did GIB. LHO opened 1 and partner overcalled 2. RHO now raised the stakes with 2. You have some useful looking values. What do you call?

Double (technically a "responsive" double here, but I'm treating it as just another competitive double) would suggest sufficient points for it to be our hand, four hearts and four clubs (maybe five if the suit is not headed by good honors). The other four or five cards should be one or two diamonds (else why not raise?), and two--possibly three--spades. In other words, a relatively balanced hand with both the round suits, and about 10 or more points.

Why do I say at least two spades? Because experience has taught me that when you are short in the opponents trump suit, if partner leaves it in, those little trumps that you do not have will be in declarer's (or dummy's) hand. And it's slightly more more likely that partner will have four and leave it in with the hope that the opponents are on a seven- (or at most eight-) card fit.

And why should you have the balance of power? Because whether you choose to play in a possibly poor trump fit, or to defend a doubled contract of the opponents', you need to have some reasonable expectation of making or setting them, as the case may be. This is especially true at IMPs where an injudicious contract our way can get doubled and go for quite a number, or where doubling their part-score into game could be very expensive. You should aim to have at least 22 hcp between your two hands if you're forcing partner to bid at the three level (or defend). In this case, we can assume partner has 12 hcp (he did make a 2/1 overcall) so we should have 10.

Here, we have too few hcp (eight) and too few spades (singleton). Not a hand to double, therefore.

What about 3? Well, if you do bid here, you should have a good honor sequence so that partner's lead of your suit (if we end up defending) isn't an instant own goal. And, you should have at least a decent five-card suit with some tolerance for partner's suit in case he has to go back to his suit. 3 would be the perfect bid here! You have the very best possible lead-directional suit, headed by AK. And a doubleton diamond on the side.

Eschewing the obvious 3 bid (you don't want to miss a 4-4 heart fit!) you double and partner, trusting your judgment, passes. It's unfortunate because 5 our way is cold (partner has Qxx and they split 2-2). Partner leads a trump, not his diamonds. What do you deduce from that? Possibly a broken suit, so the hand has scattered honors. After drawing trumps, declarer starts to eliminate clubs. You win and lead your 9 covered by ten, jack, ace. A second club comes which you win and now there's one obvious card in your hand that will set the contract--your other diamond. But being the GIB, you don't worry about partner getting endplayed and you lead a heart. Game over.





Saturday, September 16, 2017

You can squeeze, too

One of the first things we learn about the playing of the cards in Bridge is how to take a finesse. If you lead towards your AQ combination and the person on the left has the K, you will win two tricks. Otherwise, one. Of course, there's a little more to finessing. There are also ruffing finesses, two-way finesses, intra-finesses, “obligatory” finesses, and so on.

The situation is similar with squeezes. What was that you said? Surely, squeezes are much harder than finesses? Well, they are a little bit harder but it really isn’t rocket science. The first bit of added complexity is that, whereas in the case of the finesse, where your LHO must follow suit if able, a squeeze operates on two, or sometimes three, suits at a time. In fact, when you lead the "squeeze card” to the squeeze trick, your victim will be out of that suit and have to discard. But that’s where the fun (for you) begins. If the squeeze is working, your victim will have to discard either a winner, a guard to a winner, or a card that he could have led to his partner’s winner. Either way, he loses out.

Note that I didn’t say “LHO” but victim. That’s because sometimes your victim is on your right, although that’s a bit harder to pull off because he will be discarding last, after everyone else. But that’s when you play your “ace in the hole,” so to speak. More on this later.

For now, let’s stick to squeezing LHO, because that’s the easiest situation—hence its name: “simple squeeze”. In the following situation, either the enemy trumps are all drawn or it’s a no-trump contract and you are in your hand. RHO doesn’t have any cards in the three critical suits (he can be playing with Happy Families cards for all you care). Here’s the layout at trick 11 when you play the 2, the squeeze card:

         A3
         2
 
         
54              
A               
—               
—               543
         2
         
         2
         2

According to the rules of bridge, LHO will have to play one of his cards before you play from dummy. His A is a winner, of course, and if he discards it, your lowly deuce will be promoted to master rank. What if instead he decides to pitch the 4? Now—and here’s the fun part—you pitch that little deuce of hearts because it’s no longer useful. When you cross to the A, with your deuce, your trey will now win the last trick. So, we just swindled LHO out of his useful cards simply by virtue of playing after he does. These cards which can be potentially promoted into tricks are called “threats” or, “menaces”—the word the Brits use to describe them. Note that LHO didn’t make a mistake—he had no legitimate choice because all his plays were going to give something away. That’s the essence of a squeeze! Sometimes you can gain a trick by a defender pitching the wrong card just because they don’t remember what’s out. That’s called a "pseudo-squeeze" and it's better than nothing (although don’t run a pseudo-squeeze against an expert—it’s a little bit insulting).

So, let’s look at this situation and see what really happened. There’s a reason that there are only three cards each in this layout. If there were any more, LHO would have a spare card to throw and you wouldn’t know which card to pitch from dummy. Basically, you have to be in the situation where you’ve lost all the tricks you are willing to lose but you don't want to claim yet because there’s a chance the squeeze will operate for an extra trick. You also had to have a (single-card) entry to dummy's two card holding. Your threats have to be positioned in the upper hand (the one that plays after your victim). And, finally, your victim has to be the one guarding the two suits where you have threats—he said to be busy in both suits.

From these conditions described in the previous paragraph, we get the famous BLUE law for squeezes: Busy-Lost-Upper-Entry. In this short article it’s impossible to go into all the details of why you need these four conditions. But deal out some cards yourself and try it. You’ll soon see what happens.

I will just add some observations on situations that are a little more complicated than what we just had (you knew it wasn’t really that simple, didn’t you?). You might have noticed the presence of the 2 in your hand in the situation above. It didn’t seem to play a part, certainly not a starring role. It’s a spare card. In this case, it could equally have been a small heart. And RHO’s cards didn’t all have to be clubs: they could have been any small cards as long as there weren’t two spades (the suit of your two-card threat). If the spare card happens to be in the upper hand, and you have a threat in your own hand, you have what’s called an “automatic” squeeze. The title doesn’t mean that it will play itself (although actually, if you’re paying attention, it will). What it means is that it operates against either defender. So, in the diagram above switch your 2  (the spare card) for dummy’s 2. Now, when you play the squeeze card from your hand, you know ahead of time that you’re going to pitch from dummy—the spare card. But you do have to look out in case someone (either player) discards the A because if that happens, and you've been paying close attention to the hearts, you now cash the 2 in your hand, and then cross to dummy’s A for your last trick. In other words, in this situation, you could swap all of LHO's cards for RHO's cards—in an automatic squeeze you don’t care who your victim is—lefty or righty.

What you can do with two suits, you can sometimes do with three. These situations are a bit more complicated, but not much. For one thing, their end positions (i.e. at the squeeze trick) will have more cards in each hand. Just as with the two-suit squeeze above, there are two major forms: the triple squeeze where only one player is busy in all three suits and the double squeeze where both players are busy (each of them in just two suits). The triple squeeze itself has at least two forms: in one, you can actually gain two tricks! Triple squeezes aren’t common but they are even more fun than a two-suit squeeze. Some double squeezes are easier to pull off than others. There are even ruffing squeezes. But, for these you need to do some reading. There are lots of good books on squeezes. The classic (and not too long) is Clyde E. Love's Bridge Squeezes Complete or, if you can find it, its predecessor: Squeeze Play in Bridge. Perhaps the ultimate is Hugh Kelsey's four volume set (now published in one very thick volume as Kelsey on Squeeze Play). But my favorite, with the inimical style of David Bird—and with lots and lots of examples—is Squeezes for Everyone.

But whether you want to read about squeezes in books or not, here is a pretty simple way to try for one of the simpler squeezes on your next hand. Don’t worry too much about preparing the ground (yes there are some techniques you can use but they are a bit more advanced). Just follow this checklist:

  1. Do you have exactly one loser? If you have more, try to lose the tricks you must lose early on, for example ducking from a suit where you have Axx opposite Kxxx. If you no losers, claim: don't torture your opponents.
  2. Are there any trumps out? If there are draw them. If you can't, or if you need the trumps yourself for a cross-ruff perhaps, then this is not a squeeze hand.
  3. Run all your winners—Yes, even your last trump—I know you'll feel naked without it, but it's essential.
  4. If your single threat (2 in the example above) has come good, cash it and claim. If not, use your entry to the other hand and hope that its companion will be good. If it isn't, then there was no squeeze but you've done your best. Note that in the example where we had Axx opposite Kxxx, and we ducked a trick early, if that suit splits 3-3, you will end up making the same number of tricks as everyone else. But if that suit was 4-2 and the one with the four had to guard another suit, Bingo! you just made an extra trick. Believe me that this will score a lot of matchpoints in your average club game.

I'm sure I'm going to regret publishing this article when you are making extra tricks against me! But you deserve to have more fun at bridge. Happy squeezing!

Monday, September 11, 2017

A leading question

The problem of which card to lead from a sequence or near-sequence of honors has been with us since the early days of whist. "Standard" leads are the top of any sequence, except that from AK, the K is led. At first glance, this may seem inconsistent but of course there was a good reason, historically speaking. In whist, there was always a trump suit so, typically, only one or two rounds of a suit would be cashing (it's unusual in a suit contract that you happen on the lead of a suit where all follow to three rounds). So, therefore, it was important--if you held the lead--that you don't give up a trick on the second round in case there was a useful third round. That's why the "come on" signal was developed to help you know whether to continue or to switch.

Originally, this was called a "peter" after "Blue Peter," the flag that told sailors in port that their ship was about to set sail. Failure to heed the Blue Peter would result in a charge of desertion. Not good. At bridge (or whist) the consequences might be expensive but you wouldn't (normally) lose your life over it. These days, in North America, we call it an echo or, less poetically, an attitude signal.

A minor diversion on a subject I think I've addressed here before: sequences. Bridge (the play) and whist are, to a large extent, a game of sequences and position. If you are the last to play to a trick, you have the luxury of being able to cover whatever is the highest card (assuming you have such a card). You don't have to worry too much about it. But if you are one of the early position players, especially leader and third-hand, you want to try and make the best use of your sequences if you have any. You don't normally lead unsupported honors, at least not unless you have good reason to think your partner might have a touching honor (thus giving our side a sequence).

This is why defense is so much harder than declarer play. Defenders cannot directly see where they might, between them, have a sequence. A suit might be distributed KJ53 opposite AQ42. If declarer has that suit, he knows for sure that there are four tricks to be had, however he plays it. But if the defenders have that suit, each partner may be reluctant to lead the suit for fear of giving away a trick. Just last week I gained two overtricks in a suit contract because my LHO was twice reluctant to lead from KJTx (his partner having AQxxx).

Returning to opening leads: if the K is led, partner knows, holding the A, that he always wants to encourage a continuation because the K promises the Q (or perhaps Kx). This guards against the so-called Bath coup (a whist term) where fourth hand has AJx of the suit and holds off the first trick. But what if opening leader's partner holds the J but not the A? If partner led K from KQ, then the card will form a sequence with leader's remaining card and, therefore, another lead in the suit is safe. But what if partner holds the AK and declarer has the Q. A continuation would now be unfortunate because the declarer has the Q for the third round. But as mentioned already, the rules were set up to keep us relatively safe for just two rounds of any suit. Recall that in whist, there is no declarer. The hand playing last to the opening lead may not have sufficient high cards ever to draw trumps and cash that queen.

The problem with this cozy little scheme is that, in bridge, we do have a declarer. That player, fourth to play on the opening lead, likely does have sufficient high cards and/or trumps to eventually make use of that queen on the third round of a suit.

So, various schemes over the years have tried to clarify opening lead sequences. One rather complex scheme was developed by Norwegian Helge Vinje and was later reported by the Bridge Journal in 1964. Such leads became known as "Journalist" leads. They tend to be more popular as leads against notrump contracts, although Vinje actually described two somewhat different schemes for trump and notrump contracts. There is much to be said for these lead conventions, but they are too complex for most casual partnerships.

A very common revision of "standard" leads is always to lead the highest of any sequence, even ace from AK. One problem of course is that when you want to lead an ace, you don't always have the king and partner may well go wrong (this is more typically true of higher-level contracts). It's common, especially in Europe, to agree that against suit contracts at the five-level or above, ace still asks for attitude (but now it's only the king that is good) and the king asks for count. There's also a perhaps minor problem that has to do with UI (unauthorized information) when you agree to play ace from AK. How long does it take you to select a lead from a hand with an AK combination in a side suit? About half a second. So, when you quickly lead an ace, your partner knows that you have the king to back it up. But when you think for thirty seconds and then lead an ace, partner knows that you don't have the king. He's not authorized to know this and so must go out of his way to avoid taking advantage of such information. Is this really a big problem in reality? I don't know. But experts like to avoid giving their partners any kind of problem so that brings us to one more opening lead convention: Rusinow Leads.

Originally banned by the ACBL, they were adopted by Blue Team greats Belladonna and Avarelli and sometimes therefore called "Roman Leads." The principle is simple: you lead the second of touching honors. Typically, it is agreed to play this only against suit contracts and not in partner's suit. When asked about honor leads, the defenders will say "shows the honor above or shortness." Usually, the ten is considered the lowest "honor" for this purpose so that the 9 could be from T9(x) or it could simply be the 9 from 9(x). There will always be a little ambiguity with any such convention but isn't it much better to have the ambiguity over the 9 than over the ace?

It's easy to forget that you're playing Rusinow. We have all done it. Even though I've played Rusinow (or Vinje) leads perhaps more than any other lead convention, I've still forgotten it occasionally, sometimes with disastrous consequences. But, if you are part of a regular partnership that wants to improve, Rusinow leads are a very sensible and simple lead convention.

Incidentally, whether or not you play Rusinow yourself, you should be aware that some people play it. If you are declarer and you need to know the location of a particular high card (that's to say most contracts), you may need to know what their lead conventions are. You can simply ask "leads and carding?" or you can pick up their convention card and look for yourself. You probably shouldn't ask "is that lead standard or Rusinow?" as that tends to give away the fact that you don't have one of the adjacent honors.




Sunday, August 13, 2017

Exclusion Roman Keycard Blackwood and other animals

"Exclusion" is normally defined as a jump to a non-trump suit above the keycard-ask. So, for example, if the auction goes 1--2--2--2--3--5, that would be the keycard ask with spades as trumps, excluding the ace of hearts (because responder is void in hearts). Whether or not you play 1430 or 3014, the responses to exclusion are typically 3014 (although obviously that would be a question of partnership agreement). In this case, if opener has zero keycards, the response would be 5 so responder had better have at least one keycard himself

What about the situation where a void has been shown and then that hand asks for keycards? Let's look at an example: 1--2NT*--3**--3--4***--4--4NT. In this sequence, 2NT* is Jacoby; 3** shows shortness in diamonds; 4*** shows a diamond void. Obviously, the 4NT bidder can't want to know about the A, so you should answer with the appropriate "exclusion" response, even though, in this case, 4NT isn't a jump.

This case was clear. But there is also the situation of an inferred void. Howard Piltch taught me this many years ago and it's stuck with me. Here's a real life situation:

You pick up: AQJ2 9432 KJ K94 and open 1. Partner responds 1 and you raise to 2. Partner now bids 3, showing a splinter in spades. Whoops! Most of the value of your hand just went out the window. In my opinion, you should now put the brakes on and bid 3NT (unless that would be treated as serious or non-serious 3NT). In any case, you decide to sign off in 4. But partner's not done, much to your dismay. 4NT quoth he. Now, look at it from his point of view. He didn't see the need to bid controls, rather he bid 3. That sounds like he's asking your opinion: do you have wastage in spades. You surely do so you signed off! But partner doesn't care. He's hell-bent on bidding a slam provided we have three of the four significant keycards. Get that: he doesn't care about spades--he has a void there. If he only had a singleton, would he first ask your opinion and then ignore it? No. He'd bid 4NT directly over 2--or maybe he'd make a control bid or something. So, you can safely infer that he has a spade void. No other bidding sequence makes sense.

So, now you tell him you have zero key cards (using the exclusion responses if you've agreed to play exclusion).

At the table, my partner didn't pick up on the inference (I know it's subtle) and told me about the A. I needed one keycard (the K) for the contract of 6. I still had chances though: I needed the king to be doubleton. It wasn't :( But my slam was still a 70% slam: 50% that he'd have the "right" key-card and, 20% in the zone. So I don't feel too bad about it.

Sunday, August 6, 2017

Elementary deception

Continuing my series on common beginner mistakes, I want to talk about perhaps the most elementary case of deception, or hiding your assets, in bridge.

But, first, let's look at a hand. Matchpoints, all vulnerable, you are in fourth seat holding: K962 KQ KT7654. Partner opens 1, playing more or less standard methods, you bid 1 and partner raises to 2. What now? You have a decent 11-count with a side singleton and a double fit. I presume you aren't planning to pass. Make your plan:

  • invite game?
  • bid game?
  • invite slam?
Have you made up your mind? Is there anyone out there who's going to invite slam? It seems a stretch to me given that partner raised only to 2. But, I suppose it's possible: AQxx xx Axx Axxx. Of course, many would have already upgraded this perfect hand to 1NT, especially if non-vulnerable. If the black suits split favorably, you have twelve easy tricks. Or, if there's a stiff quack in the clubs, you might still be able to pick up the suit. But, presumably, if you were making a slam try, you would bid 4 to simultaneously establish a game-force (with slam aspirations) and to let partner know that any lower honors in diamonds would be wasted. With the perfecto sample hand given, you would bid 5 and partner should take it from there.

I'm sure you would simply raise to 4. But, suppose for a moment, that you decided to invite game. How would you go about it? I assume you would bid 3 and then pass whatever partner decided on. But, couldn't you have your cake and eat it too? Why not use the good old three-way invitation? Bid 3 and if partner bids 3 immediately (or passes) you'll play it there. If he bids 4, all well and good. But if he thinks for a bit and then bids 3 he's obviously got some sort of intermediate hand and now you can raise to 4. Sound like a plan?

No! The laws of bridge expressly forbid you to take any inference from the manner in which partner makes his bid. You must act as if you were absent from the table during the period between LHO's pass and partner's bid. To use the jargon of the rule book, that information is unauthorized. When partner's bid is 3, you must respect his decision, however much soul-searching he/she might have gone through. Remember: you didn't see (or hear) any of it.

I only bring this up because it happened in a recent club game. A pair who certainly should know better had this exact auction and, guess what, the spade bidder raised to 4, making. Yes, I called the director but, despite the fact that the opponents didn't dispute the hesitation (that's the usual defense), the director ruled that the 4 was OK because "maybe he was looking for slam". Most tables played in 4, as you'd expect, although there were a few 3 and club contracts too. We got slightly below average for holding it to ten tricks (some declarers actually managed to go down in 4♠ although I don't see how that was possible).

OK, back to the real point of this blog.

The opening lead is a small diamond (fourth best) and dummy has K9. You play the 9 and RHO wins the J. The 9 comes back. Which card do you play?

Some of you, the ones this article is aimed at, will say it doesn't matter. The one nearest my thumb. Or, worse, you will automatically play the lower honor, the queen in this case. I see this mistake all the time. You are probably expecting LHO to have the ace and, therefore, it won't make any difference. When we played this hand, declarer played the queen and it won the trick. The RHO (moi) holding A98x, now knew that his partner did not hold the king and so I was able to work out exactly which cards declarer did hold. It really didn't matter in this case. There was no defense to set the contract. But why give the opponents that information? They don't deserve it!

Monday, May 29, 2017

Opening a strong, artificial two clubs

I think it may be a general truism that the more experienced at bridge you become, the less likely you are to open any given hand 2. Many hands that look like something you want to shout about, especially if they are somewhat balanced with a good minor suit, probably shouldn't be opened 2 but will be by many inexperienced players. Minors and 2 openings don't go well together because you will be at the three-level by the time you've mentioned an actual suit and 3NT, the favorite landing spot, is rapidly approaching. Once you go beyond 3NT, you need good systems to land in the right spot. So much is conventional wisdom.

But I want to talk about something a little different: overall shapeliness and suit quality. I can't emphasize sufficiently how important it is to have a twice-biddable suit when you open 2. You simply don't have a lot of room to show a two- or three-suited hand. You can happily open a no-suited hand that falls in the appropriate range (usually something like 22-24 hcp) because partner will be able to take charge and steer you to the right contract. But, when you open 2 and hear the expected 2 from partner (heaven help you if you play 2 artificially as the bust hand), your rebid, assuming it's a major, will be at the two-level--and it's forcing. So, partner's second bid is usually going to be at the three-level so any third bid that you might make, to suggest a second suit for instance, will have to be on (at least) the three-level too. Again, 3NT is rapidly approaching.

Life becomes a lot easier for responder if he can rely on your primary suit being a good one: that's to say playable opposite a small doubleton (or even a singleton). I've had several arguments with the BBO robots when they just wouldn't raise my suit with, say, xx. They end up bidding their own lame suits and we often get to the wrong spot. They should assume that my suit is a good one. Part of their problem is that they don't play any kind of puppet Stayman to eke out a five-three major fit in opener's major. So, therefore they tend to assume that 2 opener simply has a five-card suit.

Of course, I am also a believer in the raw power of a good suit. Making a game with a good six-bagger (or longer) is just going to be so much easier than with even an excellent five-card suit. The latter will need compensating values.

Here to illustrate this point is a hand I held in a robot tournament on BBO recently: AQ87435 AKJ8 AJ 8. Only 19 points but a decent six-card suit, three aces, a singleton and, by the losing trick count, only four losers. By my estimation, this was a 2 opener. I expected to make game opposite not very much. Here's what happened at 25 tables: 1 passed out, making between 9 and 11 tricks (10 mostly). Two players made sure of reaching game by simply opening 4, making with an overtrick. The auction at the other three tables, including mine, was identical: 2-2-2-3-4. We made either 10, 11 or 12 tricks (I made 12, the proper number--three tricks more than should properly be made in spades).

So, did I get lucky? Was this an anomaly? It was marginally on the light side perhaps. But I would bid the same way again.

One other hand of interest cropped up in this 12-board set. Put yourself in the position of my robot partner: Your hand is: J43 J653 KJ AJ63. We are not vulnerable vs. vulnerable and LHO deals and opens 2 (weak). Partner doubles and RHO bids 4. This is followed by two passes and partner doubles again. Your call.

I've seen so many -530, -710, -990 etc. result in my bridge career that I make it a point of honor never to make a penalty pass when I have an asset that partner is seeking. Here I have four hearts. I also happen to have seven cards in the other two suits that partner is promising. It's no big surprise that partner is short in diamonds given their bidding and my holding. Clearly, I have no diamond tricks at all unless I'm lucky. But I have a working ace and two, maybe three working jacks. This hand cries out for a 4 bid.

But not the robot. He converted the double to penalties, despite knowing that it was for takeout. So, despite our 25 hcp, we suffered the ignominy of -910 (not one of the commonest bad scores) and I lost more IMPs on this hand than I had won on the hand I mentioned above. They are actually cold for 600 their way, but of course they weren't going to bid it until I (re)-opened my mouth :( But the robot could have at least attempted to save the day by bidding 4. That would have scored 420 for the good guys and we have a cheap save over 5 (if we should take it) at -100. The first of these scores would have gained about 13 IMPs (a positive swing of 24); the second 5 (a swing of just 16).

So, that silly robot cost me either 16 or 24 IMPs! Fortunately, it made no difference to the overall result, as I still managed to win, barely, after flubbing the final board, costing myself another 14!




Sunday, May 21, 2017

Happy families are all alike; every unhappy family is unhappy in its own way.

The so-called Anna Karenina principle, quoted in the title, applies to so many walks of life. This weekend, playing with Symphony Pro Musica, I am reminded forcefully that, while music performance is not a perfect example of the AK principle--especially for professional musicians--nevertheless it is close. A perfect performance is indeed very much like another perfect performance. Again, I hasten to add that there is yet plenty of scope for differences among "perfect" performances at the highest levels of the art. But, for a non-professional orchestra, where mistakes can and do happen, there are many more ways to mar a performance than there are ways to play it perfectly. In ensemble playing, every note must be:
  • properly formed (not a squeak or croak) and with the appropriate degree of vibrato;
  • the correct note (duh!);
  • in tune (and an orchestra's domain is not the equi-tempered scale of keyboards so playing in tune requires constant attention);
  • in time (not more than a few milliseconds early or late);
  • at the proper dynamic level;
  • exhibiting the correct emphasis/style.
When you have forty or fifty independent players in a (non-professional) orchestra, each playing say a thousand notes, the chances of a perfect performance are obviously small. That's not to say that the performance cannot still be very much enjoyed by the audience. On the contrary, one of the most essential aspects of musical performance is the freshness and immediacy of the performance.

So, what does all this have to do with bridge? A hand of bridge is another endeavor in which perfection is possible, but the number of ways to achieve the perfect result (from both sides' point of view) is small, whereas the number of ways to err is vastly greater. In other words, a hand of bridge is another exemplar of AK. I note with interest that we bridge players have another thing in mind when we use the letters "AK."

One of my favorite tools for researching this idea is BBO's GIB analyzer. At each player's turn, you can check to see which cards are the correct ones to play and which the incorrect. The only flaw is that GIB will tell you the immediate ("proximal") effect of the play of a card, but it cannot (or doesn't) tell you the "distal" effect, taking into account the reaction of the player's partner (this in the case of a defender). As an example, defending a suit contract, partner opens with the appropriate card of his "AK" holding in a side suit. Dummy has two cards in the suit. You have both the queen and the jack and perhaps some other cards. From GIB's point of view the play of the Q or J is entirely equivalent. But which you choose can have a major difference to the result when partner leads to the next trick. Still, the GIB tool is an invaluable resource.

As an example, let me share this defensive problem (note that it wasn't necessary to ask GIB this time). It's the second board of the session and you hold 75 9432 AK65 982, not vulnerable vs. vulnerable, dealer. You pass and LHO opens 1NT (15-17). Partner passes and RHO bids 3NT, alerted. Opener now bids 4S which is passed out. Partner asks before leading and we are told that dummy has four spades and five hearts. "No," dummy says, 3NT means I want to play in 3NT.

Partner leads the deuce of diamonds (third and lowest) and dummy comes down with: T843 AKT T98 AT3. Well, he does have four spades. You can see that this is going to be a tough session. Even when the opponents have a major misunderstanding, they land on their feet. Anyway, you win the king and now contemplate your lead to trick two.

You have twelve cards in your hand and I can tell you now that there are only two correct cards to play (and they are absolutely alike) while there are ten possible errors to make. A perfect example of the AK principle.

This is how your thinking should be going. Partner has either one, three or five diamonds. Five is ruled out because LHO opened 1NT. That leaves one or three. If it's one, that means opener has five which is quite possible. We have no outside entry so it seems reasonable to cash the ace and give partner his ruff. Perhaps we should give him the ruff now as the diamond ace can't go to bed. What if he led from Qx2, which appears probabilistically to be the most likely holding (he wouldn't tend to lead from a xxx side suit after this auction)? Then we'll always get our three diamond tricks. This is true whether he started with three or one (where the Q will be replaced by a ruff). Does anything else jump out?

Could he also have Jxx of trumps? Yes, this is entirely possible (we expect opener to have four spades). If partner's trumps are any better than this, he'll be getting a spade trick anyway, whatever we do. Providing that we are on lead at trick four, that hypothetical trump trick can be promoted by playing the thirteenth diamond!

The problem might have been easier at IMPs where all that matters is defeating the contract. At MPs, it's important not to give away silly overtricks, chasing some chimera. Yet, this problem is no harder than playing a scale in a major key. Unfortunately, I didn't think it through quite as well as I've shown here. I picked one of the ten unhappy cards (A) instead of either of the two happy cards: the five or six of diamonds.