Free Weekly Instant Tournament - October 25 - Board 7

Board 7
Both sides vulnerable

♠ 8 7 5  ♥ A Q 10 6 2  ♦ 9 8 5  ♣ K Q

I'm in first seat. Under normal circumstance, I wouldn't open a "5332" with 11 HCP. But this is a "best-hand" tournament, which means if I don't open, there is a good chance the hand will be passed out. If so, do I mind? Was I likely to go plus had I opened?

If I open with one heart and partner bids one notrump, I can pass. It's forcing, but partner doesn't have a cup of coffee to throw at me. If that happens, we will probably go plus in one notrump. 

If partner bids one spade, however, I have a problem. If I pass, we could easily miss a spade game. But if I rebid one notrump and partner raises to two, I suspect I'm a favorite to go down. All in all, I don't think opening rates to be a winning decision. 

With a balanced ten-count, I would be more tempted to open. I would be less worried about passing a one-spade response. And knowing everyone has exactly 10 HCP is a huge advantage in the play.

I pass. As does everyone else.

Passing the board out is worth 64%. Those who opened typically rebid one notrump over their partner's one spade response and played it there. Let's pretend I did the same and try to predict what would have happened.

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

West	North	East	South
Robot	Robot	Robot	Phillip in an Alternate Universe
			1 ♥
Pass	1 ♠	Pass	1 NT
(All Pass)

West leads the five of clubs. East takes the ace, dropping my queen, and returns the four of clubs to my king, West following with the eight.

Partner has 9 HCP, so the outstanding high cards must be distributed 10-11.

How should I continue? It looks natural to attack the diamond suit. I can lead the nine and let it ride. If it loses to the king or queen, I can lead the eight next, picking up the suit for three tricks if East holds king-seven or queen-seven doubleton. But that's still only six tricks. I'll need to find the heart king onside to come to seven.

If I need the heart finesse anyway, perhaps I'm better off forgetting about the diamond suit and trying for four heart tricks. I can start with a low heart from my hand, giving West a chance to make a mistake and hop with king doubleton. If that doesn't happen, I get to dummy and lead a heart to the queen, hoping for three-three hearts. 

This offers a better chance to make my contract than playing on diamonds. The problem is, I go down more if it doesn't work. I won't even score the heart ace, because I'll have no entry to my hand. I don't think that's a serious consideration, however. The board will be passed out at most tables, so I need to go plus to get a decent score.

I lead the five of hearts from my hand. West hops with the jack and leads the six of clubs to his partner's jack. I pitch the five of diamonds.

East cashes two more clubs. I pitch a diamond and a spade from each hand. West pitches the three of diamonds and the six of spades. We've reached this position:

	NORTH Robot ♠ A 9 3 ♥ 8 ♦ A J 6 ♣ --
	SOUTH Phillip ♠ 8 7 ♥ A Q 10 6 ♦ 9 ♣ --

East shifts to the spade queen, and West overtakes with the king. I see nothing to gain by winning this trick, so I duck on principle.

West continues with the spade jack. I take dummy's ace, and East follows with the four. East has shown up with the spade queen and ace-jack of clubs. There is room in his hand for the heart king. I lead the heart eight from dummy--five--queen--king. I score only the diamond ace after that. Down four.

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

The pairs who played one notrump all floated the diamond nine at trick three. So I would have gotten a zero for down four. I'm happy I didn't open.

Free Weekly Instant Tournament - October 25 - Board 6

Board 6
Opponents vulnerable

♠ 5 4  ♥ K 9  ♦ K Q J 5 4 3 2  ♣ K 4

RHO opens with one spade, and I overcall with two diamonds. LHO bids three diamonds, showing a limit raise or better in spades, and partner bids four diamonds. RHO bids four spades.

I have five losers. Can partner cover three of them? No doubleton is useful, so partner needs three working high cards. That's a limit raise. With a limit raise, partner would have bid three spades rather than gently raising to four diamonds. So five diamonds does not rate to make.

Next question. Can we beat four spades? I have one trick on defense (no tricks in diamonds and half a trick for each side-suit king). Partner needs three tricks to beat this, and I've already expressed my doubt that he has them. So four spades rates to make, and five diamonds should be a good save.

I bid five diamonds. LHO and partner pass, and RHO doubles, ending the auction. LHO leads the seven of spades.

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

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

I can't avoid losing the two red aces. Is there any way I can pitch my spade loser? Spades are three-five, but West doesn't know that. From his point of view, East might have six spades, giving the defense no spade tricks. Further, West doesn't know his partner has no small trumps. So perhaps if I give him a chance to give his partner a heart ruff, he will take it rather than try to cash a spade.

This shouldn't work. Why would I play hearts before drawing trump if East might be ruffing hearts? But it's my only chance. I rise with the ace of spades. RHO plays the six; I play the four. Now deuce of hearts--six--king--ace.

West doesn't fall for it. He lays down the queen of spades, then shifts to the jack of hearts. I win in dummy and claim down one.

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

Minus 100 is worth 57%.

We have a shot at beating four spades. Declarer must pick up the jack of spades. Will he do so? Say I lead the diamond king. Declarer wins in his hand and plays a spade to the queen and ace. Partner returns a spade. What's the right play?

If North has ace doubleton of spades, it makes no difference, so let's assume he has ace third. (Yes, he might have ace fourth. Let's ignore that possibility for now.) Some might think this makes the odds three to two that North has the spade jack. Actually whether that's true or not depends on what we knew about the ace.

There are three ways for North to hold ace-jack third, and there are three ways for South to hold jack doubleton. So if North was known to hold the ace (from the auction perhaps), then it's a toss-up whether to finesse or not. Once you factor back in the possibility that North holds ace-jack fourth, finessing becomes the percentage play, but only by a small margin. If North can't have four spades or if you can't handle the hand if he does, then your play is indeed a toss-up. It's worth remembering this combination, since a lot of players don't realize that. 

In this case, however, North was not known to hold the spade ace until he played it. That makes a difference. Holding ace-empty third, he might duck the queen, but with ace-jack third, he can't afford to duck, since his jack will pop up next. In other words, with ace-jack third, his choice is restricted; with ace-empty third, it isn't. So, when he wins the ace, ace-jack third is more likely by restricted choice. Finessing is now a favorite even without factoring in the four-one break. So if we sell to four spades, declarer should finesse and make his contract.

One player almost managed a top. He bid an imaginative four diamonds over one spade. West bid four spades, and North bid five diamonds. After two passes, West decided to take the push to five spades, making this South the only player who found a way to get a plus score. But then he turned his top into a near zero. After two passes, he saved in six diamonds. There are two good reasons not to bid six diamonds:

1. It violates captaincy. A pre-empt describes your hand, so it's partner's job to make any further decisions. In this case, overruling partner is especially bizarre, since South has more defense than he has promised.
2. If you don't care about theoretical matters like captaincy, there is a practical reason not to save: Not everyone will take the push to five spades. Five diamonds doubled will be the contract at some tables--perhaps many tables. If you compete to six diamonds, you are automatically losing the board to everyone who bought it in five diamonds. So even if you are right, your gain is small. The odds favor defending even if you think five spades is a moderate favorite to make.

Free Weekly Instant Tournament - October 25 - Board 5

Board 5
Our side vulnerable

♠ K 10 4  ♥ 10 7 3  ♦ A 6 5  ♣ A K 5 4

Two passes to me. I have only 14 HCP, but I have three and a half honor tricks and two tens, so I open with one notrump. (No. I'm not "upgrading." I'm evaluating. I'm judging that this hand rightly belongs in the strong notrump category. I find the term "upgrading" annoying, because it implies there is something canonical about the Work point count. There isn't. It's just one method of evaluation.)

LHO bids two clubs, showing a one-suiter but declining to identify the suit. Partner bids two hearts, a transfer to spades. I bid two spades and everyone passes. LHO leads the king of hearts.

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

West	North	East	South
Robot	Robot	Robot	Phillip
	Pass	Pass	1 NT
2 ♣	2 ♥	Pass	2 ♠
(All pass)

It appears West's suit is hearts. I have three hearts losers, a diamond loser, and probably two losers in spades. Unless I can avoid one of those losers, I'm going down.

East plays the deuce of hearts, and I follow with the three. At trick two, West shifts to the three of clubs. That's a strange play. I'm guessing that's a singleton club. But he knows at least one more heart is cashing. What does it cost to cash it? Is he hoping his partner can ruff hearts twice and give him two ruffs? That's awfully greedy. Is he really going to lead a low heart next, risking his partner has a doubleton? Robots don't signal at trick one, so there is no reason his partner couldn't have a doubleton heart.

I might as well win this trick in dummy and try a spade to the ten. I play the club queen, and East follows with the deuce. When I play a spade from dummy, East hops up with the ace. I play the four; West, the three.

East shifts to the jack of clubs. I play the ace, and West ruffs with the nine of spades.

West will presumably cash a heart. When his partner shows out, he will lead a low heart for his partner to ruff, then get a second club ruff, reaching this position:

	NORTH Robot ♠ J 8 7 6 ♥ -- ♦ Q 9 ♣ --
	SOUTH Phillip ♠ K 10 ♥ -- ♦ A 6 5 ♣ 5

At least West will be endplayed at that point if he has the diamond king. So I'll get out for down one.

West cashes the ace of hearts. To my surprise, East follows with the nine. Oh? Here I was assuming West's suit was hearts. Apparently it was diamonds. West presumably has three hearts. With a doubleton ace-king, he would have led the ace, and with four, he might have overcalled two hearts, showing hearts and a minor. So he's either 3-3-6-1 or 2-3-7-1.

I follow with the heart seven. West continues with the five of hearts to East's queen, and East plays another club, which West ruffs with the spade queen, reaching the above position in a different way than I anticipated.

At least I'm right that West is endplayed after taking his ruffs. He has only diamonds left. He leads the king of diamonds, and I claim down one.

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

This result is worth 93%! The opponents can make three diamonds and most of the field played it there. A typical auction was

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

I'm not sure I would have found that three-diamond bid. One diamond, then two diamonds seems like enough bidding to me, especially when the opponents are probably in a four-three fit. (North will usually bid two spades himself with five.) But West was right to bid on, so who am I to complain?

Our auction gave the West at our table a different problem. The way it timed out, West didn't know my two-spade bid was getting passed out, and he didn't want to bid at the three-level in a live auction. My one-notrump opening gained, as it sometimes does, by making the auction harder for the opponents. Although, interestingly, it wasn't my opening per se that gave West the problem. It was partner's transfer, which kept West in the dark about partner's intentions.

Free Weekly Instant Tournament - October 25 - Board 4

Board 4
Both sides vulnerable

♠ K J 10 9 7 6  ♥ A 7  ♦ K  ♣ A K Q 5

Partner opens with one heart in second seat. I have twenty high-card points and a good six-card suit. We're bidding some slam. It's just a question of finding the right one. The robots play two spades as a strong jump shift, but this isn't the hand for it. Strong jump shifts are for slam invitations, not slam drives. It's a way to let partner know you have about an ace more than a minimum game force, which can be hard to do in Eastern Science Fiction. And it surrenders captaincy. If you know you want to be in slam, you shouldn't jump shift, because you want to maintain captaincy yourself.

I bid one spade, and partner bids two hearts. After a one spade response, two hearts guarantees six, since there are no awkward patterns that might require a rebid in a five-card suit. In my style, it also denies three spades. Unfortunately, the robots don't play that way.

I bid three clubs, and partner bids three diamonds. In my methods, I would know I was facing a singleton or void in spades. The two-heart bid denies three spades, and the failure to take a preference denies a doubleton. This is one of the advantages of my approach. Discovering partner is short in your suit can be useful.

Perhaps I should just ask about partner's keycards. If he has king-queen sixth of hearts and two aces, we have thirteen top tricks if hearts break. And if hearts don't break, we have the spade suit in reserve. 

Actually, we don't necessarily need the heart queen. The spade queen is just as good. And grand slam is worth bidding at matchpoints even if partner has ace doubleton of spades and ace-jack of hearts. Opposite that hand, I can try to drop the heart queen first, then try to run spades if that fails. The combined chances are better than 50%. But I don't know how to find out about either of those hands. So I'll settle for six notrump if we're missing a key card or the heart queen.

Four notrump now would agree the last bid suit, diamonds. So I bid three hearts to set the trump suit. Partner bids three spades, a cue-bid, showing the spade ace. I bid four notrump, and partner bids five clubs, showing three keycards. I bid five diamonds to ask about the heart queen. Partner bids five notrump, showing the heart queen but no kings. I bid seven notrump. Everyone passes, and West leads the deuce of spades.

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

West	North	East	South
Robot	Robot	Robot	Phillip
Pass	1 ♥	Pass	1 ♠
Pass	2 ♥	Pass	3 ♣
Pass	3 ♦	Pass	3 ♥
Pass	3 ♠	Pass	4 NT
Pass	5 ♣	Pass	5 ♦
Pass	5 NT	Pass	7 NT
(All pass)

If either major suit runs, I have all the tricks. If neither major runs, I have eleven tricks: three spades, three hearts, two diamonds, and three clubs. If I had twelve tricks, I might have a squeeze. But with only eleven tricks, a simple squeeze isn't going to work. So I need to run one of the majors to make this.

I suspect spades are running. A spade lead from a singleton or from queen fourth would be strange on this auction. The lead is probably from three small.

I play low from dummy; East plays the four, and I win with the six. If the lead is from three small, why didn't East play the queen? I can't believe West would lead a spade from queen third, so I suspect he actually did lead a singleton, and East withheld the queen from queen forth.

I don't see how it hurts to cash the spade ace to see if I'm right. I lead a spade to the ace. West plays the five, and East discards the club deuce. West led from queen fourth. That's a strange choice.

We've reached this position, with the lead in dummy.

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

Now I need hearts to run. East's discard of the club deuce is probably from a five-card suit, since robots like to discard count cards. Could East be 1-4-3-5? If so, then I'm going down, since I'm not about to finesse him for the heart jack. But I need to make sure I'm down only one if that's the case. This was not a tough hand to bid, so most of the field should be in seven no trump. If this makes, it will be a little above average. My best chance for a good board is that it doesn't make and I go down fewer than the rest of the field.

I have only eleven tricks if hearts don't break, and I can't set up hearts and get back to dummy without burning a diamond trick. So I'll need an endplay to take twelve. If East is 1-4-3-5, can I arrange that? I can play a diamond to my king and cash the spade king, pitching a diamond from dummy. East can't afford a heart or a club, so he must pitch a diamond, coming down to a singleton. Now I start hearts. If West shows out on the second heart, we'll be down to this position:

	NORTH Robot ♠ -- ♥ Q 10 9 8 ♦ A 6 ♣ 10
		EAST Robot ♠  ♥ J x ♦ x ♣ J x x x
	SOUTH Phillip ♠ J 10 9 ♥ -- ♦ -- ♣ A K Q 5

I can cash the ace of diamonds, stripping East of his last diamond, then play four rounds of clubs to endplay him.

Wait a minute. None of this is necessary. I get two extra tricks if I set up hearts, so I can afford to overtake the diamond king to get back to dummy. Is that right? Five hearts, three spades, and three clubs makes 11 tricks. So yes, I need only one diamond trick to make twelve. I had a blind spot for a moment. Good thing I caught it before leading a diamond to my king.

I play the three of hearts from dummy. East plays the jack. I win with the ace and claim. Making seven.

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

A little above average? Shows what I know. This result is worth 100%. Most of the field is in six heart or six notrump. Those who could count to thirteen bid seven hearts instead of seven notrump. That's not a good choice. Not only does seven notrump score higher; it is also more likely to make.

Does Restricted Choice Work Against Robots?

[Below is an excerpt from a series of Substack articles I'm working on. It wasn't until I started writing this article that the conclusions I draw concerning playing against robots occurred to me.]

In the April 1954 issue of Contract Bridge Journal, Alan Truscott concluded a discussion of a bridge deal with the words "This line of thought, allowing for the defenders having had a choice of plays, crops up in many disguises."

The line of thought Alan was referring to had been introduced to the bridge world three years earlier in an article in Bridge Magazine by the mathematician A. O. L. Atkin. But it did not become widely known until Terrence Reese discussed it in his book The Expert Game, in 1958. There he devoted an entire chapter to the concept and gave it the name The Principle of Restricted Choice.

The applications of Restricted Choice go well beyond the game of bridge. It can provide intuition for understanding a number of apparent probability paradoxes. But its proper application can be tricky at times. In this series of articles, I shall examine some of these "paradoxes" and show how Restricted Choice can shed light on them.

[At this point, I introduce Restricted Choice, using Atkin's original example:

	NORTH ♠ K 10 7 6 2
	SOUTH ♠ A 9 6 3

After explaining enough of the rules of play to allow the non-bridge player to follow the argument, I show how, when East drops the jack under ace and West follows low at trick two, Restricted Choice shows that finessing is better than rising by a factor of 2 to 1.]

There is an issue one might raise with applying Restricted Choice to this problem. Let’s consider what happens on trick two. You lead the three and West plays the eight. If West has both the queen and the eight, he has a choice of cards to play. Thus, when he plays the eight, you might reason that he is only half as likely to have queen-eight as to have just the eight. We already decided that Case QJ is half as likely as it was before play began. Now the same is true of Case J. So it’s even money whether to finesse or rise.

What’s wrong with this argument? It doesn’t hold because of two constraints that were not explicitly stated but that a bridge player would assume: 

Constraint 1: The defenders (East and West) have a goal of preventing declarer from taking five tricks. 

Constraint 2: The defenders assume that declarer cannot see their cards.

When declarer leads toward dummy’s ace-ten and West holds queen-eight, West’s only hope of scoring a trick is to play the eight and hope declarer plays the ace. He has no way to win by playing the queen. Because of Constraint 1, his choice is just as restricted as East’s but for a different reason. So there is no adjustment to the likelihood of Case J.

Constraint 2 is also important. If West thinks declarer can see his cards, he will think it makes no difference which card he plays. So if we remove Constraint 2, West has no reason to prefer one card over the other, and his choice is no longer restricted.

We took these constraints into account when we specified the rules for the defenders’ play in the proposed simulation. It’s worth noting, however, that Constraint 2 does not necessarily hold any more. We now have robots that play bridge. While robots are programmed to try to thwart declarer in his goal (satisfying Constraint 1), they make an assumption humans do not: that declarer can see their cards.

Why? Because programming computers to play bridge is hard, and no one has yet found a suitable algorithm that does not require this assumption. The code for robot play is not open source, so I can’t say for sure that a robot West would randomize his play with queen-eight. But I have seen them rise with the honor in this situation, so I believe they do. If so, then the odds for rising and finessing are approximately the same. 

I say approximately because, as we said earlier, Case QJ is slightly more likely than Case J. The difference didn’t matter when the odds were 2 to 1, but now it does. So rising is actually the percentage play by a small margin.

It's worthwhile emphasizing the reason Restricted Choice applies differently to robots and to humans. Some assume, before hearing my analysis, that I am going to claim they don't randomize properly. As we saw in the Prior Strategy discussion [part of the redacted section above] whether an opponent in fact randomizes his play doesn't matter. What matters is that he might. 

The critical factor in deciding whether to apply Restricted Choice is whether an opponent has a choice of plays or whether his play is restricted. This restriction could be because he has no other card to play, or it could be because his alternative is proscribed by the logic of the situation. The latter consideration is what matters here. Because of a glitch in the way robots "think" about bridge, they have a choice of plays in situations where a human does not. That's where the difference lies.

Free Weekly Instant Tournament - October 25 - Board 3

Board 3
Opponents vulnerable

♠ J 5 4  ♥ K 10 9  ♦ A Q 8 6 3  ♣ A Q

I open with one notrump. LHO bids two hearts, showing hearts and a minor. Partner bids two notrump, lebensohl (a puppet to three clubs). I dutifully bid three clubs and partner passes.

RHO now comes to life with three hearts. I can't imagine why he let me find out what partner's suit was before bidding three hearts. Bidding three hearts immediately must be better.

I pass, LHO passes, and partner balances with three spades. See? If RHO had bid three hearts the first time, partner wouldn't be able to bid three spades. The delayed raise gave partner a chance to show both his suits.

RHO doubles three spades. Partner should have four spades and longer clubs. The four-three spade fit doesn't look appetizing, so I'll correct to four clubs. But, with ace-queen of clubs to fill out partner's suit, I might as well bid three notrump on the way. If partner doesn't fancy three notrump, he can always pull to four clubs.

I bid three notrump and partner pulls to five clubs. Partner had no game interest originally but now, after his LHO showed a stack in his second suit and I've shown wastage in hearts, he's suddenly willing to try for eleven tricks?

For some reason, the opponents don't double. Everyone passes and West leads the three of spades.

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

West	North	East	South
Robot	Robot	Robot	Phillip
			1 NT
2 ♥	2 NT	Pass	3 ♣
Pass	Pass	3 ♥	Pass
Pass	3 ♠	Double	3 NT
Pass	5 ♣	(All pass)

Pulling to five clubs was an unfortunate decision. We have nine tricks off the top in three notrump. Ten if they lead a heart. This contract I'm not making.

The spade lead is either a singleton or three-deuce doubleton. If it's a singleton, I can hop with the ace and draw trump, reaching this position with the lead in dummy:

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

Now I lead a heart to the nine. If West has the heart ace, as seems likely, he is endplayed. He must give me a tenth trick in one red suit or the other for down one. If the lead was a doubleton, however, the endplay won't work. He will win the heart and exit with a spade for down two.

What happens if I duck the spade at trick one?  If it's a singleton, East can win and give his partner a spade ruff. But that's OK, since he's ruffing a loser. After ruffing, West will exit with a trump. I will win, draw trump, and execute the same endplay. I still get ten tricks.

What if I duck the spade and East wins and returns a heart to get his partner off the endplay? I play the nine, and West wins with the queen or jack. We've now reached this position with West on lead:

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

If West plays another spade, I can win in dummy and play a third spade, eventually ruffing a spade to my hand for my tenth trick. And if, instead of a spade, he shifts to a trump, then I can draw trump and lead a spade toward my jack for my tenth trick.

In short, hopping with the ace works if West has a stiff spade, but ducking works whether he has a stiff or a doubleton. (Assuming you define "works" as getting out for down one, which seems like the best I can hope for.)

I play a low spade from dummy. East wins with the queen and returns the deuce. I hop with the jack and West ruffs with the four of clubs. He doesn't give me the satisfaction of endplaying him, however. He cashes the heart ace, then shifts to a club. I claim the rest. Down one.

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

Minus 50 is worth 64%. That's pretty generous for going minus when we're cold for a game. Five clubs was played at a few tables. Most declarers rose with the spade ace at trick one. While not best, that works as the cards lie. Or it should work. After that start, they failed to find the endplay and went down two. 

Some sat for three spades doubled, which seems like an odd decision. Two of them made it after a poor defense. But most were down several.

Free Weekly Instant Tournament - October 25 - Board 2

Board 2
Our side vulnerable

♠ K J 10 2  ♥ A Q 5  ♦ Q 9 6 3  ♣ A Q

RHO passes. I open with one diamond and partner bids one spade. 19 support points is worth a four-spade bid. But the ace-queen in my short suit isn't pulling full weight, nor is the unsupported queen of diamonds. And I have six losers. This hand doesn't merit driving to game. I bid three spades.

Partner bids four notrump. I bid five clubs, showing my three keycards. Partner bids five diamonds to ask about the queen of trumps.

If I bid five spades and partner passes, then we are off an ace and the queen of spades. Am I unhappy if that happens? Should I lie and say I have the queen to make sure we get to slam?

One doesn't normally lie about holding the trump queen unless you are known to have a ten-card fit. But holding the jack and ten of spades may make this hand an exception. If partner holds five spades and passes, we've missed a 52% slam. If he has four spades and passes, we've missed a 50% slam. 

Actually both of those percentages are slightly overstated. The opponents might start the defense with ace and a ruff. Or, if partner has four trumps, we might run into a five-zero break. So slam is good, but only marginally so, if partner has five spades and against the odds if he has four. 

He is more likely to have five spades than four, since, with four, he needs a better hand in high cards to bid Blackwood. In other words, the set of hands where he will bid Blackwood is larger when he five spades. So if small slam were the only consideration, it's probably right to lie. But I can't be sure we're off an ace. If we aren't, partner is intending to bid seven if I show the spade queen. I certainly don't want to get to a marginal grand slam. It's close, but I think the odds favor telling the truth.

I bid five spades, and partner passes. RHO leads the three of hearts.

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

West	North	East	South
Robot	Phillip	Robot	Robot
Pass	1 ♦	Pass	1 ♠
Pass	3 ♠	Pass	4 NT
Pass	5 ♣	Pass	5 ♦
Pass	5 ♠	(All pass)

Partner has five spades, so slam is a favorite. Nothing I can do about that now. All I can do is take as many tricks as I can in this contract.

Some players think that, in a situation like this, you should take an anti-percentage play in spades, taking a finesse rather than playing for the drop, in the hope that six spades is going down. That's faulty reasoning. You aren't competing against the pairs in six spades. You are either going to beat them or lose to them, and nothing you do at your table will change that. You are competing only against the other pairs who didn't reach slam, and your best chance to beat those pairs is to take your percentage play. So I'm winning the heart and cashing two top spades.

What do I do after that? What I would like to do is sneak a diamond through. If I can, then I can pitch my last diamond on dummy's hearts and avoid a diamond loser. I don't want to play a side suit to get to the right hand for the diamond lead. That would give the defense unnecessary information. So I need to decide now which hand I want to be in after I cash the spades.

Which hand is more likely to have the diamond ace? West might have led the diamond ace if he had it. He might be afraid to lose it, or, if he has diamond length, he might hope his partner has a singleton. That's not a lot a go on, but it's all I've got. So I want to end up in dummy to lead toward my hand after cashing the spades.

I play low from dummy on the heart lead. East plays the nine, and I win with the king. I cash the spade ace--six--deuce--five. Now a low spade--three--king--queen. Unfortunately, six spades is making. I lead a low diamond from dummy as planned--four--king--five. I claim the rest. Making seven.

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

A few players did reach slam by lying about the trump queen. And perhaps they were right to do so. It's close.

The overtrick was important. Plus 710 is worth 46%. Plus 680 would have been worth 29%. Only one player decided to finesse the spade because he missed a slam. He scored 650 for a zero.

Free Weekly Instant Tournament - October 25 - Board 1

Board 1
Neither vulnerable

♠ K 5  ♥ A 9  ♦ K 7 6 4 2  ♣ A 9 4 2

Two passes to me. I open with one diamond, and partner bids one spade.

We had a similar deal a few weeks ago where I chose to rebid one notrump instead of my lower-ranking four-card suit, and I received a number of objections. (Not well-articulated objections. Simply comments like "One notrump is wrong.") On that deal one could argue for either rebid. This time I think one notrump is clear. Half my high cards are in my short suits. And I have decent secondary support for spades. If we belong in two spades in a five-two fit, we won't get there if I bid two clubs.

I bid one notrump, and partner bids two hearts, non-forcing. I correct to two spades. I'm glad I rebid one notrump. If partner is 5-4-1-3, he would pass two clubs, and if he's 5-4-2-2, he would correct to two diamonds. I suspect in either case I would rather play in two spades.

But partner doesn't pass two spades. He bids three hearts, invitational with a fifth heart. I have a maximum in high cards and two fitting honors in partner's suits. But having no third card in either suit is a liability. We are probably high enough at the three level. I correct to three spades and partner passes. RHO leads the five of hearts.

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

West	North	East	South
Robot	Phillip	Robot	Robot
			Pass
Pass	1 ♦	Pass	1 ♠
Pass	1 NT	Pass	2 ♥
Pass	2 ♠	Pass	3 ♥
Pass	3 ♠	(All pass)

It looks as if I'm going to lose two hearts, the diamond ace, a club, and possibly a spade. I could conceivably set up the diamond king for a club pitch. But I'm in danger of losing control. It's not clear I can afford the tempo of playing a diamond to the king.

I don't think East is leading a low heart from honor doubleton or from both honors. So there isn't much point in finessing the nine. I rise with the heart ace. East drops the queen, and I play the deuce.

The four of hearts is still out, so it appears the lead was from a five-card suit and the queen was a singleton. If I pick up the trump suit and drive the king of hearts, I can take five spades, three hearts, and the club ace for nine tricks. If West has five hearts, it's likely East has spade length, so my best play in spades is to cash the king and finesse the ten. That is, assuming I can handle four-two spades. Can I?

After four rounds of spades, say I lead a heart to the nine and West ducks. I need to get to my hand twice--once to drive the heart king and a second time to cash my heart. I have only one trump left, so I'll need to find the club king on my right and use the club queen as one of my entries. If spades are three-three, I don't need the club king onside.

Even though the finesse is the right play in the spade suit, the fact that it works only half the time even when it's right makes playing for the drop better. Playing for the drop works any time the suit is three-three or when West has jack doubleton and the club king is onside.

I cash the king of spades--eight--three--nine. Why are they playing their high spades? I lead the five of spades from dummy, and East plays the six. Would East play 86 from J864 or J862? What if his partner had Q9? I would win the second trick with the ace and could now drive East's jack with my 107x. If East had saved the eight, he would have a second trick.

East can't afford the eight from jack fourth, so even if I were intending to finesse the ten, I would now change my mind. 

I play the ace; West plays the four. I cash the queen--jack--diamond from dummy--deuce. This deal is a good illustration of the folly of blindly falsecarding. You must decide whether you want to feign shortness or length. If you want to feign length, as here, you must play low cards, since you often can't afford high cards from length.

While East's play of the eight was an error, he might have gotten away with it if his partner hadn't outed him. East could afford the eight from J98x, but West's play of the nine told me that wasn't the case. While it turns out I wasn't going to finesse, the defense didn't know that. The two errors combined might have talked me into the winning play.

I'm up to nine tricks. Can I afford to try a diamond to the king? Say I lead a diamond to the king and ace. East taps me with another diamond. I lead a heart to the nine and it holds. Nope. I'm in trouble. I can't afford to play a diamond. I have to lead a heart to the nine.

I play a heart. Dummy's nine wins and East discards the three of diamonds. That looks like a five-card diamond suit, making West 3-5-2-3. Here is the current position, with the lead in dummy:

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

West presumably has K87 of hearts, two diamonds, and three clubs remaining. If he has the club king, can I endplay him? I can't let East in to put a club through, so I must hope West has the diamond ace as well. Say I lead the diamond king off dummy to West's ace. If he exits with his diamond, I can endplay him. So he must exit with a low heart instead. I win and cash a spade, and West pitches a club. We've now reached this position:

	NORTH Phillip ♠ -- ♥ -- ♦ 7 6 ♣ A 9 4
WEST Robot ♠ -- ♥ K 8 ♦ x ♣ K x
	SOUTH Robot ♠ 7 ♥ J 6 ♦ -- ♣ Q 7

When I cash my last spade, West simply pitches a heart, giving me a heart trick I can't make use of. There is no way I can extract that last diamond, so I can't endplay him.

Perhaps I'm better off hoping East has ace-queen or ace-queen-jack of diamonds and the club king. If I play a low diamond from dummy, he may hop with an honor for fear I'll score my singleton, then exit with a club, playing his partner for the queen.

That seems like my best option. I play the four of diamonds from dummy. East plays the five. That didn't work.

West wins with the jack and continues with the queen of diamonds. I doubt if he would do that with the club king, but it's my only chance. I ruff in my hand and lead the jack of hearts. West wins and plays a club. I duck, and East wins with the king. Making three.

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

Plus 140 is worth 79%.

If you rebid two clubs instead of one notrump, partner bids three hearts, natural and invitational. So three spades would appear to be the normal spot. Strangely, only one other declarer played it, and he went down three. Half the field was in three notrump, which three declarers managed to make. The other half was in three or four hearts, down several.

Free Weekly Instant Tournament - August 30 - Board 8

Board 8
Neither side vulnerable

♠ A K 6 5 3  ♥ J  ♦ A K Q 9  ♣ 7 6 2

RHO opens with one diamond. I overcall with one spade, hoping to make a take-out double of hearts on the next round. LHO makes a negative double, and RHO bids two clubs.

This auction didn't go as planned. Partner's best suit is apparently opposite my singleton, and I have excellent defense. So it looks right to stay out of the auction.

I pass, and LHO corrects to two diamonds. Good. I have even better defense against this contract.

Partner bids two spades. Presumably he has three-card support and not enough for a raise on the previous round. RHO and I pass. LHO, however, hasn't had enough. He bids three diamonds. Partner hasn't had enough either. He bids three spades. So he has four spades and not enough to raise on the first round? With that hand, he should have bid three spades over the double. If he's willing to bid to the three level, getting there as quickly as possible puts more pressure on the opponents. This auction should not exist.

Partner obviously has a stiff diamond. With that, plus four-card spade support, he can hardly have anything else, given his first-round pass. The opponents presumably don't have an eight-card heart fit, so partner's likeliest shape is 4-5-1-3. That means we are off three club tricks and a heart. If West leads a diamond, I may be able to dispose of some of dummy's club losers. But a diamond lead is unexpected on this auction. It's normal to lead opener's second suit, where you are more likely to develop tricks.

It feels strange to take one bid and subside with such a good hand. But three spades rates to be the limit. I pass. West leads the three of diamonds.

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

West	North	East	South
Robot	Robot	Robot	Phillip
Pass	Pass	1 ♦	1 ♠
Double	Pass	2 ♣	Pass
2 ♦	2 ♠	Pass	Pass
3 ♦	3 ♠	(All pass)

4-5-1-3 as expected. We have four losers off the top. But after this lead, I may be able to dispose of at least one of them.

As I said, a diamond lead is unexpected on this auction. North isn't competing to the three level with a flat hand. He rates to have diamond shortness, so one diamond trick is all the defense is getting, and there is no hurry to cash it. There is a serious danger, however, that I will be able to pitch dummy's losers on diamond tricks in my hand. It's true, my diamonds don't have to be as good as they are. But even if I have as little as the king of diamonds, a diamond lead will set up a pitch. So a club, partner's second suit, rates to be a better choice.

One reason West might choose not to lead a club is that he holds the club ace. Now a diamond lead, hoping to reach partner with the ace for a club play, makes some sense. But on this particular auction, a heart lead is a better choice. 

If dummy weren't broke, you might worry that declarer could take pitches on dummy's hearts. But dummy is broke, so it is unlikely to provide any discards. The most likely way for defense tricks to disappear is for declarer to pitch dummy's losers on his diamonds. If partner has the diamond ace, declarer must lead the suit himself to take his pitches, so there is no need for you to lead the suit. But if partner has the heart ace instead, a heart lead for a club through may be necessary.

Back to the deal. East plays the diamond ten, and I win and cash two more diamonds, pitching clubs from dummy, and ruff the fourth round. Diamonds were four-four as one would expect. East began with jack-ten fourth.

It's unlikely I'll manage to ruff two clubs in dummy, but I can ruff one to bring me up to ten tricks. I play a club from dummy. East wins with the queen; West drops the jack. East now, predictably, switches to a trump.

Here is the current position, with South to play after East's trump shift.

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

Is there any way I can ruff both clubs? I need East to have a stiff trump so he can't play another one. Since West has four hearts for his negative double, that would make East 1-3-4-5. Some would open one diamond with that shape, although I don't think the robots would. Even if he has that shape, however, he can duck the heart when I lead one off dummy and let his partner gain the lead for a second trump play.

I don't have much hope this is going to work, but I ride the trick around to dummy's queen and play a heart. East hops with the king and plays a second trump. I get only one ruff. Making four.

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

Plus 170 is worth 64%. A couple of players did reach four spades (one made it; one went down), but on auctions starting with a double. I don't understand doubling one diamond with a 5-1-4-3 pattern. When you can bid a five-card major at the one level, that is probably the right start. At some point your hand may be good enough that you have no choice but to double for fear of missing a game. But this hand isn't even close to that point.

West did have the club ace as I suspected. With ace-jack-ten, I can hardly blame him for not leading the suit. But I believe a thoughtful West would choose a heart over a diamond on this auction, so I don't mind "missing" this game.

Free Weekly Instant Tournament - August 30 - Board 7

Board 7
Both sides vulnerable

♠ J 8  ♥ K J 9 6 3  ♦ A 10 9 3  ♣ K 8

I open with one heart in first seat, and partner bids one spade. I can rebid either two diamonds or one notrump. If I bid two diamonds and partner corrects to two hearts with a small doubleton, I'll wish I had bid one notrump. If I bid one trump and partner passes with a stiff heart and a diamond fit, I'll wish I had bid two diamonds. It's not at all clear what to rebid for purposes of playing the right partscore. But if we reach game in notrump, it's probably better to play from my side. So I bid one notrump.

Partner raises to two notrump. I have no reason to accept; we may be too high already. I pass, and West leads the four of diamonds. It's usually a good sign when you decline to bid a suit and the opponents lead it.

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

West	North	East	South
Robot	Robot	Robot	Phillip
			1 ♥
Pass	1 ♠	Pass	1 NT
Pass	2 NT	(All pass)

Partner has the hand I was worried about. If I had bid two diamonds, I suspect partner would have raised to three and we would play it there. Fortunately partner has enough in high cards that notrump may actually be a better strain than diamonds. If his hand were weaker, diamonds would be better.

I play low from dummy, and East discards the three of spades. I guess we did well to avoid three diamonds.

I have four diamond tricks and two club tricks. I need two more. I can get two spade tricks if I find the nine of spades. The spade discard is probably from a five-card suit. So it's tempting to let dummy's five of diamonds hold and lead a spade to the eight. If the finesse works, I'm home. 

The downside to this approach is, if the finesse loses, I've not only conceded a trick to the opponents they were never going to score themselves, but I've also lost a tempo. I'll need to give up the lead twice more in spades, and, with three entries, the opponents have plenty of chances to find three additional tricks.

The alternative is to drive the ace and king of spades and hope to find an eighth trick somewhere. There are several possibilities. The nine of spades may drop. I may guess the hearts. And the opponents may present me with a heart or a club trick trying to develop tricks for themselves.

In general, it's not a good idea to stake the contract on a committal play at trick two when you have other chances. Still, the odds are five to two that this particular committal play will work. Perhaps better than that, since East, with ace-king of spades, may hop when I play a spade from dummy. In addition, if we assume East discarded his lowest spade from five, then the only way the finesse can lose is if West has specifically nine-deuce.

I decide the odds merit a committal play at trick two. I win the first trick in dummy and play a low spade. East plays the spade six, and my eight drives West's ace. I have eight tricks once I drive the spade king. Now I'm playing for overtricks.

West shifts to the club queen. That would be a strange play from queen-jack, looking at ace-ten in dummy. It could be from queen doubleton, or it could be from queen-nine third. (If you choose to lead the suit from that holding, the queen is the correct card. Leading low or the nine costs a trick if I have king-eight doubleton or third.)

I play a low club from dummy and win in my hand with the king as East plays the six. Now the jack of spades--four--seven--king. The four? That's strange. That means East gave false count at trick one. He either started an echo with K9632 or played low from K963. In the middle of the hand, the robots tend to discard honest count cards. But I have noticed that they don't always do so when showing out at trick one. Since I doubt East would have discarded a spade from K963, I'm going to continue to assume he started with five. Although I have to take back my claim that the finesse of the eight would lose only to nine-deuce.

We've reached this position, with East on play:

	NORTH Robot ♠ Q 10 ♥ 8 ♦ K Q 6 ♣ A 10 7
	SOUTH Phillip ♠ -- ♥ K J 9 6 3 ♦ A 10 9 ♣ 8

East shifts to the seven of hearts. If I guess the hearts, I'll make an overtrick. Do I have any clues? West's shift to the club queen suggests he believes he has an entry. There is no point to that play unless he can get in to lead another club. That's an indication he has the ace of hearts. It's not much of an indication, since the heart queen could be an entry as well. But it's all I have to go on. I play the jack, and West takes the ace. I have nine tricks now.

Surprisingly, West doesn't continue clubs. He returns the deuce of hearts. I pitch the seven of clubs from dummy, East plays the heart queen, and I win with the king. 

Why is West continuing hearts when a club shift looks routine? Perhaps he is trying to cash his partner's king of hearts? But he knows my shape. Even if his partner does have the heart king, it isn't going anywhere unless I have king-jack tight of clubs. It's fine to cater to an unlikely layout if you can afford to. But West can't afford to unless he holds the heart ten. If I hold king-ten, a heart continuation will pick up East's queen and given me the rest of the tricks. So, unless West is being careless, the heart shift marks him with the ten.

We have now reached this position:

	NORTH Robot ♠ Q 10 ♥ -- ♦ K Q 6 ♣ A 10
	SOUTH Phillip ♠ -- ♥ 9 6 3 ♦ A 10 9 ♣ 8

I have all but one of the remaining tricks. Can I execute a squeeze to take the rest? Not if East holds the club jack and West holds the heart ten, as I assume. What if I'm wrong about the heart ten? If East holds both of them, I can cash all my tricks ending in my hand, and he is squeezed. But West could see the squeeze coming. If that were the layout, he would have shifted to a club to break it up. 

My only chance, then, is that I'm wrong about the club jack. Could West have shifted from queen jack of clubs after all, perhaps hoping his partner had the king? I can't afford to risk the finesse, but I don't have to. I can run my winners, coming down to:

	NORTH Robot ♠ -- ♥ -- ♦ -- ♣ A 10
	SOUTH Phillip ♠ -- ♥ 9 ♦ -- ♣ 8

with the lead in my hand.

West must keep his putative heart ten. So if he has the club jack, he's caught in a show-up squeeze. 

I play a diamond to dummy. East discards the five of hearts. On dummy's spades, East follows, and West discards the seven of diamonds and the nine of clubs. 

On the diamond queen, East pitches the heart four. The ten is the only heart remaining. I play a diamond to the ace. East pitches the club four, and West follows with his last diamond.

We've reached the two-card ending above. West presumably holds the heart ten and one club, and East holds two clubs. I lead the eight of clubs, and West follows with the deuce. 

If my construction is correct--and I am quite confident it is--then it makes no difference what I do. If I finesse, East will win and return a club to dummy's ace. But you should never assume your play makes no difference. So I must assume I'm wrong about the heart ten and East holds it after all. If that's the case, is East's remaining club the jack or a small one?

I already decided West would have broken up the squeeze if one existed. So if East does unexpectedly hold the heart ten, his club must be a small one. I finesse. East takes his jack and returns a club. Making three.

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

Plus 150 is worth 82%. Note West did indeed make the nice play of shifting to the club queen from queen-nine third.

No one is in three diamonds. If you rebid two diamonds, partner bids two notrump instead of raising. So, except for the three optimists who raised two notrump to three and the one pessimist who passed the board out, everyone is in two notrump, taking anywhere from six to nine tricks.