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.