Using All the Information Available

Today we look at a 1 NT hand. We will see how declarer uses all the information that is available to make a brilliant choice for a play at the end of the hand.

Norm is the dealer and bids 1 diamond with this hand:

Norm
♠ J8
♥ T986
♦ AT74
♣ AK8

He has 12 points in high cards and can count one distribution point for the second suit. He has no 5-card major, so he bids his better minor suit.

Edie is sitting to Norm’s left and was surprised that Norm opened the bidding. She holds:

Edie
♠ AK6
♥ A75
♦ KQJ85
♣ 64

Without Norm’s opening bid, she was trying to decide whether to open 1NT or 1 diamond. She has the balanced distribution and the required 15-17 HCP to open 1NT, but she was worried about the worthless doubleton. One diamond is a safer opening bid, and she could jump in diamonds at her next turn to show her strength. Then, Norm goofed up her plan with his opening 1 diamond bid. Now, bidding diamonds is out of the question. Her only alternative to bidding 1NT is to pass. She decides to risk the club trouble and overcall 1NT.

Sally happily passes with this hand:

Sally
♠ Q954
♥ QJ3
♦ 9
♣ QT753

Had there not been the 1NT overcall, Sally would have a demand bid with 6 points opposite her partner’s opening bid. She doesn’t have the required 10 points to show her club suit, therefore she would have responded with 1 spade. The overcall, however, relieves her of her obligation to keep the bidding open for her partner. A free bid over the 15-17 points shown by Edie would need to guarantee that her side had the balance of the strength in the hand. She should probably pass any hand weaker than 10 points. She does.

Walt holds

Walt
♠ T732
♥ K42
♦ 632
♣ J92

With 8 or more points, Walt might look for a 4-4 spade fit with the Stayman convention, but he has only 4 points. He passes.

Norm has just a minimum opening hand and no help from his partner. He passes also.

Sally is on lead against 1NT. She wants to find the strongest combined holding for her side. He first consideration is the suit that her partner bid, but his 1 diamond opening might be based on a 3-card suit, Sally holds only one card there, and the 1NT overcall should have the opener’s suit stopped at least once, preferably twice. Sally needs to look elsewhere for an opening lead.

Sally
♠ Q954
♥ QJ3
♦ 9
♣ QT753

The clubs are longer than the spades, however, Norm indicated that his diamonds were better than his clubs. A spade lead is worth considering, but Sally decides to go with clubs. Having just two honors at the top of the suit, she leads her 4^th highest—the 5.

Now, Edie pauses to make a plan for the hand before playing to the first trick.

Dummy ♠ T732 ♥ K42 ♦ 632 ♣ J92

Declarer ♠ AK6 ♥ A75 ♦ KQJ85 ♣ 64

In no trump, Edie starts by counting how many tricks she can take off the top. Well, with the club lead, that number is zero. She can use the rule of 11 to reason that there are 6 cards outside of Sally’s hand that are bigger than the 5 of clubs. Edie can see 3 of them, so Norm must have 3 clubs bigger than the 5. That’s good. It means that the opponents can run at most 5 tricks in clubs. Let’s suppose that happens and the 6^th trick is either a spade or a heart. Now, Edie can take 4 tricks in spades and hearts. She will need to take 3 more tricks to make the contract. The diamonds look like a good choice. Edie will need to force out the ace of diamonds, probably losing one of her top honors in the process. Then, she will have two good honor cards to cash. If the diamonds break 3-2, her two small diamonds will be set up, and she’ll make an over trick. If they break 4-1, the 4 cards will certainly be in the hand that opened 1 diamond. She might be able to win a trick with a deep finesse. She might not have the luxury of just losing the 4^th diamond trick to set up the little one. If the opponents lead hearts at every opportunity, Edie will be out of heart winners after losing the 4^th heart, and she will lose a bunch of hearts before she can cash her little diamond. Still, it looks like she has a solid plan.

Edie decides to play a small club from dummy. Norm wins with the ace, cashes the King of clubs and returns a small club.  Edie discards a small heart. Sally takes with the Queen and continues to cash her small clubs. Now Edie needs to be careful with her discards. She needs to preserve two small diamonds in dummy in case she needs to try a fancy finesse if there is a bad diamond break. She discards a small heart from dummy. Norm plays a small heart and Edie finds herself in a tight spot. She discards a spade, and plans to discard a small diamond on the next club. There goes one chance for an extra trick in diamonds.

Sally leads her last club. Edie discards a spade. Norm plays a heart, and Edie plays a diamond as planned.

Here is what is left for Edie before Sally leads. Edie hasn’t taken any tricks, yet.

Dummy ♠ T73 ♥ K4 ♦ 632 ♣

Declarer ♠ AK ♥ A7 ♦ KQJ8 ♣

Sally continues with the Queen of hearts. Edie needs to preserve the entry to the board, so she takes with the ace in her hand. Now, she needs to force out the ace of diamonds. She leads the King. Sally follows with her 9. Dummy plays small, and Norm takes the ace. He hopes that he can eventually take the ten of diamonds after the Queen and Jack are played. He decides to take out declarer’s only entry to the board by leading a heart.

The King wins on the board. Now, Edie leads a small diamond from the board and Norm plays the 7. If the diamonds break 3-2, Edie can take the Q and J of diamonds and cash the last diamond. However, if they break 4-1, this play will set up the ten for the opponents. She could, instead, decide to finesse the 8. That play works against a 4-1 break, but loses if Sally started out with the T9 of diamonds. Edie knows that Norm started with at least 3 diamonds based on his opening bid. The only way that the finesse fails is if Norm opened 1 diamond with the A74 of diamonds and the AK8 of clubs. Since Edie was paying such careful attention to the cards that Norm had played and could visualize the hand he started with, she bravely plays the 8 of diamonds on this trick.

The 8 holds. Edie breathes a sigh of relief, and she cashes the rest of her tricks to make 1 NT.

Where did these point ranges come from?

Most bridge players are familiar with "rules" like these:

• 13 points to open the bidding
•  6 points to respond to opener
• 26 points for a game in a major suit or no trump
• 28 points for a game in a minor suit
• 33 points for a small slam
• 37 points for a grand slam

Did you ever wonder where they came from?

Consider the whole deck. Each suit has 10 HCP in it, so there are 40 HCP in the deck. There are 13 tricks in the hand, so one trick is worth about 40/13 points, or slightly more than 3 points. In other words, you can expect to take a trick for every 3 points in your hand. Let's examine some of these familiar point ranges:

 

13 points to open

With 4 players and 40 points in the deck, a 10-point hand is an average hand. If your hand is 1 trick stronger than average, you should open the bidding: 13 points.

6 points to respond to opener

Your partner's opening bid is at least 13 points, but it might be much stronger. If you have 6 points, it is likely that your side has the balance of the strength (more than 20 points between you). If that is the case, the contract likely belongs to your side. You should keep the bidding open for your partner.

26 points for a game in a major or no trump

If your side has 26 points, that is 2 tricks more than average. Half the tricks is 6 1/2. Two more is 8 1/2. OK, that's a little bit of a stretch. Still, it seems to work out. Also, the scoring bonus for making a game is usually better than the penalty for going set. It's worth stretching to bid and maybe make a game.

We require the same total strength for a no trump game (9 tricks) as for a game with a trump suit (10 tricks). That is because it is usually a little easier to take tricks with a trump suit.

28 points for a game in a minor

That seems like only 2/3 of a trick more than the requirements for a game in a major. We're stretching, again.

33 points for a small slam

With 40 points in the deck, you don't expect to be missing 2 aces if your side has 33 points. You might notice that we require 5 more points to bid six than we needed to bid five. Here, the bonus for making a slam is not so lucrative that you want to risk losing the bonus for making a game.

33 points for a small slam

With 40 points in the deck, you don't expect to be missing 1 ace if your side has 33 points. 
 

An Easier Slam

Today, we’ll look at a small slam that is easy to bid and easy to make. Not every lesson needs to be hard.

	Norm ♠ A98 ♥ A983 ♦ AK6 ♣ A52
Walt ?	Dealer: Edie W N E S     Pass 1D Pass 2H Pass 3H Pass 6H Pass Pass Pass	Edie ?
	Sally ♠ QJ6 ♥ KQJ7 ♦ Q9875 ♣ 4

 

The Bidding

For this lesson, the opponents’ hands don’t matter much, so I haven’t shown them. Edie passes and Sally chooses to open the bidding. She has 11 HCP and can count 2 points for distribution—one for the 5^th diamond and one for the second suit. Walt passes. Norm has 19 HCP and balanced distribution. No trump seems like a good place to land, but he doesn’t have an appropriate no trump response. Remember that no trump bids generally show a specific point range. Point ranges for trump responses to a one-of-a-minor opening are:

1NT: 6-9 HCP

2NT: 13-15 HCP

3NT: 16-18 HCP

Another issue is that Norm should be thinking about slam. His partner should have at least 13 points and he has 19. Just one more point makes the 33 points needed for slam. Sally could easily pass the 3NT response with a minimum opener. The solution is to make a jump shift. The jump shift by responder shows 19+ points, is forcing to game, and strongly suggests that slam is possible. Norm bids 2H. Edie passes.

Sally is happy with hearts. She responds 3H. That bid doesn’t give any additional information about the strength of her hand. It just shows a heart fit. Walt passes.

Now, Norm has a difficult decision to make. He is happy to have found a heart fit, but he still likes no trump. The problem with no trump is that 3NT is likely to be passed and 4NT is the Blackwood ace-asking convention. Norm needs to give up on no trump. He needs to be happy with finding a heart fit.

Now, how does Norm figure out if their side has enough strength for a small slam? How about a grand slam? What about all the holes in his hand? Let’s consider the last question first. Sure, it looks like his hand is only good for 5 tricks, but Sally has to have her 13 points somewhere. That’s why you consider the combined strength of the two hands when deciding where to place the contract. There really isn’t any way to figure out if Sally has the 14 points needed to add to Norm’s 19 to get to 33. He just needs to take a chance. There isn’t any way for him to figure out if Sally has the 18 points needed to get to the 37 needed for a grand slam either. Norm is going to need to leave that up to Sally.

What about Blackwood? Well, there is no information to be gained from asking for aces. Norm has them all. He could bid 4NT, knowing what Sally’s answer will be, and then bid 5NT to see if she has the missing 3 kings. What does he do if she answers with 2 kings? It’s hard to say. Norm just skips all the rigmarole and bids a small slam.

The bidding isn’t over, yet. Sally knows that Norm has at least 19 points. If she happens to have 18 or more points, she can raise the 6H to 7H on her own. She knows that Norm hasn’t learned about her extra strength yet, and his first jump-shift bit has told her that the team has enough strength for grand slam.

The Play

Edie is on lead and starts with a small club. Norm starts by counting his losers. He can only afford to lose one trick.

	Dummy ♠ QJ6 ♥ KQJ7 ♦ Q9875 ♣ 4
	Norm ♠ A98 ♥ A983 ♦ AK6 ♣ A52

There shouldn’t be any trump losers unless the missing 5 trumps break 5-0. Norm might lose the K of spades. He shouldn’t lose any diamonds, but clubs might be an issue. He could try to ruff two clubs in dummy, but he will need dummy’s high hearts to pull trump. A better plan is to discard the club losers on dummy’s small diamonds after that suit is established. That will work if the 5 outstanding diamonds break 3-2. If all else fails, there is a spade finesse for the king. So, there are multiple ways to make the hand.

Plan A: Pull trumps. If they break 5-0, Norm will lose a trump trick. If that’s the case, he’ll need the favorable diamond split and the spade finesse. Knowing which hand holds the trumps, might open up some fancier plays, but we won’t get into those.

Plan B: Pull trumps. They break 4-1. Now, Norm needs only the favorable diamond break OR the spade finesse. He can try the diamonds first by leading the A and K. If they break 4-1, he leads to the diamond Q and tries the spade finesse.

Plan C: Pull trumps. They break 3-2. That’s the easiest way. After pulling trumps, each hand has a little trump left. If the diamonds break 4-1, Norm can ruff the 4^th diamond in his hand and return to the board by ruffing a club. Then, he can cash the 5^th diamond to discard his losing club.

Let’s play through one of those scenarios. Norm starts by taking the A of clubs and cashes the A of hearts. Both players follow. He leads a small heart to the K on the board. Walt follows and Edie shows out. Norm is on the board with this holding:

	Dummy ♠ QJ6 ♥ QJ ♦ Q9875 ♣
	Norm ♠ A98 ♥ 98 ♦ AK6 ♣ 52

Norm is going to need the QJ of hearts to pull Walt’s trumps. There is still a spade to lose. If Norm finishes pulling trumps before knocking out the K of spades, he can lose a club—having no more trumps in his hand. Therefore, he needs to switch gears and work on the spades.

He tries a finesse. The finesse doesn’t need to work, but it doesn’t hurt to try. Norm leads the spade Q from dummy. Edie follows small. Norm ducks in his hand and Walt takes the K. Now, Walt leads the K of clubs. Darn! Norm can’t afford to use one of the high trumps to win this trick, and he can’t afford to lose the trick either. He’s sunk.

Let’s back up. Suppose on trick 2, Norm leads a small trump to the KQJ on the board. He takes the K and leads the Q. Edie shows out. Now, Norm is in almost the same position, but he still has a small trump on the board that he can ruff a club return with.

	Dummy ♠ QJ6 ♥ J7 ♦ Q9875 ♣
	Norm ♠ A98 ♥ A9 ♦ AK6 ♣ 52

 

Now, Norm tries the spade finesse that loses. Walt returns the K of clubs, but Norm can ruff with the 7 of hearts. Walt still holds 2 trumps. Norm is on the board and cannot afford to lose any more tricks. Once trumps are pulled, he has two spade winners and can run the diamonds as long as they split 3-2. He leads the J of hearts and the 6 of spades to the A. As long as Walt didn’t start with 4 trumps and the singleton K of spades, Norm will win this trick with the A and can pull Walt’s last trump. Then, it is a simple matter of running the diamonds. This play works with an unfortunate 4-1 trump break and a losing spade finesse, as long as the diamonds break 3-2. Those are pretty good odds.

	Dummy ♠ J6 ♥ J ♦ Q9875 ♣
	Norm ♠ A9 ♥ A9 ♦ AK6 ♣ 5

 

Bidding a Slam and Working Hard to Make it

In this post, we will look at how a slam was bid, looking at just one hand at a time. North holds the following hand

North ♠ AK86 ♥ Q98 ♦ QT976 ♣ 8

 

It has 11 points in high cards and can count 2 points in distribution. That is good enough for an opening bid. North bids 1 diamond. South responds with 2 hearts. North thinks, “Wow! That’s a jump shift. My partner has 19 or more points. We are forced to game and could easily be in slam.” Now, North needs to choose a rebid. The best way to approach the rebid is to make the same bid that she would have chosen after a 1 heart response, only at a higher level. South is only promising a 4-card heart suit, so North can’t support hearts. South could have both major suits, though. North chooses to bid 2 spades. South responds with 3 spades. OK, good deal. They have found a trump fit. Now, the team needs to figure out if they have enough strength for a slam. North doesn’t know. If South has a minimum jump shift of 19 points, the team might be a little short of slam. North chooses to show a minimum opening hand by bidding 4 spades. Remember that the jump shift forces North to continue bidding until a game contract is reached, or the opponents are doubled for penalties. The opponents have been silent.

The auction could end here, but South continues with 4 no trump. That is the Blackwood ace-asking convention. North doesn’t have any flexibility in her responses. She just bids according to her ace count. She has 1 ace, so she bids 5 diamonds. South continues the Blackwood convention by bidding 5 no trump to ask for kings. North has 1 king, and bids 6 diamonds. South signs off at 6 spades. North is required to pass, here. South was in charge of the bidding, and has set the contract.

Now, try to forget what North held and look at the bidding from the South hand. South holds

South ♠ QT52 ♥ AJT6 ♦ AK ♣ AQJ

That's 21 points in high cards and 1 in distribution. South was thinking of opening the bidding with 2 no trump, but to his surprise and delight, his partner opened 1 diamond. South is thinking, “Wow! With partner’s 13 points and my 21, we should at least have a small slam. I need to jump shift to show my strength and make sure the bidding stays open.” South responds with the same suit he would have bid with a weaker hand, but one level higher—2 hearts.

North responds with 2 spades. South is delighted. Now, they have found a trump fit. Since the jump shift keeps the bidding open until a game contract is reached, there is no reason to hurry. He bids 3 spades to show that he has 4 of them to go along with partner’s 4. South intends to bid on to reach a slam at his next turn.  North is forced to bid, again, so South will get another turn. North responds with 4 spades. At this point, South expects that his partner has just a minimum opening hand. The team has already reached a game contract, so North can pass at her next turn if she wants. South already knows that the side has enough strength for 6 spades, and he knows that the side is not off two aces. He could just bid 6 spades at this point. However, South tries to picture the sort of hand that North might hold. Her 13 points need to fit in somewhere. If she holds the missing ace and all 3 missing kings, a grand slam is possible. A missing queen of hearts might require a finesse, but grand slam is worth investigating.

South bids 4 no trump, the Blackwood convention, to ask for aces. Even if partner responds with 5 clubs showing zero aces, South will bid a small slam. North replies 5 diamonds, showing the missing ace of spades. South continues the Blackwood convention by bidding 5 no trump. He wants to know if North holds all the missing kings. Some readers might be thinking, “Suppose North responds 6 clubs. Then, you are missing three kings and would need two finesses to make the small slam.” Well, yes, that is true, but North needs to fit 13 points somewhere. With 1 ace and no kings, there are still 9 points to put somewhere. There aren't enough queens and jacks out to make up the difference. 

North responds 6 diamonds. Well, nuts. You are missing 2 kings. South is committed to a small slam at this point, but he was planning to bid at least a small slam from the moment his partner opened the bidding. He bids 6 spades and North passes.

Playing the Hand

West leads the 4 of diamonds. Let’s plan the play of the hand. It might be a challenge.

Dummy ♠ AK86 ♥ Q98 ♦ QT976 ♣ 8

 

Declarer ♠ QT52 ♥ AJT6 ♦ AK ♣ AQJ

 

Declarer starts by counting his losers. With a reasonable split, he doesn’t have any losers in spades. From the view of the declarer hand, he has a potential loser in hearts, and one in clubs. That’s one too many. There is an extra winner in diamonds in dummy, but that won’t get rid of enough losers in the declarer hand. There are 7 diamonds. Should they break 3-3, declarer could discard the queen and jack of clubs on the small diamonds, allowing him to give up a heart loser and making the contract. That possibility is somewhat less likely because West chose to open with a diamond. She might have led from 3 to the Jack. In any case that is one possibility that allows the contract to make. What else?

If the trumps break 3-2, both hands will hold a trump when the 4^th diamond is led. Declarer could ruff the 4^th diamond and get back to the board by ruffing a club.

If East holds the king of hearts, dummy can lead the queen. If East holds the king and covers, all the hearts are good. Declarer can afford to lose the club king. If West holds the king and ducks, declarer ducks with the Jack. We’ll see why he doesn’t play small in a minute. Next, he can lead the 9 of hearts. If East ducks again, declarer plays his carefully preserved 6 of hearts so he can stay on the board for one more heart lead. He leads the 8 of hearts. If East started with 4 cards to the king, he can duck again. No problem. The king will fall on the ace whenever declarer wants to play it. Even if East started with 5 or 6 hearts to the king, declarer will make 4 heart tricks. He should be able to  ruff when East tries to take his king later. OK, that’s one more way to make. What else?

Picking up the king of clubs in a finesse is tricker. Dummy can only lead clubs once. If East holds the king, he can duck the club lead. Declarer plays the queen in the declarer hand and still has a club loser. That’s not a big problem. There should be a trump left in dummy to ruff the jack. OK, that’s one more way to make.

There is probably a better way to avoid a club loser. Declarer could play the ace of clubs and ruff both the queen and jack in dummy. That plan requires both opponents to hold at least 3 clubs, or for the hand with short clubs to also hold at most 2 spades. Declarer will not be able to finish pulling trumps before ruffing the clubs. The opponents hold 9 clubs between them, so getting them to follow suit to three rounds seems reasonable.

There is also the possibility of a ruffing finesse in clubs. Declarer could cash the ace of clubs and lead the queen, intending to discard an off-suit if West doesn’t cover. If West covers, dummy ruffs, setting up the jack.

So far, it looks like declarer will need a lot of bad luck for all these plans to fail. The next thing we need to consider is transportation and pulling trump. We can’t run the diamonds until all the trump are out. We also need an entry to dummy to run them. Likewise, we can’t try the heart play until all the trumps are out, and we need an entry to dummy.

Of all the plays, the choice of ruffing two clubs in dummy seems most likely to succeed. It only requires the 9 clubs that the opponents hold to break no worse than 6-3. On the other hand, it requires two leads to the declarer hand before trumps are pulled. The finesses are only 50-50 chances, but they both need to fail for the contract to fail. That makes a 75% chance of success, if you have the entries. Running the diamonds works if trumps break 3-2 and the diamonds break no worse than 4-2, or if the diamonds break 3-3. The advantage of the latter plays is that we try each of them and go on to plan B and plan C if they don’t work. I can’t do all of the probability calculations in my head. Let’s just pick one and get started.

Recall that we are facing an open lead of the 4 of diamonds. We play the 6 from dummy. East follows with the 3! If we only had a small diamond in our hand, we could play that to win the trick, and the hand would be easier, but no such luck. Declarer plays the ace.

We start by pulling trump, leading the queen. We want to preserve our entries to the board. Both opponents follow. Now, declarer leads the 2 to the ace on the board. Both opponents follow suit.

We know, now, that the trumps break 3-2. Can we afford to pull the last one before trying our other ideas? If the heart finesse works, the contract makes. We’ve already discovered that by preserving the 6 of hearts in the declarer hand, we can stay on the board to repeat the finesse as many times as necessary. If the finesse loses, we will usually create an entry in hearts to try to run the diamonds. If West holds Kxx in hearts and waits until the 3^rd round to take the king, we don’t have the extra entry. Can West figure that out? It’s more likely that West would take the king at the first opportunity. I think we can finish pulling trumps. Dummy leads the king. East discards a small club. The trumps are gone. Now it is time to try the heart finesse. Dummy plays the queen. East follows with the 3. Declarer wisely follows with the jack. West takes the king. Now, declarer needs to try one of his other chances.

West returns the 2 of hearts. There goes one of the entries that declarer so carefully created. Dummy plays the 8. East follows suit, and declarer takes the ace, maintaining one entry in hearts.

The remaining cards are

Dummy ♠ 8 ♥ 9 ♦ QT97 ♣ 8

 

Declarer ♠ T ♥ T6 ♦ A ♣ AQJ

 

and declarer is in his hand. The plan, now, is to see if we can set up the diamonds, and if that doesn’t work, try a club finesse. Declarer cashes the ace of diamonds. West follows with the 2. Dummy plays the 7. East discards a small club. Crud! Declarer plays the ace.

There is no way to set up diamonds. The only chance left is a club finesse. There are two ways to try it. We could try a normal finesse by leading the 8 from dummy, planning to play the queen if East ducks. If that works, we can cash the ace and ruff the jack. If it doesn’t work, c’est la vie.

The other alternative is to try a ruffing finesse by cashing the ace. There is a tiny chance that an opponent started with the singleton king. Don’t back on it, but that is another chance to make. Presuming that the king doesn’t fall. Declarer leads the queen, ruffing if West covers with the king, and discarding if she doesn’t.

Which is the better choice? Let’s consider how many cards each opponent has that we know are not the king of clubs. West started with 3 spades and 5 diamonds. She has played 2 hearts. That leaves 3 unknown cards in hearts and clubs. East started with 2 spades and 1 diamond. He has discarded 2 clubs and followed to 2 hearts. That leaves 6 unknown cards in hearts and clubs. Therefore, it is twice as likely that East holds the king of clubs.

We lead our carefully preserved 6 of hearts to the 9 on the board. West discards a club and East follows suit with a small heart. We still have a chance to change our mind. Let’s reassess the new information that we have gathered. This last trick has given us the complete distribution for all the hands. We know that West has 3 diamonds left, no hearts, and no spades. That leaves two clubs. East has no spades, 1 more heart, and no diamonds. That leaves 4 clubs. OK, we are not going to change our minds. It is still twice as likely that East holds the king of clubs. We lead the 8 from the board. Declarer plays the queen. East takes the king.

Some days you just can’t win.

Epilogue

Wait! Yes you can. As I was proof-reading this, I found the way to make the hand. Let's go back to the original planning stage.

West leads the 4 of diamonds. Let’s plan the play of the hand. 

Dummy ♠ AK86 ♥ Q98 ♦ QT976 ♣ 8

 

Declarer ♠ QT52 ♥ AJT6 ♦ AK ♣ AQJ

 

Declarer starts by counting his losers. With a reasonable split, he doesn’t have any losers in spades. From the view of the declarer hand, he has a potential loser in hearts, and one in clubs. That’s one too many. There is an extra winner in diamonds in dummy. You can use that to discard a losing club, and you can ruff a club in dummy. We can try the heart finesse, but it isn't necessary to make the contract.

We play the 6 from dummy. East follows with the 3. Declarer plays the ace.

We start by pulling trump, leading the queen. We want to preserve our entries to the board. Both opponents follow. Now, declarer leads the 2 to the ace on the board. Both opponents follow suit.

We know, now, that the trumps break 3-2. Declarer pulls the last trump by leading the king on the board.

Next, we move on to the clubs. Declarer leads the 8 of clubs from dummy and takes with the ace. Trumps are gone, so it is safe to cash the K of diamonds. East shows out. No problem, really. Had East followed suit to three rounds of diamonds, we could have discarded 3 hearts on the good diamonds in dummy. You can't have everything. 

Now, lead the jack of clubs and ruff it on the board. Declarer has no more diamonds, and the lead is in dummy. Declarer plays the queen of diamonds and discards the last club.

The hand now looks like this:

Dummy ♠  ♥ Q98 ♦ T9 ♣

 

Declarer ♠ 2 ♥ AJT6 ♦  ♣

 Declarer can lead the queen of hearts to try the finesse, but he doesn't need it to work. It doesn't. Now, all the hearts are good and the 2 of spades can take any non-heart return that West plays. Contract made.

One More Variation

Suppose the trumps break 4-1. What is the plan, then? We start out as before, playing for the 3-2 trump break.

Dummy ♠ AK86 ♥ Q98 ♦ QT976 ♣ 8

 

Declarer ♠ QT52 ♥ AJT6 ♦ AK ♣ AQJ

 

Declarer starts by counting his losers. With a reasonable split, he doesn’t have any losers in spades. From the view of the declarer hand, he has a potential loser in hearts, and one in clubs. That’s one too many. There is an extra winner in diamonds in dummy. You can use that to discard a losing club, and you can ruff a club in dummy. We can try the heart finesse, but it isn't necessary to make the contract.

We play the 6 from dummy. East follows with the 3. Declarer plays the ace.

We start by pulling trump, leading the queen. We want to preserve our entries to the board. Both opponents follow. Now, declarer leads the 2 to the ace on the board. East discards a club. Now, you have a spade loser. There is no way to keep from losing a spade trick. We'll stop pulling trumps for now. You will see why later.

Now, we need to keep from losing a heart. We continue with the plan as before. Lead to the ace of clubs, and cash the king of diamonds. East shows out on the diamonds. Too bad, but it doesn't matter. Even a 3-3 break in diamonds would not create enough winners to discard 3 hearts and a club. We need the heart finesse to work.

It doesn't. That was still a pretty good try. It only required a 3-2 trump break (a 68% chance) or for the king to be in the East hand (a 50% chance). You miss both chances just 16% of the time.

Now, why did we stop pulling trumps? The reason was to keep from going down more than one if the heart finesse fails. Suppose we pulled a third round of trumps. Now, West will hold the top trump when she gets in with the heart. After taking the king of hearts, she can pull your last trump. She still has the jack of diamonds that she can cash, and she might even hold a club so that the defense can take the king.

If we just let her hold her two trumps, she no longer holds the highest one. We let her ruff in whenever she wants, or she can lead a trump after she takes the king of hearts. Now, however, that is just the third round of trump and declarer will still hold a small trump to take the lead back after East takes her ruff. Should she ruff in before taking the king of hearts, you pull her last trump with the king You still go down, but only by 1.