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Par Zero Deals

Thomas Andrews; © 1999-2009.

Introduction

Are there any deals where neither side can make any contract?
This comes up in bridge discussions now and then, and the answer is, yes. The easiest example is:
A K Q J
x
x x x x
x x x x
x x x x
x x x x
x
A K Q J
x x x x
x x x x
A K Q J
x
x
A K Q J
x x x x
x x x x
In any contract, the defense can take eight top tricks, which means that not only can nobody make any contract, but all contracts end up down two.
This collection is a set of deals where no declarer can make more than six tricks in any contract. We start by exploring "symmetric" examples, like our first example above, then move on to wilder specimens.

Themes

The obvious, perhaps tautological, theme in these deals is tempo. Indeed, the Bridge World's glossary defines tempo as simply having the lead, but in most cases, it is used to describe an advantage to having the lead. (Nobody says that an endplayed player has a tempo.)
The tempo can be used for many purposes:
  • Setting up tricks - this is why we often lead from our longest and strongest against notrump, to set up our tricks before they set up theirs.
  • Pitching losers - if you hold xx opposite AKQx in one suit and xxx opposite xxxx in another, you can pitch a loser in the weak suit on the run of tricks.
  • Taking tricks - the converse of pitching losers. Having the lead lets us take our tricks before the other side gets to pitch in the suit(s).
  • Trump promotions - if the other side had the lead, they could simply draw trumps, but with our side on lead, we can manufacture an extra trump trick.
  • Killing entries - you can force an opponent to use entries in the wrong order.
In many of these deals, we'll see mixes of each type of tempo advantage. In an extreme example, there is one deal where a single lead has the simultaneous effect of trump promotion and entry killing.

Par-zero suit distributions

The first bunch of deals we'll discuss are "symmetric," like the above example. Given any specific suit distribution, say:
10 7 2
A
K Q 5 4 3
J 9 8 6
we might ask whether the symmetric deal:
10 7 2
K Q 5 4 3
J 9 8 6
A
A
10 7 2
K Q 5 4 3
J 9 8 6
K Q 5 4 3
J 9 8 6
A
10 7 2
J 9 8 6
A
10 7 2
K Q 5 4 3
is par-zero. In this case, the full symmetric deal is par-zero. Such a suit distribution is called par-zero.
When dealing with non-symmetric examples, we'll find that most of the cases at least one of the suits is par-zero.
We'll define the "complexity" of a par-zero deal to be the number suits which are not parzero. So in a symmetric deal, the complexity is zero. I've only found one example where the complexity is the maximum of four - that is, none of the suits are par-zero.

Finding deals

As with my collections of Double Asymmetries and Bad Fit Deals, this collection used a combination of two programs to find interesting examples:
  • Deal 3.0, my own flexible dealer and deal filter.
  • The double dummy solver from GIB.
In order to cut down on calls to GIB's double-dummy solver, I used some filters - I rejected deals where either side has a 9-card fit or longer, and any deals where one side has a significant amount of playing strength. It is possible that this misses some interesting examples, but some expediency is necessary. The algorithm I used found one match out of every 500,000 deals or so.
Searching for symmetric examples, I've used the same software. There are fewer than 17,000,000 distinct symmetric deals, and I've done a fairly systematic sample of over 1,000,000 examples.

Open Question

What is the worst case? There are many different ways to define this. One way to define it is to say, "What is the value of the lead?" In one case having the lead is worth six additional tricks in all suits (and 11 tricks in notrump.)

About the analysis

A number of shorthands are taken in the analysis, because there are, technically, 20 different contracts to analyze per deal - five denominations (notrump, spades, hearts, diamonds, and clubs) and four possible declarers. In none of the cases I've found is the line of play "positional" - that is, the same suit lead which sets the contract played by South also sets the contract when played by North. So I simply refer to "North/South" as declaring the contract, rather than being more specific.
I also skip some analysis, for the sake of brevity. Usually the correct lines are obvious, but in a few cases, they are not obvious.
The last section is filled with deals that are currently unanalyzed or partially analyzed. These are interesting and sometimes fascinating little puzzles you might want to try to solve. If you want to write up an analysis for any of these, I'll include it, with attribution.
As usual, I encourage all feedback, particularly if you find errors in analysis, but even if I've left in typos or bad grammar. I'm a bad copy editor.

Thomas Andrews; © 1999-2009.

Six Plus Promotion

A K Q
10 9 8 7
J 3 2
6 5 4
6 5 4
J 3 2
A K Q
10 9 8 7
J 3 2
6 5 4
10 9 8 7
A K Q
10 9 8 7
A K Q
6 5 4
J 3 2
If South declares notrump, the defense starts with three top diamonds, then three top clubs and a long diamond.
In spades, East/West start with three rounds of clubs, then three rounds of diamonds, followed by a fourth club at this position:
A K Q
10 9 8 7
—
—
6 5 4
J 3 2
—
7
J 3 2
6 5 4
7
—
10 9 8 7
A K Q
—
—
The lead of the last club promotes East's ♠ J.
If East/West declare spades, North/South draw trumps immediately, killing the promotion.
All the other suits follow similarly.
In fact, there are many deals of this sort above, where the defense takes the first six tricks and then sets up a seventh trick by a trump promotion of some sort. These are common enough that I devote the first set of non-symmetric deals to this idea. It's very rare to have a situation like the example in the introduction, with each side able to simply take seven or more tricks off the top in every contract.

Thomas Andrews; © 1999-2009.

Held to Five

K 2
8 7 6 5
A Q 9 4
J 10 3
J 10 3
K 2
8 7 6 5
A Q 9 4
8 7 6 5
A Q 9 4
J 10 3
K 2
A Q 9 4
J 10 3
K 2
8 7 6 5
In this case, the defense can always hold declarer to five tricks in every contract.
If South declares notrump, hearts, or clubs, the defense can take eight tricks off the top, playing four trumps first in the trump contracts.
If South declares spades, the defense starts with three top clubs, East pitching a diamond on the third round. West then leads a fourth club at this position:
K 2
8 7 6 5
A Q 9 4
—
J 10 3
K 2
8 7 6 5
9
8 7 6 5
A Q 9 4
J 10
—
A Q 9 4
J 10 3
K 2
8
If North ruffs low, East overruffs and the defense runs three hearts and promotes another trump trick with a fourth heart.
If North ruffs high, East pitches a diamond, leading to the position:
2
8 7 6 5
A Q 9 4
—
J 10 3
K 2
8 7 6 5
—
8 7 6 5
A Q 9 4
J
—
A Q 9 4
J 10 3
K 2
—
What are North/South to do? The high ruff promoted one trick for West. Attacking diamonds now lets East ruff.
So the best thing for declarer is to play the ace, queen and low spade, leading to this position:
—
8 7
A Q 9 4
—
—
K 2
8 7 6 5
—
8
A Q 9 4
J
—
9
J 10 3
K 2
—
with West on lead. Now the defense takes three hearts and the ♥ 9. South can ruff, but that promotes East's trump eight.

Thomas Andrews; © 1999-2009.

Massive Wastage

A K Q J
—
10 9 8 7 6
5 4 3 2
10 9 8 7 6
5 4 3 2
—
A K Q J
5 4 3 2
10 9 8 7 6
A K Q J
—
—
A K Q J
5 4 3 2
10 9 8 7 6
Srinivasa Damarla sent me this example with 9-card fits where nobody makes. It's interesting to note that this is the ultimate case of wasted values; every honor is held opposite a void.
This is also a worse case for declarer than our first diagram - South can be held to five tricks in notrump, clubs and diamonds.
In spades, West starts with four clubs, East ruffing the fourth and taking his diamonds. West still gets his long spade, holding South to four tricks.
In hearts, West again starts with four clubs, and East pitches spades. West then exits a heart. East still gets his four diamonds and a long heart, again holding declarer to four tricks. So in notrump, having the lead is worth three tricks, while in each suit, having the lead is worth four tricks.

Thomas Andrews; © 1999-2009.

More Wastage

A K Q 9 8 7
J 6 5 4
—
10 3 2
10 3 2
A K Q 9 8 7
J 6 5 4
—
J 6 5 4
—
10 3 2
A K Q 9 8 7
—
10 3 2
A K Q 9 8 7
J 6 5 4

Spades

Against spades (declared by South) West starts with ♥ A-K-Q, then a heart, ruffed by East. East then cashes his ♣ A-K-Q, then leads another club, West ruffing with the ♠ 10, which promotes the ♠ J in East's hand, giving the defense eight tricks.

Hearts

Against hearts, West leads the ♥ A-K-Q then exits in hearts to North. North can take three spades, but has to give East an entry, and the defense gets five hearts and three clubs.

Diamonds

Against diamonds, West starts with two top hearts, then a heart ruff by East, and three top clubs, followed by a club ruff and a fourth heart, ruffed by East with the ten - essentially the same promotion as when defending the spade contract.

Clubs

Against clubs, the defense takes three hearts, East pitching diamonds, then a diamond ruff and three top clubs, ensuring East/West score five trumps and three hearts.

Notrump

Against notrump, West starts with three top hearts, East pitching diamonds, West then exits a heart to North, and both North and East have only black suits. East eventually gets three clubs and a spade, for a total of six tricks.

Thomas Andrews; © 1999-2009.

A Worst Case?

J 9
6 5 4 3 2
K
A Q 10 8 7
A Q 10 8 7
K
J 9
6 5 4 3 2
K
A Q 10 8 7
6 5 4 3 2
J 9
6 5 4 3 2
J 9
A Q 10 8 7
K
South makes three tricks declaring in either major, and five tricks in either minor. South also makes only three tricks in notrump. That means that, in each suit, having the lead is worth five tricks, while having the lead in notrump is worth seven tricks.

Notrump

East/West can run 10 tricks in the majors. Due to the doubleton jack-nines, the defense can affort to overtake the stiff kings.

Spades and Hearts

East/West run the same ten tricks as against notrump, starting with the five trump tricks.

Diamonds or Clubs

We'll argue from the point of view of the diamond suit. The defense against clubs is just done by swapping.
Against diamonds declare by South, West starts with three top spades, East pitching two clubs. Assume North ruffs the third round of spades with the ♦ K in dummy, leading to:
—
6 5 4 3 2
—
A Q 10 8 7
8 7
K
J 9
6 5 4 3 2
—
A Q 10 8 7
6 5 4 3 2
—
6 5
J 9
A Q 10 8 7
K
How does declarer get out of the North hand? If North leads a club, East ruffs and leads a heart to West's ♥ K. West cashes one more spade, and East ruffs the last round of spades. East then cashes the ♥ A and plays the ♥ Q at this position:
—
6 5
—
A Q 10
—
—
J 9
6 5 4
—
Q
6 5 4 3
—
—
—
A Q 10 8 7
—
West is assured of a diamond trick, by promotion. North/South score a total of five tricks - the ♦ K and four diamond tricks at the end.
So maybe, after ruffing with the ♦ K, North should exit with a heart? Then West wins, takes the rest of his spade, and gives East a club ruff for the entry. East then plays off his top hearts as before.
And if North doesn't ruff the third or fourth spades, West shifts to a club, East ruffs, takes two high hearts, and leads the ♥ 10 for a slightly queasier promotion:
—
6
K
A Q 10 8
7
—
J 9
6 5 4
—
10 8
6 5 4 3
—
6
—
A Q 10 8 7
—
East/West have taken the first seven tricks, and South can't keep the defense from scoring an eighth trick. Obviously, South must ruff high, and West pitches the club. South now cannot ruff his spade loser, and the diamond king is the only entry to the clubs to pitch the spade loser, but if declarer uses the diamond entry now, East can ruff a club lead.

Thomas Andrews; © 1999-2009.

Another Candidate

7
8 4 3 2
A K Q J 10 9
6 5
6 5
7
8 4 3 2
A K Q J 10 9
8 4 3 2
A K Q J 10 9
6 5
7
A K Q J 10 9
6 5
7
8 4 3 2
If South declares notrump, hearts, or clubs, the defense can take twelve tricks off the top.
If South declares spades, West starts with three top clubs, East pitching a diamond on the second round.
If North ruffs the third round, East overruffs and plays two hearts, West ruffing the second round, leading to:
—
8 4
A K Q J 10 9
—
6
—
8 4 3 2
J 10 9
4 3 2
Q J 10 9
6
—
A K Q J 10 9
—
7
8
With West on lead, he can cash a club, East pitching another diamond, then give East a diamond ruff, for seven tricks.
Obviously, the same defense works when South declares diamonds, only in reverse.
So, is this a "worse" case than our previous candidate? In the previous candidate, we could not make more than five tricks in any suit, and having the lead was worth five tricks against all suit contracts. In this case, everybody can make six tricks in some suit, but having the lead is worth an additional six tricks.

Thomas Andrews; © 1999-2009.

Not So Impure

9 6 5 4 3 2
10 7
A K
Q J 8
Q J 8
9 6 5 4 3 2
10 7
A K
10 7
A K
Q J 8
9 6 5 4 3 2
A K
Q J 8
9 6 5 4 3 2
10 7
In most cases where either side has a fit, the opponents make up for it by holding the top trumps. In this case the trumps are not nearly as weak. Despite the strength, the defense can always score three trumps.
Against spades, the defense starts by cashing the ♥ A-K, then the ♣ A-K, followed by a heart from West at:
9 6 5 4 3 2
—
A K
Q
Q J 8
9 6 5 4
10 7
—
10 7
—
Q J 8
9 6 5 4
A K
Q
9 6 5 4 3 2
—
If North ruffs, East overruffs and leads a club, which gives West another trump trick.
If North pitches the ♣ Q, East ruffs and still leads a club:
9 6 5 4 3 2
—
A K
—
Q J 8
9 6 5
10 7
—
10
—
Q J 8
9 6 5 4
A K
—
9 6 5 4 3 2
—
South can't ruff, or it promotes a third trump trick for the defense, so South pitches a diamond and West ruffs high. This is just West's natural trump trick, but West then promotes another trump trick in his hand by leading a fourth heart:
9 6 5 4 3 2
—
K
—
J 8
9 6 5
10 7
—
10
—
Q J 8
9 6 5
A K
—
9 6 5 4 3
—
East ruffs with the ♠ 10, promoting a third trump trick for the defense.

Thomas Andrews; © 1999-2009.

A Subtle Promotion

A K
A 8 7 4 2
10 8 4
10 7 2
Q 7 6
9 6 5
6 3
A Q J 8 5
8 3 2
Q 10 3
A K Q 7 2
9 6
J 10 9 5 4
K J
J 9 5
K 4 3

Notrump

In notrump, each side can run at least seven top tricks. East/West can take 10 tricks by leading a diamond to the ace, club finesse, run four diamonds, then finesse clubs again and run clubs. North/South get seven tricks by leading a spade to the king, heart finesse, heart king, another spade, and then running three good hearts.

Spades

Against a spade contract, East/West must get a spade, so they simply take their six top tricks in the minors.

Hearts

Against a heart contract, East/West start the same way - diamond, club finesse, two diamonds, club finesse, club ace, then a fourth club. East ruffs high with the queen (or 10) and this promotes West's ♥ 9.

Diamonds

If East/West declare in diamonds, the defense takes two top spades, finesses in hearts, cashes the ♥ K, ruffs a spade, cashes the ♥ A, and finishes up with a fourth round of hearts, promoting South's ♦ J.

Clubs

The trickiest case is when East/West declare clubs.
North/South take their two spades, three hearts, and a spade ruff as before. On the third round of hearts, South pitches a diamond, and, when North leads a fourth round of hearts, South pitches a diamond again at this position:
—
8 7
10 8 4
10 7
—
—
6 3
A Q J 8 5
—
—
A K Q 7 2
9 6
J 10
—
J 9
K 4 2
What should East/West do? If they ruff in East, they can only finesse once in trumps. But if East pitches a diamond and ruff with the West hand, they only have one entry to dummy, since South can ruff a second diamond entry, so again they can only finesse in clubs once.

Thomas Andrews; © 1999-2009.

Two en passant Coups

A K Q 5
K 2
Q 10 8 7
6 4 3
J 8 7 2
10 9 7
K 5
A K Q 9
9 6 4
Q 8 4
A J 6 3
10 7 2
10 3
A J 6 5 3
9 4 2
J 8 5

Notrump, Hearts, and Clubs

Any heart, club or notrump contract fails easily, with simple promotions necessary in some of the suit contracts.

Spades

If East/West declare spades, the defense starts with three spades, then three hearts. The fourth heart is led at this position:
5
—
Q 10 8 7
6 4
J
—
K 5
A K Q 9
—
—
A J 6 3
10 7 2
—
6 5
9 4
J 8 5
North's ♠ 5 scores en passant.
If North/South declare spades, East/West start with three clubs and three diamonds, West pitching a heart. On the fourth diamond, West overruffs South for the seventh trick.

Diamonds

If North/South declare diamonds, East/West start with three diamonds, then three clubs. West's ♣ 9 is led at this position:
A K Q 5
K 2
Q
—
J 8 7 2
10 9 7
—
9
9 6 4
Q 8 4
6
—
10 3
A J 6 5 3
—
—
East scores the ♦ 6 "en passant" again.
If East/West declare diamonds, the defense starts with three hearts, then three spades. When North leads his ♠ 5 at this position:
5
—
Q 10 8 7
6 4
J
—
K 5
A K Q 9
—
—
A J 6 3
10 7 2
—
6 5
9 4 2
J 8
East must ruff with the ♦ J to prevent South from scoring the ♦ 9, and that promotes a second diamond trick in the North hand.

Post mortem

The en passant coup is a minimalist trump promotion - where our side has only one trump and it is not a natural winner.

Thomas Andrews; © 1999-2009.

Top Tricks

10 7 2
8 6 5 4 3
J
A J 9 3
A K Q 9 3
Q
10 9 8 4
10 8 6
8 4
A K J 10
6 5 3 2
Q 7 5
J 6 5
9 7 2
A K Q 7
K 4 2

Notrump

East/West have a running nine tricks, while North/South have eight. How eight? North wins the ♦ J, then leads the ♣ J, forcing East to cover. South then runs his diamonds and finesses in clubs against West's ♣ 10.
In fact, North/South have the first seven tricks off the top in any contract, since East/West are forced to follow to three clubs and four diamonds. In every contract, they can manufacture an eight tricks as well, as we'll see.

Spades

If East/West declare spades, North/South takes their seven tricks, then North plays a fourth round of clubs, promoting an eighth trick in trumps when South ruffs with the ♠ J.
If North/South declare spades, West just pull trumps and takes the four heart tricks, for nine defensive tricks.

Hearts

If East/West declare hearts, North/South again take the top seven tricks, North pitching three spades on diamonds. At trick eight, North exits passively with a heart. East/West never score a spade, because North's long trump keeps East from pulling all of them, and when North gets in, he leads his last club, promoting another trump for himself. East/West are left with the four trump tricks he started with.

Diamonds

If North/South are on lead, they can take four diamonds and four clubs, for eight tricks.
If East/West are on lead, they take six top tricks and lead a fourth round of hearts, South forced to ruff. Now East/West get two natural trump tricks, for eight tricks. (West sheds clubs on East's hearts, so if South doesn't ruff the fourth heart, East just gives his partner a club ruff.)

Clubs

As usual, North/South take the first eight tricks.
But if East/West are on lead, they take the first six tricks and promote a seventh by leading a fourth round of hearts.

Post Mortem

East/West can't make more than five tricks declaring any contract here. In hearts or notrumps, only four.
North/South can make as many as six tricks only in clubs and hearts.
The play of leading the club jack from A-J-9-x opposite K-x-x is called a "backward finesse," if done by declarer, or a "surrounding play" if done by the defense. Peculiar that the terminology would be so different.

Thomas Andrews; © 1999-2009.

Weakened Trumps

A K J 6
9 8 3 2
Q 3
8 6 4
8 7 3
A K J 6 4
8 7 4
Q 10
10 9 5 2
7
J 9 5 2
A K J 3
Q 4
Q 10 5
A K 10 6
9 7 5 2

Notrump

East/West have four clubs and three hearts off the top, while North/South have four spades and four diamonds off the top.

Spades

If West is on lead (East on lead is easier), East overtakes the first club, and then leads a heart, West taking three hearts. Then East overtakes another club, cashes a third club, then, on the fourth club, promotes a trump trick in his hand.
If North/South are on lead, they draw trumps immediately then take four diamonds.

Hearts

If North/South are on lead, they take three spades then three diamonds. They have an eventual natural trump trick.
If East/West are on lead, the defense play four rounds of clubs immediately, pitching two diamonds. North can ruff but if so, West can get at least four trump tricks.

Diamonds

If East/West are on lead, they defend as they did against spades, ovetaking a club, taking three rounds of hearts, two more top clubs, and a fourth round of clubs, West ruffing high. North must overruff with the queen, setting up another trick for the defense.

Clubs

If East/West are on lead, they just draw four clubs then take three hearts.
With North/South on lead, life is a little subtler. East/West's trump suit does not seem succeptible to a promotion, but the suit does possess a weakness. Notice how, on defense against a number of contracts above, with West on opening least, East sometimes needs to overtake twice in clubs. East has an entry problem. North/South can make East's problem a bit worse.
North/South take three top tricks in both diamonds (finessing the ten) and spades, South pitching a heart, leading to this position:
J
9 8 3
—
8 6 4
—
A K J 6 4
—
Q 10
10
7
J
A K J 3
—
Q 10
A
9 7 5 2
North leads the ♠ J, and West must ruff, South pitching a heart. Now East/West's entries are all tangled. East's hand cannot overtake the ♣ Q without promoting a trump trick in the South hand. Neither can East cash the ♣ Q and get to his hand with a high heart rough, or South just pitches his high diamond and scores a long trump. If East/West try a cross-ruff, South can over-ruff the first heart ruffed.
What would happen if South was on lead after six tricks in the above position, and led his good diamond? West again would have to ruff, but now South still has two hearts. East/West take the next six tricks by playing the ♥ A, ruff a heart with the ♣ 3, ruff a spade in West and East is left with ♣ A-K-J, scoring the last three tricks. East/West score six trump tricks and a heart.

Thomas Andrews; © 1999-2009.

Trump Magic

J 9 3
A K 10 9 7 3
9 7 6
9
6
6 5 2
K Q 8 5
8 7 5 4 2
A K Q 5 4
Q 4
A 4 2
10 6 3
10 8 7 2
J 8
J 10 3
A K Q J
This is the first non-symmetric example with a six-card suit, and also the first with eight-card fits.

Notrump

East/West have three spades and four diamonds off the top. North/South have four clubs and six hearts.

Spades

With East/West declaring spades, North/South start with two top hearts followed by four clubs, North ruffing the fourth round high. East can't avoid two trump losers.
With North/South declaring, the defense starts with three spades, then runs four diamonds, East pitching a heart. South can take his trump trick now or later, but the defense still gets eight tricks.

Hearts

Against hearts by North/South, the defense starts with three spades then four diamond, promoting East's ♥ Q.
East/West declaring is clearly pointless.

Diamonds

If North/South declare diamonds, East/West just take their four trumps and three spades.
If East/West declare diamonds, the defense starts with three top clubs, North pitching two spades, then two top hearts followed by a third high heart.
If East pitches, South also pitches and North just continues hearts, promoting a trump trick for the defense. So East must ruff.
If East ruffs low, South overruffs and leads a spade, East winning, reaching this position:
—
9 7 3
9 7 6
—
—
—
K Q 8 5
8 7
K Q 5 4
—
A 2
—
10 8 7
—
J 3
J
East is on lead. If he draws trumps, he can't get back to his spades. If he tries to cash a spade to pitch a club now, North ruffs. There is no way of keeping East/West from scoring a seventh trick.
So perhaps East should have ruffed the third heart high. South pitches a spade, arriving at this position:
J
9 7 3
9 7 6
—
6
—
K Q 8 5
8 7
A K Q 5 4
—
4 2
—
10 8 7
—
J 10 3
J
North/South have promoted a natural trump trick, and West again needs to get rid of his clubs, but can't attack spades. Nor can he pull one round of diamonds and ruff a club, because North can ruff ahead of him without costing the natural trump trick. There is, in fact, no way for West to avoid losing two more tricks.

Clubs

If North/South declare clubs, the defense starts with three spades (West pitching hearts) followed by three diamonds. West is destined to get a long trump, as well.
If East/West declare clubs, the defense runs four clubs then three hearts. East is out of hearts, and West must follow, for seven tricks.

Post Mortem

This deal, perhaps, does not belong in this section - for example, North/South don't have six top tricks against diamonds. Still, the basic pattern "take a bunch of top tricks then score your trumps by promotion" is followed. We're on a cusp of a more complex type of Par Zero Deals.
The end position in the diamond contract is interesting. Here is a simpler version of it:
J
10 9 8
9 7
—
6
2
Q 8 5
7
A K 5 2
—
A 4
—
10 8 3
—
J 10
J
North to lead, diamonds trumps, North/South need two tricks. The only lead to score two tricks is a heart lead.

Thomas Andrews; © 1999-2009.

Repeated Promotion

A K Q 10
10 9 8 5
Q 3
9 3 2
7 4 2
A K Q 2
10 4
A Q 10 8
J 8 6 5 3
6 4
9 8 7 2
K 6
9
J 7 3
A K J 6 5
J 7 5 4

Notrump

Against notrump, North/South have three top spades and five top diamonds, while East/West have three hearts and four clubs (finessing the jack.)

Spades

If East/West are declaring spades, North/South take three top spades then four top diamonds.
If North/South are declaring spades, the defense starts with three hearts, and three clubs, East pitching two diamonds. East gets at least one more trick in trumps.

Hearts

If North/South declare hearts, East/West run three top heart, then the ♣ K and a club back, finessing the jack, and two more clubs. On the last club, if North ruffs, West's low trump is the outstanding trump. If North fails to ruff, that's the defense's seventh trick.
If East/West declare hearts, North/South cash two diamonds then three spades. Finally, a fourth spade is ruffed high in South, promoting a second trump trick for North/South.

Diamonds

If North/South declare diamonds, the defense starts with three hearts and then four clubs. On the fourth club, if North ruffs low, East overruffs, while if North ruffs high, it promotes a trump trick for the defense.
If East/West declare diamonds, North/South draws trumps immediately, taking three spades and five diamonds total.

Clubs

If North/South declare clubs, the defense simply draws trumps and takes its three heart winners.
If East/West declare clubs, North takes his three top spades, South pitching two hearts, and then the defense plays off top diamonds until West ruffs.
10
10 9 8 5
—
9 3 2
—
A K Q 2
—
A Q 10 8
J 8
6 4
9 8
K 6
—
J
K J 6
J 7 5 4
North/South have taken only five tricks. East/West appear to have lots of tricks - four clubs, three hearts, two spades. Yet the third diamond crushes them.
If West doesn't ruff, another diamond lead promotes a seventh trick for the defense.
So West ruffs with the ♣ 10, promoting one long trump in the South hand. Now West's timing is all crossed up. If he draws three rounds of trumps and then attacks hearts, South ruffs and cashes more diamonds. If West instead attacks hearts immediately, South ruffs the second and continues diamonds, promoting a second trump trick.
Whenever South gets in, he will be able to play another high diamond. If West has any more trumps, the lead promotes another trump trick for South, and if West has no more trumps, the diamonds are good.
This deal, in clubs at least, is not really a six-plus-promotion example, but rather a five-plus-promotion example.

Thomas Andrews; © 1999-2009.

A First Example

K Q 10 3
K 9 6 3
9 7 5
J 2
A 6 4
A Q 8 4
Q J
A 10 8 7
9 8 5 2
J 10
10 6 3 2
Q 6 5
J 7
7 5 2
A K 8 4
K 9 4 3

Notrump

If North/South declare notrump, the defense starts with hearts, killing the entry to the North hand before the spades are established. South gets one spade, one heart, three diamonds and, with care, a club.
If East/West declare notrump, the defense starts with spades, which establishes three spade tricks before the heart entry to North is killed. North/South get three spades, a heart, two diamonds and either another diamond or a club.

Spades

If East/West declare spades, the defense starts with trumps. West can duck the ♠ A as many times as he wants, he still cannot get to East's hand before North gets in again to finish pulling trumps. North/South get three spades, a heart, two diamonds, and an eventual club.
If North/South declare spades, West leads a low heart and East gets to ruff a heart when West wins the second round of trumps.

Hearts

If North/South declare hearts, a heart lead keeps them from scoring any extra trumps, and they are left with three spades, one trump, and two diamonds.
If East/West declare in hearts, the defenders start by attacking spades while North still has a heart entry. When North gets in with a heart, North runs spades and West is forced to overruff South on the fourth round, and North gets another heart, for two spades, two hearts two diamonds and, eventually, a club.

Diamonds

If declared by North/South, the defense starts with (low) heart leads. This allows the defense to get two heart tricks and killing declarer's entry to the long spades. That means that North/South loses a diamond, two hearts, a spade, two clubs and either a third heart or, when South ruffs the third heart, a second diamond.
If declared by East/West, South takes two top diamonds before attacking spades. West wins the second one, and plays ace and a heart, the king winning in dummy. Another two spades are led, pitching a heart and a club. Then the club jack is led. East has to cover, South plays the king, and West wins the ace and is caught in an endplay:
—
9 6
9
2
—
Q 8
—
10 8
—
—
10 6
6 5
—
—
8 4
9 4
There is no way to prevent North/South from scoring two tricks.
If West leads a high heart, and East pitches a club, South ruffs low and exits in clubs to West's hand. West, force to exit in clubs or hearts, must allow South to ruff with the nine in dummy or eight in hand, respectively, unless East ruffs high. Either way, North/South must score one of their two remaining trumps. Twist and turn as they might, East/West can't keep North/South from scoring two more tricks, for two spades, the heart king, two top diamonds and two more tricks.

Clubs

If declared by East/West in clubs, the defense starts with spades. Eventually, South gets to pitch a losing heart on a spade. South then takes two top diamonds and exits in diamonds to this situation:
K
9 6
—
J 2
—
Q
—
A 10 8 7
9
—
6
Q 6 5
—
—
8
K 9 4 3
East, stuck on lead, can't avoid two more losers. Say they lead a club to the ace, and a club back to the jack, queen, and king. South simply continues with a high diamond, and West is forced to ruff and South scores a long trump.
If declared by North/South, the defense starts with low hearts, which has the usual entry-killing effect. North wins the first heart, then attacks spades. West wins the second spade, leads a low heart to East, who then leads a club to West's ace and a club. West cashes a heart and leads a low club to the jack, queen and king.
K Q
9
9 7 5
—
4
Q
Q J
10 8
9
—
10 6 3 2
6
—
—
A K 8 4
9 4
East/West have taken two hearts, a spade and a club, and they will get a diamond and two more clubs.

Complexity

This is a rare par-zero deal of complexity three. The spade suit is the only par-zero suit:
K Q 10 3
9 8 5 2
J 7
A 6 4
A 6 4
K Q 10 3
9 8 5 2
J 7
9 8 5 2
J 7
A 6 4
K Q 10 3
J 7
A 6 4
K Q 10 3
9 8 5 2
South can make exactly six tricks in any contract.

Thomas Andrews; © 1999-2009.

More Trump Magic

K Q 6 4
Q 8
9 8 4 2
9 5 4
J 7 5
A 7 6 4
Q 5
K Q J 3
10 9 8 2
9 5 2
A K J 7
6 2
A 3
K J 10 3
10 6 3
A 10 8 7

Notrump

If East/West are on lead, they can set up two clubs, and have four diamonds, two clubs, and the ♥ A.
If North/South are on lead, they can set up three hearts, and have three spades, three hearts, and a club.

Spades

If East/West are on lead, they can take three top diamonds (West pitching a heart) and on the fourth diamond promotes a trump trick. If South ruffs low, West overruffs and sets up a pair of club tricks. So South must ruff high, and West pitches a heart. When West attacks hearts, West wins the second round and exits a high club to this position:
K Q 6 4
—
—
9 5
J 7 5
—
—
Q J 3
10 9 8 2
9
—
6
3
K J
—
10 8 7
Now if South tries to pitch a club loser from dummy on a heart, West ruffs high without costing the defense a trump trick.
And declarer can't afford to draw two rounds of trumps before attacking hearts for the same reason. And if declarer plays two top spades and exits a spade, the defense gets to attack clubs before the heart ace is knocked out.
The play of the fourth diamond is similar to the end position from Trump Magic. The lead functions as both a trump promotion and an entry killer.
If North/South are on lead against spades, they draw three rounds of spades immediately and then set up hearts. North/South have the club ace, three hearts, and three spades. (If East ruffs the third heart winner, then North simply gets another trump.)

Hearts

If North/South are on lead, they attack trumps immediately. When West wins and they take four top diamonds, South pitches a club. North/South eventually get three hearts, three spades, and a club.
If East/West are on lead, they take three diamonds (West pitching a spade) and then attack clubs.

Diamonds

If East/West are on lead, they draw trumps, then lead clubs. They score two clubs, four diamonds and a heart.
If North/South are on lead, the defense starts with three spades (South pitching a club) followed by the ♥ Q lead. This kills an entry to the West hand for setting up clubs two club tricks. To preserve his other entry, the diamond queen, the defense cannot draw trumps. West must win the second heart lead, and start with high clubs. South wins the first one, and plays two rounds of high hearts, North pitching clubs. North/South have taken three spades, two hearts and a club, so East must ruff the fourth round of hearts, at which point North must score a long trump.

Clubs

If North/South are on lead against clubs, they take three top spades, South pitching a diamond. At this point, North can lead anything and the contract is set, but the direct route is hearts, starting the queen. West wins the second heart and the defense runs two diamonds getting to this position:
6
—
9 8
9 5 4
—
7 6
—
K Q J 3
10
9
K J
6 2
—
J 10
—
A 10 8 7
Declarer has taken three spades, and one heart. He has an obvious club and additional heart, but he can also always score a second club with a heart ruff high in dummy - East's weak doubleton can't overruff the club nine.
If East/West are on defense, they start with four diamonds (West pitching a spade on the third round.) South can't pitch on the fourth diamond, because the defense has two clubs and a heart absolutely, so South has to ruff low, West pitching a second spade, leading to:
K Q 6 4
Q 8
—
9 5 4
J
A 7 6 4
—
K Q J 3
10 9 8 2
9 5 2
—
6 2
A 3
K J 10 3
—
A 10 8
Nothing declarer does here can keep West from scoring three club tricks - he cannot be endplayed, and if he ever ruffs a spade with the ♣ 3 it's all obvious.

Thomas Andrews; © 1999-2009.

Heart Blockage

7 4 3
8 7 3 2
J 5
K 6 4 2
K 10
Q J 10 9 5
A 8 6 4
7 5
Q 8 6 2
K
10 7 2
A J 10 9 8
A J 9 5
A 6 4
K Q 9 3
Q 3

Notrump

East/West have a lot of potential tricks, but the blockage in hearts is their downfall as declarer. On a diamond or spade opening lead, North/South kill one entry, and, when West wins and leads a heart to the king, South ducks. To set up the hearts, then, West needs two more entries, entries he doesn't have. East/West then get a heart, two spades, and a diamond. Can East get three club tricks? No matter how hard he tries, he cannot do so without losing two clubs first, which gives the defense three diamonds, a spade, a heart, and two clubs.
If North/South declare notrump, though, West's two entries are enough. A heart to the king is led, and South should duck. East leads a low spade, South wins the ace and can kill one entry but not the other. West wins at the first opportunity and sets up hearts, and gets a diamond, at least one spade, a club and four hearts. (The only way to keep East's ♠ Q from scoring actually lets East score a second club instead, so the defense actually has eight tricks.)

Spades

If North/South declare spades, the defense starts with hearts. South does not good to duck, so he wins and attacks diamonds. Too late, West wins the ace, pitches two diamonds on hearts, and leads a fourth round of hearts, pitching a club. Sout to play at:
7 4 3
8
J
K 6 4 2
K 10
10 9
8 6 4
7 5
Q 8 6 2
—
—
A J 10 9 8
A J 9 5
—
K Q 9
Q 3
East/West have scored three tricks already, and they have two natural tricks in trumps left, as well as the ♣ A. If South ruffs, the defenders get a third spade trick. If he doesn't ruff, then he gets the ♣ A, two spades or a heart.
If East/West declare spades, the defense starts with diamonds and East doesn't get to pitch any diamonds. The defense ultimately gets two diamonds, two spades, a heart and two clubs.

Hearts

It is somewhat difficult to believe that East/West can be held to six tricks in hearts. They have what appears to be four heart tricks and a trick in each other suit. But the spade trick is the most vulernable, and it vanishes in a puff of smoke.
The defense starts with diamonds, West either holding up one round or not. What does West do, now that he's in? The only vulnerable trick he has is spades, so he'd better set it up. South wins the spade king lead with the ace, cashes the heart ace, and then cashes his diamond, West forced to follow and North pitching his two remaining spades. South then gives North a ruff in spades and leads a club at this position:
—
8 7
—
K 6 4 2
—
Q J 10 9
—
7 5
Q 8
—
—
A J 10 9
J 9
6 4
—
Q 3
East/West need the rest, but the club lead forces them to take the entry before the trumps are drawn, so that the spade queen can't be cashed.
Of course, if North/South were declaring hearts, they can attack spades right away, and North won't have time to pitch his spade losers.

Diamonds

If North/South declare, the defense leads to the heart king. South wins and attack trumps, but too late. West wins the second trump and starts cashing hearts, East pitching spades. If South ruffs the fourth round of hearts, that promotes a trump trick for West.
If East/West declare, the defense starts with diamonds, and West wins the second round. He leads a heart to the king, and South ducks. Now East is on lead at this position:
7 4 3
8 7 3
—
K 6 4 2
K 10
Q J 10 9
8 6
7 5
Q 8 6 2
—
10
A J 10 9 8
A J 9 5
A 6
K Q
Q 3
Whatever East leads, South gets an entry to draw trumps and play ace and another spade. That kills the West hand completely, and the defense still gets two clubs, and the heart ace.

Clubs

If East/West declare clubs, the defense attacks diamonds and West wins the first or second.
If he attacks hearts, South wins and plays the rest of his diamonds, North pitching two small spades. If East doesn't ruff the fourth diamond, the defense gets three diamonds, a heart, a spade, a spade ruff and a club. So East must ruff, and we're at this end position:
7
8 7 3
—
K 6 4 2
K 10
Q J 10 9
—
7 5
Q 8 6 2
—
—
A J 10 9
A J 9 5
6 4
—
Q 3
East is on lead, but what does he lead? If he leads a spade, South wins the ace, continues a spade, North ruffs. North then leads a club, and, if East finesses, North gets another spade ruff, while if East plays the ace, the defense gets two clubs directly. Even then, there is the eventual spade loser. Basically, East can't avoid lots of losers.
If North/South are declaring in clubs, East starts with the ♥ K, and South should duck. East then leads a spade and South wins the ace and starts diamond, but it is too late - West wins the ace and leads a high heart and East pitches a diamond, and the defense is restricted to two diamond. Now South has to win. He cashes one diamond, but East can ruff the third round, cross with a spade to West and leads high hearts, pitching two spades. South can ruff, but he's ruffing with a natural trump.
North/South get only five tricks.

Thomas Andrews; © 1999-2009.

Sneak Attacks

Q 4
A Q J 3 2
J 9 5 3
10 5
10 3 2
K 10 4
K Q 7 6 2
J 2
9 7 5
9 8 7 6 5
A
A K Q 4
A K J 8 6
—
10 8 4
9 8 7 6 3

Notrump

With East/West on lead, they can unblock the diamond ace, cross to the club jack, take two more diamonds, and three more clubs.
But North/South can only take six top tricks. How do they stop East/West from taking seven tricks? They take the top five spades, North pitching two hearts and a club, then South exits a club.
This sneak attack in the opponents' club suit forces East/West to spend one club entry or another before the diamond ace is unblocked. If West plays low, East winning, then East can unblock the diamond and cross to the club jack for two more diamonds, but then they can't cash the remaining two clubs. If West wins, he has used his club entry before East has unblocked the diamond ace.
Either way, North/South get two more tricks, either a diamond and a heart, or two hearts.

Spades

With East/West on defense, they unblock the diamond ace, cross to the club jack, and take two more top diamonds. Now West must shift to a spade, another attack in an a strong suit held by the opponents.
South's spades are strong enough that he can afford to ruff a club in dummy with the queen for his seventh tricks, but this spade lead lets the defense prevent it. North appears to have a diamond trick established, but there in no entry to the North hand after drawing trumps, and East just ruffs if North plays the diamond before trumps are drawn.
If East/West get into an unlikely spade contract, North/South just defend as they defend against notrump, running five spades and exiting in clubs.

Hearts

If North/South declare hearts, the defense starts as usual - diamond ace, club to the jack, two more top diamonds. Then West leads a club to East, and East leads a third round of the suit, West ruffing high, establishing a third trump trick for the defense.
If East/West declarer hearts, the defense starts with three top spades, North pitching a club. When South leads a fourth spade, North pitches his last club:
—
A Q J 3 2
J 9 5 3
10
—
K 10 4
K Q 7 6 2
J 2
—
9 8 7 6 5
A
A K Q 4
J 8
—
10 8 4
9 8 7 6 3
Pitching the clubs has severed the East/West hands almost completely.
If East/West try to avoid giving North two club ruffs with his small trumps, they are stuck leading hearts and diamonds. On heart leads, declarer gets three natural heart tricks (with the king onside) and every time exits in diamond. If East ruffs the fourth diamond (which will be good,) that sets up a long heart in the North hand.

Diamonds

With East/West declaring in diamonds, the defense starts with a spade to the queen, ♥ A, heart ruff, two more top spades, North pitching a club, and a fourth spade at this position:
—
Q J 3
J 9 5 3
10
—
K
K Q 7 6 2
J 2
—
9 8 7
A
A K Q 4
J 8
—
10 8
9 8 7 6
North pitches a club whatever East/West do. The defense has taken five tricks, and North has a natural trump trick coming, so they just need one more trick. If East ruffs the spade with the diamond ace, that promotes a second trick for North. Bad plan. Similarly if West ruffs high. So West must ruff low.
West can't lead a heart (it gives South another ruff) or a club (North ruffs and gives South a ruff) so he must lead a diamond, but then East is stuck leading clubs or hearts, and the defense scores its seventh trick.
The position above is interesting. If West ruffs low and North chooses to overruff, East ruffs with the ace, crosses to the club jack (since North still has one), takes two top diamonds and exits a diamond, retaining a club entry still left to East and no more losers. Only the club pitch works here.
If North/South declare diamonds, the defense starts with a diamond to the ace, a club back to the jack, two more top diamonds and and then more clubs. Whenever North ruffs, West is left with two small trumps, and the defense ultimately takes five diamonds and two clubs.

Clubs

If North/South declare clubs, the defense attacks in its usual fashion - diamond ace, club to jack, two top diamonds, and three more clubs.
If East/West declare clubs, the defense starts with two top spades, a spade ruff, the heart ace, South pitching a diamond, and finally the club ten.
—
Q J 3 2
J 9 5 3
10
—
K 10
K Q 7 6 2
K 2
—
9 8 7 6
A
A K Q 4
J 8
—
10 8
9 8 7 6 3
This has the usual affect of forcing East/West to win a club entry early, but it seems like East/West should be able to make seven tricks some way. How will North/South manufacture three tricks?
Say East wins in hand, cashes the diamond ace, and leads a heart. South doesn't take the bait, and instead pitches his last diamond and West is wins the king to this position:
—
Q J 3
J 9 5
—
—
10
K Q 7 6
J
—
9 8 7
—
K Q 4
J 8
—
—
9 8 7 6
If West plays a top diamond, South ruffs and exits a club. On a second top diamond, South ruffs again and exits a spade, East force to ruff high.
Other lines are similar.

Complexity

This is another complexity three deal, with only the diamonds suit being par-zero on its own.

Thomas Andrews; © 1999-2009.

Who Gets Stripped?

K 8 2
A K Q 5
7
A 8 7 4 3
A J 10 9 7 5
10 8 6 2
A K Q
—
4
7 4 3
J 10 3
K Q 10 9 5 2
Q 6 3
J 9
9 8 6 5 4 2
J 6
Another deal with a void, this one with three hands holding six-card suits.

Notrump

If East/West declare notrump, North/South strip four hearts from West, then cash the club ace and exit a diamond. West, stuck leading from his hand, can't avoid two spade losers.
If North/South declare notrump, West starts with exactly one top diamond, then exits with a heart. The diamond play seems innocuous, but it strips North of his satisfactory exit after the hearts are taken.
If South wins the first heart and exits a spade, West finesses, and North wins and is stuck at:
8 2
A K Q
—
A 8 7 4 3
A J 10 7 5
10 8 6
K Q
—
—
7 4
J 10
K Q 10 9 5 2
Q 6
J
9 8 6 5 4
J 6
North can either set up four more tricks in West or three more tricks in East.
In general, whatever South exits after winning the first heart, North is stuck.

Spades

If East/West declare spades, the defense starts with a diamond lead. The defense inevitably gets four hearts, two spades, and a diamond ruff.
If North/South declarer spades, West just plays ace and another spade, killing the ruff threat immediately.

Hearts

If North/South declare hearts, the defense starts with the spade ace, spade ruff, club ruff, another spade ruff and a diamond. Finally, West exits with a heart at this position:
—
A K Q 5
—
A 8 7 4
J 10 9
10 8 6
K Q
—
—
7
J 10
K Q 10 9 5
—
J 9
9 8 6 5 4
J
The South hand is useless - it can't ruff a club, because West will ruff the club ace, and the diamonds can't be set up because there is no entry to get back to them. Indeed, North/South take can only take four hearts and the club ace, and must concede the rest to East.
If East/West declare hearts, the defense just plays four top trumps, then North defends as she defended against notrump - cash the club ace and exit with the stiff diamond.

Diamonds

If North/South declarer the diamonds, the defense starts with ace of spades, spade ruff with the three and five more high cross-ruffs.
If East/West declarer diamonds, the defense starts by leading four rounds of heart. South pitches a spade on the third round and, if East ruffs the fourth round low, South overruffs. If East ruffs the fourth round high, South pitches another spade. Down to one spade, South can keep East from ever scoring the diamond three, and North/South get three hearts, three diamonds and either a spade or a club.

Clubs

If East/West declare clubs, the defense starts with a diamond. West wins, plays the spade ace and spade ruff, then leads a top club. North wins, leads a low heart to South's jack, who gives North a diamond ruff, then North plays the rest of his hearts, South pitches a spade on the third round, leading to this position on the last heart:
K
Q
—
8 7 4
J 10 9
10
A
—
—
—
J
Q 10 9 5
—
—
9 8 6 5
J
The defense has taken five tricks - three hearts, a diamond ruff and the club ace. If East pitches a diamond, South pitches as well, and North plays a spade, still promoting South's club jack. If East ruffs low, South overruffs and gives North a diamond ruff. And if East ruffs high, the defense has another natural trump trick.
If North/South declare clubs, the defense starts with the spade ace, spade ruff, diamond ace, spade ruff, and a heart at this position:
—
A K Q 5
—
A 8 7 4 3
J 10 9
10 8 6 2
K Q
—
—
7 4 3
J 10
K Q 10 9
—
J 9
9 8 6 5 4
J 6
If South wins the jack and gives partner a diamond ruff, North can run his herats, but on the fourth heart, East can pitch a diamond:
—
A
—
A 8 7 4
J 10 9
10
K
—
—
—
10
K Q 10 9
—
—
9 8 6
J 6
East is assured of getting three clubs now.
Unlike before, there is no way for North/South to promote the club jack and get three more trump tricks. When North/South had the opening lead, they could control the spade situation by killing the entry to the West hand with the diamond lead, making sure East could not ruff two spades in the East hand and strip the suit.
Compare this position, for example, with one spade left in each of North and South's hand:
K
A
—
A 8 7 4
J 10 9 7
10
K
—
—
—
10
K Q 10 9 5
Q
—
9 8 6
J 6
When North leads a heart here, if East pitches a diamond, South pitches a spade and North leads a spade, promoting South's jack. North still has the ♣ A, so East/West get only three of these tricks.

Thomas Andrews; © 1999-2009.

A Complexity Four Example

A 7 5 4
J 9 7 2
3
Q 6 4 2
Q 9
K 4
K 10 4
J 10 9 8 7 5
J 8 6 3 2
10 8 6 5
A Q 2
A
K 10
A Q 3
J 9 8 7 6 5
K 3
This deal has complexity four, as described in the Introduction.

Notrump

North/South can set up two clubs, three hearts, and two spades. East/West can set up three spades, one heart, three diamonds, and a club. The key, then, is timing.

North/South

If North/South declare notrump, the defense starts with a diamond to East, followed by a spade lead (spades cannot be effectively established by leading from West.)

East/West

If East/West declare notrump, the defense starts with a club. The best declarer can do is lead a spade to try to break up the communication between the North/South hands. South takes the ♠ K, and cashes the ♣ K.
A 7 5
J 9 7 2
3
Q 6 4
Q
K 4
K 10 4
J 10 9 8 7
J 8 6 3
10 8 6 5
A Q 2
—
10
A Q 3
J 9 8 7 6 5
K
What does East pitch? If East pitches a heart, then South can play the ♥ A and then the ♥ Q, setting up two hearts in dummy, along with the three tricks taken and the ♣ Q and ♠ A in dummy.
If east pitches a low diamond, south plays the ♥ A and the ♥ Q. West wins, but what does West play?
A 7 5
J 9
3
Q 6
Q
—
K 10 4
J 10 9 8
J 8 6 3
10 8
A Q
—
10
3
J 9 8 7 6 5
—

Spades

North/South

????

East/West

????

Hearts

North/South

????

East/West

????

Diamonds

North/South

If North/South declare diamonds, the defense starts with the ♣ A, then a heart shift. Whatever declarer does, he cannot avoid a heart loser, a club loser, and five seperate trump losers in a cross-ruff.

East/West

If East/West declare diamonds, North/South start with diamonds, keeping East/West from cross-ruffing diamonds.

Clubs

East/West

If East/West declare clubs, North/South start with diamonds. The goal is to score four trump tricks via ruffs, in addition to a heart and two spades. Say declarer wins in the East hand, and cashes the ♣ A. Unfortunately for East/West, there is no entry to the West hand to continue clubs:
A 7 5 4
J 9 7 2
—
Q 6 4
Q 9
K 4
10 4
J 10 9 8 7
J 8 6 3 2
10 8 6 5
A Q
—
K 10
A Q 3
J 9 8 7 6
K
If a spade is led from dummy, South wins the king, gives North a diamond ruff, a heart is returned to South's ace, and North ruffs another diamond. North cashes the spade ace and then leads a spade at this position:
7 5
J 9 7
—
Q
—
K
—
J 10 9 8 7
J 8 6
10 8 6
—
—
—
Q 3
J 9 8
K
South ruffs high, and West can either pitch his heart king or underruff. In any event, the defense scores two spades, a heart, two low diamond ruffs, a spade ruff high, and North's ♣ Q.

North/South

If North/South declare clubs, the defense starts with the ♣ A, then a heart. Say South wins and leads a diamond, won by West. South loses a trump to the king in South, and North can get one diamond ruff, but declarer is held to two top trumps, a diamond ruff, two spades, and a heart.

Comments

What does complexity four mean? Essentially this means that none of the individual suits are parzero. For example, if we take the spade suit:
A 7 5 4
Q 9
J 8 6 3 2
K 10
and construct a symmetric deal from this distribution:
A 7 5 4
J 8 6 4 3
K 10
Q 9
Q 9
A 7 5 4
J 8 6 4 3
K 10
J 8 6 3 2
K 10
Q 9
A 7 5 4
K 10
Q 9
A 7 5 4
J 8 6 4 3
this resulting deal is not par-zero.

Thomas Andrews; © 1999-2009.

Untitled I

K J 9 6
10 6
Q 7 4 3
A 10 2
10 7 5 3
Q 8 4 2
8 6 5
Q J
A 8 2
J 5
A K J 2
K 6 4 3
Q 4
A K 9 7 3
10 9
9 8 7 5
INCOMPLETE

Notrump

Spades

Hearts

Diamonds

Clubs

If North/South are on lead, South cashes two top hearts and leads a third heart, with North ruffing with the ♣ 10. East can't do any good by over-ruffing, so he pitches a spade. North leads a spade which East ducks, South winning the queen and leads another heart, North ruffing with the ♣ 2, and East overruffing.

Thomas Andrews; © 1999-2009.

Untitled III

Q
Q 7 6 3
10 8 6
K J 8 6 3
K 7 5
K
K Q 4 3 2
10 7 5 4
9 6 4
A J 9 5 4 2
J
A 9 2
A J 10 8 3 2
10 8
A 9 7 5
Q

Notrump

If South is on lead, the defense starts with the ♠ A and a spade. West wins the second or third round of spades, and cashes the ♥ K and exits in diamonds, South winning the ace. South then runs his spades reaching this position:
—
Q 7
10 8
K J
—
—
K Q 4 3
10 7
—
A J 9 5
—
A 9
—
10 8
9 7 5
Q
South has taken six tricks, so when he leads the ♣ Q, East must win, but then has to concede a winner to North's hand.
So perhaps East/West should have killed South's club exit before leading diamonds? East wins the club shift, cashes the ♥ A and then leads the ♦ J. After South runs his spades, this end position is reached:
—
—
10 8
K J
—
—
K Q 4
10
—
J 9
—
9 2
2
—
9 7 5
—
West has to pitch on the last spade, but if he pitches a diamond, he has to give up a club trick to North at the end, and if he pitches the club, he has to give up a diamond trick to South.
If North/South declare notrump, the ♥ K is cashed, then a diamond is led to the jack.
If South ducks this, East cashes his ♥ A and leads a high heart, setting up his heart suit before the ♣ A entry is taken.
So South must win the diamond. The problem is, that was South's only entry to his spades, and North's hand is one trick shy to make up for it.

Spades

If North/South declare spades, the defense gets two diamonds, a diamond ruff, two hearts, a club and a spade. The North hand is made completely irrelevant.

Hearts

Diamonds

Clubs

Thomas Andrews; © 1999-2009.

Untitle IV

A 6 2
K Q 9 8 2
7 6 5
J 2
K Q J 10 8
10 6 5 4
Q
A 10 5
5 4
A 7 3
K J 10 9 4
8 6 3
9 7 3
J
A 8 3 2
K Q 9 7 4

Notrump

If East/West declare notrump, the defense starts with a heart. If East ducks the heart, North/South set up four clubs, and have a heart, a spade, a diamond and four clubs. So East must win.
Attacking diamonds is useless, because East has no entry - South just wins the first diamond and sets up clubs.
So East must attack spades. North wins the first spade, takes a high heart, then attacks clubs. West can win and take four spades, but those are his last tricks. North/South get a spade, a heart, a diamond and four clubs.
If North/South declare notrump, the defense is much the same, with the suits reversed - East/West attack diamonds, and South must win, or East/West shift to spades, setting up four spade tricks, plus a trick in each suit.
If South attacks clubs, West wins the second, then shifts to the ♠ K. If North ducks, West leads a heart to East's ace and East takes four diamonds for a total of seven tricks. So North must win the spade, but now is stuck with no entry to the good clubs, and the heart suit un-established.
So, after winning the first trick, South could lead only one club, then shift to hearts. East wins, cashes one diamond (West pitching a club) and exits a spade:
A 6 2
K Q 9 8
6
2
K Q J 10 8
10 6 5
—
A
5 4
7 3
J 10 9
8 6
9 7 3
—
8 3
K Q 9 7
If North ducks or wins, it doesn't matter, he has only two heart tricks along with his earlier diamond ace and club.

Spades

If North/South declare spades, East/West start with spades until North wins. The best North can do is win the second spade.

Hearts

Diamonds

Clubs

Thomas Andrews; © 1999-2009.

Untitled V

A K 3 2
9 7 6 2
Q 10 9 5
Q
Q J
A 8 5
A 8 3 2
A 9 7 3
10 9 8 4
J 10 3
K 7 6
J 6 2
7 6 5
K Q 4
J 4
K 10 8 5 4

Thomas Andrews; © 1999-2009.

Untitled VI

10 9 3
A 8 5 3
A K 10 9 8
3
K 6
Q 9 6 4 2
3 2
A 8 7 2
A Q J 7 2
K
J 7 6 5
10 5 4
8 5 4
J 10 7
Q 4
K Q J 9 6

Thomas Andrews; © 1999-2009.

Untitled VII

K 9 8 2
A
A Q 10 4 3
Q 10 9
A 7 6
J 10 8 3
7
K J 8 6 2
J 10 3
7 6 5
K J 9 8 5 2
A
Q 5 4
K Q 9 4 2
6
7 5 4 3
This example was found by Jan Stroem.
This deal has the odd property that everybody can make six tricks in any denomination, played from any seat.

Thomas Andrews; © 1999-2009.

Untitled VIII

9 7 6 3
10 4
K J
K J 8 6 4
K Q 8 5 2
K 5
10 7 3
A Q 7
A
Q J 8 2
9 8 5 4
9 5 3 2
J 10 4
A 9 7 6 3
A Q 6 2
10

Thomas Andrews; © 1999-2009.

Untitled IX

6 5 2
4 2
A Q 10 6 2
A 10 3
8 7
9 7 5 4
K Q J 10 7
Q 8
A K 4 3
8
J 8 3
K J 7 5 2
Q J 10 9
A 9 6 5 3
K
9 6 4

Thomas Andrews; © 1999-2009.

Untitled X

J 10 4
5 3 2
3 2
K J 10 5 3
A Q 8 2
A K Q J 7
J 9 8
8
9 5 3
6
7 6 5 4
Q 9 6 4 2
K 7 6
10 9 8 4
A K Q 10
A 7

Thomas Andrews; © 1999-2009.

Untitled XI

9 5 2
7 4
K 10 8 6 2
10 5 3
7 3
A 10 9
J 9 7 4 3
9 7 4
Q J 10 8
K J 2
Q 5
A Q 6 2
A K 6 4
Q 8 6 5 3
A
K J 8

Thomas Andrews; © 1999-2009.

Untitled XII

6 2
A Q 8 7 4
10 7
9 7 4 3
J 9 7 5
K J 6
A K 6 4 2
5
K 10 3
10 9 5
9 3
A K 8 6 2
A Q 8 4
3 2
Q J 8 5
Q J 10

Thomas Andrews; © 1999-2009.

Untitled XIII

Q 10 9 2
Q 7 6
A K Q 3 2
2
4
A K 10 9 5
J 8 4
K 7 6 5
A K 7 6 5
8 2
10 7 6 5
J 8
J 8 3
J 4 3
9
A Q 10 9 4 3

Notrump

If North/South declare notrump, the defense starts off with a low heart, eventually getting four hearts, two spades, and either a club or a diamond. South's immediate tricks are three top diamonds, a club, and the first round heart.

Spades

If East/West declare spades, the defense starts with three top diamonds and a diamond ruff (with the eight,) followed by the ♣ A-Q, North ruffing. North leads the last diamond at this position:
Q 10 9
Q 7 6
2
—
4
A K 10 9
—
K 7
A K 7 6 5
8 2
—
—
J 3
J
—
10 9 4 3
North/South have taken the first six tricks, with a natural trump trick remaining, but the diamond lead brings the defense total to eight tricks.
If North/South declare spades, the defense starts off with three rounds of spades. South gets an entry too early.

Thomas Andrews; © 1999-2009.

Untitled XIV

K 3
Q
A Q 7 6 4 3 2
10 7 2
Q J 6 5
8 6 4
J 10 9 8 5
A
8 4 2
A 10 9 2
K
K Q 5 4 3
A 10 9 7
K J 7 5 3
—
J 9 8 6
A case with one hand holding a 7-card suit.

Diamonds

If North/South declare diamonds, the defense starts off with the ♣ A, the ♥ A, and two more top clubs, for four tricks. East then exits the ♦ K, and West has three natural trump tricks.
If East/West declare diamonds, North starts with the ♥ Q, destroying East's entry to the club winners. North has two spade tricks and four natural trump tricks. If East ducks the heart, that is North/South's seventh trick, while if East wins it, South has a spade entry to two good hearts, and can pitch two of North's club losers.
This is purely a case where one side (East/West) needs to take its entries in a particular order, and the other side can keep it from happening if on lead.

Clubs

If North/South declare clubs, East/West start with the ♣ A, then the ♥ A, and two more top clubs. East then exits a heart to South, leading to:
K 3
—
A Q 7 6 4 3
—
Q J 6 5
8
J 10 9 8 5
—
8 4 2
10 9
K
5 4
A 10 9 7
K 7 5
—
J
North/South get another heart, a club, a diamond, and two spades, for six tricks total.
If East/West declare clubs, the defense starts with the spade king, the diamond ace, the spade ace, and a spade ruff by North, then the heart queen, again killing the heart entry before clubs are unblocked. Say East wins, and crosses to the club ace.
—
—
Q 7 6 4 3 2
10
Q
8 6
J 10 9 8
—
—
10 9 2
—
K Q 5 4
10
K J 7
—
J 9 8
There is no way to prvent South from getting two hearts and a club, and, in fact, North/South can make two club tricks via promotion. Suppose West exits in diamonds, North covering. If East ruffs high, South has two trump tricks. If East ruffs low, South overruffs, takes his hearts, then exits in spades or hearts, North ruffing with the ten, promoting a second trick. Similar promotions occur if West exits in spades or hearts.

Hearts

If East/West declare hearts, the defense starts with two top spades and a spade ruff, then the ♦ A and a diamond. North/South have taken four tricks, and East/West can't prevent them from scoring three trumps.
If North/South declare hearts, West takes his club ace, then the ♥ A, two top clubs and a club ruff, leading to:
K 3
—
A Q 7 6 4 3
—
Q J 6 5
8
J 10 9
—
8 4 2
10 9 2
K
5
A 10 9 7
K J 7 5
—
—
East/West have the first five tricks, and have a sure heart trick.

Thomas Andrews; © 1999-2009.

Untitled XV

Q 10
9 7 6 2
J 5 3
9 8 4 3
K 9 8
5 3
10 9 8
K Q J 7 5
J 7 6 2
A J 10 8
7 6 4 2
A
A 5 4 3
K Q 4
A K Q
10 6 2

Notrump

Fundamentally, this is about the timing for the unblock in clubs. West's only entry to run his clubs is in spades. East/West must unblock the club ace before the spade king is taken.
If North/South are declaring, then, the defense immediately takes the club ace, and then East leads a low spade. If South ducks, the defense has five clubs, a spade and the ♥ A. But if South takes the trick, he is no better off.
If East/West are declaring, then South wins a diamond and immediately leads a low spade to the queen. If West wins, his hand is finished, and declarer leads a spade to dummy's queen, leads a heart which East must duck, and then takes all his winners, exiting with the fourth spade. East gets a long diamond, a spade, and the ♣ A, but must eventually lead from the ♥ A, giving South a second heart trick. So West must duck the spade, North wins the queen, and leads a heart. If East flies the ace, that sets up seven tricks for South (three diamonds, two spades, and two hearts.)
So East must duck. South takes the rest of his diamonds, then plays ace and a spade, again killing West's entry before the clubs are unblocked.

Thomas Andrews; © 1999-2009.

Untitled XVI

J 7
A 4 3
K 8
9 8 5 4 3 2
A 10 9 4
9 2
10 5 2
K Q J 7
8 6 3
K Q J 8 6
A 7 4 3
10
K Q 5 2
10 7 5
Q J 9 6
A 6

Thomas Andrews; © 1999-2009.