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:
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.