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:

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.

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.

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: 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.

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.

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.)

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.