This column gets to the meat of my review of Jeff Rubens’ book *Expert Bridge Simplified: Arithmetic Shortcuts for Declarer, *2009, The Bridge World Press. My previous column used his introduction to prior probabilities and restricted choice as an excuse for me to give my flavor on those topics. Here, I’ll introduce a topic I’d never encountered until I saw it in this book and I’ll wager you’ve not seen it either: short shorts. If you’re not comfortable with simple squeezes, stop here.

*Simplified*, as in the book’s subtitle, does not mean simple. Short shorts is not one of the easier topics in the book, so don’t relax. Moreover it looks at suit combinations and suit splits in ways that you may find counter-intuitive. Let’s start with an example, from page 242 of the book, to get your attention, the way it got mine.

Imagine that you’re playing 7NT, where you have seven tricks combined in clubs and diamonds, with no entry difficulties. You hold hearts of AKxx opposite xxx and spades of AKQx opposite xx. That’s five more top tricks giving a total of twelve. If the hearts break worse than 3-3, then the heart guards are all in one opponent’s hand and for sure the spade guards are all in one opponent’s hand. If the same opponent holds the guards in both suits, then we can squeeze that opponent for one trick. How likely is the squeeze to work? That’s bridge arithmetic of the sort this book addresses and simplifies.

Here’s my answer from the olden days, BSs, that is Before Short shorts. I believe that a suit, such as hearts here, where I hold seven cards, with opponents holding six, will have those six split 3-3 about 35.5% of the time. That means the split will be worse than 3-3 about 64.5% of the time. About half that, 32% of the time, the spade guard will be with the long hearts. I think the squeeze will work a bit less than 32% of the time. Why less? Because when I cram four or more cards into one defender’s hand (but two or fewer into another), fewer places remain in the first hand to hold longer spades. Rubens introduced me to the terminology of describing a bridge hand as having 13 places. We use four or more of those with the hearts, leaving nine or fewer for the spades in one hand while using only 2 or fewer in the other opponent’s hand for hearts leaving eleven or more for the spades. The difference in free places to put the non-heart cards makes the surprisingly, to me, large difference.

From Rubens, I learn that the two guards will be together, not a bit less than 32%, but waaay less, only 22%.

Now to work. First, the term *remainder*. Rather than focus on our holding, say the seven cards AKxx opposite xxx, we focus on the six cards the opponents hold, which are collectively called the *remainder*. We are interested in the remainder’s splitting 4-2 split or worse, but we don’t look at the four (or more) that guard the hearts, we look at the two or fewer, and call them the short. In describing the desired outcome for our seven-card heart suit we say we want a 6-card short. We want the six cards to split so that the lesser length is two cards are fewer and if we get that, we call it a “6-card short” or a “6 short”. That is, we have a holding with a remainder of 6 and we want it split so that the smaller part of the suit split is two or fewer cards. Our focus in conceiving the squeeze may have been on the partner’s hand with the longer hearts, the four or more, the hand holding the guard, but for terminology purposes, we label the joint holding with the length of the short piece, not the longer one. Because the sum of the cards in the two hands must sum to the remainder’s total we can know the number of cards in one opponent’s hand by knowing the other’s. We have been accustomed to focusing on the hand with the (heart) guards, the hand with the four or more cards, but we could just as well say we want a split with the short holding to have two or fewer hearts. [Until now, if you described a squeeze position that way—I want East to have two or fewer cards so I can squeeze West—most people would think you are prolix, or even nuts.] That’s the terminology here: a 6 short means a suit where we hold seven (= 13 – 6) cards, the opponents hold six and we are interested in a split where one opponent holds four or more (and guards a suit or cannot overruff a dummy), but we use the nomenclature based on the partner/opponents’ short holding of two or fewer. The power of this terminology derives from its generalizing to holdings of different lengths. Consider that a 5 short means a suit where we hold eight cards, the opponents hold five and we are interested in a holding where one opponent holds three or more to guard a suit or has to follow suit, so that his partner has two or fewer. We call that a 5 short, seeking a distribution where the opponents’ five cards split 3-2 (or 4-1 or 5-0). Later, I think we learn that we don’t count the 5-0 case.

For the spade suit, in the example deal at 7NT, with a remainder of 7, we want it split with any short, because there is no way for that suit to split where more than one opponent has a guard, but we can still say we want a 7-card short. [I find the descriptions in the book of a short deficient because it doesn’t say how many cards need to be, or can be, in a short. I’ve struggled in writing this because sometimes I don’t know whether the short means just the doubleton or includes doubleton and singleton et al. The book does address the case of the so-called “long short short” but I find those discussions so hard to follow that I won’t try to weave its lessons into this introduction—as if I could. See Postscript II for *reductio ad absurdum.*]

Take any card holding in a given suit for two partners. The cards held by the opponents are collectively the remainder. Their number is the label on the short, which exists if the suit splits so that the smaller number in the split is a doubleton, typically. Sometimes a tripleton and sometimes a singleton. Think doubleton for clarity.

So we can say that the 7NT deal in the illustration depends on the success of a squeeze that will work if and only if there is a “6,7 short short” in Rubens’ notation. Sometimes he puts the larger number first.

Apparently, many deals come up whose success depends on the success of a short short. Rubens has computed, or caused to have computed, a table of the most common short shorts. It appears on page 231. This alone is worth the price of admission, but there is such uniformity in its results that you don’t need the table to memorize its key message. Rubens explains why **odd shorts are much more likely than even shorts, so that odd/odd shorts are twice as likely as odd/even ones, which are almost twice as likely as even/even ones**. Odd/odd short shorts (such as 7,7 or 5,7 or 5,5) all are about 40% likely; odd/even short shorts (such as 6,7 or 8,5 or 4,5) are about 23% likely; and even/even short shorts (such as 6,8 or 6,6 or 4,6) are about 12% likely. If one of the remainders is only 3 cards long, then add 3% for each such remainder to the total. So, for example, a 6,3 short short would be about 26% (= 23% + 3%) likely. If all you take away from this entire review is the **boldface **of this paragraph and the three numbers in the sentence following it, you’ll have the useful main message.

In the example above, involving the squeeze, we didn’t face a decision. We were curious about the probability of its success. We’re going to try the squeeze independent of whether it’s likely to work or unlikely. Below, I’ll show a hand from Rubens’ book where you need to choose between two lines of play and knowing how to recognize that you are in short short territory helps you to decide which.

A typical short short situation involves a case where you have one suit where you need to ruff some losers, but fear an over-ruff. You can’t, however, draw trumps to prevent the over-ruff because then you’ll have nothing in dummy to ruff with. You want one of the opponents to be long in the suit you want to ruff and simultaneously to be long in trumps. That opponent won’t be able to ruff. His partner who will be void in the suit you are ruffing will be void of trumps, so that opponent also won’t be able to ruff. That’s the opponent where we want to focus the short short laser. How likely is it that one opponent will simultaneously be short in trump and short in the suit we want to ruff? The short short table tells us. We need to know the remainders of the two suits. Assume, for example, we hold eight trumps (remainder of five), five opposite three, and a seven-card side suit (remainder of six) five opposite two, where we’d like to ruff once. We propose to draw two rounds of trump, play AK of the side suit and hope to ruff a third card in the side suit with the remaining trump in dummy, without an over-ruff. How likely is that play to work? We’re hoping for a 6,5 short short. The table tells us the success probability is 24%. But, don’t overlook that we also succeed if the side suit splits 3-3, 35.5% of the time, so we get the sum of those two chances, for a total of just under 60%. We get the benefit of the short short, but not just that.

Next is a deal where your having learned about the existence of the short short and the bare bones of the relative likelihoods will help you to the better line of play. This deal comes from page 232 of the book. Challenge yourself first to see if you can develop two competing lines of play to make 7 hearts, if clubs do not break 3-3, against West’s lead of the spade Ace.

From p. 232 of Rubens'

Did you think of two lines? This is the sort of problem chock-a-block throughout Rubens’ book: problem, challenge to find competing lines of play, then analysis to decide the better/best of the lines.

(a) Ruff a club in dummy. This requires the opponent with four or more clubs to hold three or more hearts. Can you give it the Rubens label? Clubs has a remainder of six and hearts has a remainder of four, so we want a 4,6 short short.

(b) Squeeze an opponent who holds three or more diamonds and the long clubs. Diamonds has a remainder of five, clubs still has a remainder of six, so we want a 5,6 short short.

What do you do? You have learned the **boldface **lesson above or if not, refer to it now….

We are comparing two lines of play, one of which involves an even/even short short and the other an odd/even short short. The latter is almost twice as likely to succeed as the former, so go with (b), the squeeze. I wonder if your bridge intuition, developed BSs, led you there anyhow.

This fantastic book will teach you as much as you can absorb. I’ll guess there are few who will say, I’ve mastered all in this book and I’m ready for more. Rubens did not exhaust my capability to learn this material. Rather the reader is likely to say Rubens took me to the frontiers of my ability to learn material of this sort. I’ll be able to return to these pages time and again, continuing to learn.

**Postscript**

The night after I completed the first draft of the above, I encountered the following board in a local club matchpoint game, which I declared at 4♥.

North Deals, Both Vulnerable:

On seeing the dummy I thought: if the opponents don’t lead a club, prematurely cutting me off from that solid suit, and if hearts don’t break, I’m facing a 5,6 short short to run enough clubs to pitch my losing spades before the long heart will ruff my clubs. That’s an odd-even short short, so has about a 24% chance to add to my 68% chance of breaking the hearts 3-2.

Nuts, after BBOer Trysche69 found the good lead of a diamond, cashed two diamonds and a spade, she wisely shifted to clubs and doomed me. Still, I was the only one in the room who conceived of computing the chances of success of that contract if the defenders slipped up. Even if they had, the short short didn’t materialize and the contract would have failed. Trysche69 may have cost me a trick, but not this postscript.

**Postscript II — This Can’t Be Serious**

The next afternoon, next after the short short described just above, I played the hand below in 5♦, with a club lead, and got so delighted on seeing dummy that the other three players thought I was having a seizure. How about an overtrick?

Migod, I thought, If one hand holds five or six clubs and all five diamonds, I can toss the losing hearts in dummy on the long clubs and ruff at least one of the hearts from my hand in dummy; probably not both before I run out of entries or trumps or something. I’ve eight cards in diamonds, remainder of five, and seven cards in clubs, remainder of six, so I have a 5,6 short short. Page 231, whose basics I’ve memorized, tells me odd even has about 23% chance to succeed.

*Ridiculous.*

Nowhere did I see in the chapter on short shorts any limits on when the short was too short to qualify for the table, but it can’t apply in this situation. I’m confident that somewhere the book alerts me not to take the reasoning this far, but I can’t point my finger at that teaching. [Do I need to explain why? I’d better. Diamonds splitting 5-0 all by itself is 4% (less than 23%); clubs splitting 5-1 or 6-0 by itself is 16% (less than 23%) and a parlay is so small, 0.6%, as to be nearly invisible. And don’t forget, the same hand has to hold both the shorts. The book shows you the combinatorial formulas you’d need to use to compute that.] I’m better off hoping to find the heart Ace in East, 50%, and making an extra trick that way. Still, I got to practice short short thinking at the table; almost as good as thinking about squeezes. Is there any wonder that I’m developing a reputation as the slowest player in the club?

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I must admit that a lot of the stuff in this post is way above my pay grade, but some of it does make sense. One thing that my simple brain can't fathom is that odd/odd = 40%, odd/even = 23% and even/even = 12%. If one of the shorts is 3 then another 3% is added to two of the combinations bringing us up to 81%. Is there a missing 19% somewhere ? If there is a 3,3 short short do we add another 3% ??