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Improve Your Backgammon
by Paul Lamford

Did you know that the English word “hazard” comes originally from a dice game of that name? The Arabic az-zahr meant “luck or chance,” a good name for a dice game. Our modern meaning, “to risk or venture” evolved from that. It is believed that the English expression “at sixes and sevens” referred to the game of hazard. Sixes and sevens were hard points to make in hazard. (Sevens especially, with only one die!) Six hundred years later the expression means “to be confused.” And that is how I am – at sixes and sevens – as I begin this review of Paul Lamford’s new book Improve Your Backgammon. (Paul’s is the only name on the cover, but he credits Simone Gasquione and Stefanie Rohan as co-authors in his introduction.)

To whom, precisely, am I recommending this book? That’s my dilemma.

Black’s dilemma – fewer shots, or an extra man off? – is a common one. In this case, fewer shots is correct, since White cashes when Black has four checkers off, but otherwise is near a very efficient cube turn when Black has born off five.

Paul offers the rule of Sixes and Sevens as a guideline. If you can bear off a sixth checker leave one extra shot (as at left), and if you can bear off your seventh (through tenth) it is okay to leave two extra shots. This is a simple and useful rule, as long as you can remember the mnemonic “at sixes and sevens over whether to leave extra shots.”

But if you are struggling with that, are you ready for this?

Black is on roll leading 8-5 to 11. To decide the proper cube actions – Should Black redouble? Should White take if Black does? – it would be handy to know within 1% Black’s cubeless winning chances in this race. Paul has a method for determining just that.

Paul’s formula is: P = 50% + (900 + R + 400L)/(R + 7L + 25)% where P is the winning percentage, R the leader’s race length, and L the difference in adjusted pips. In the position above, you will be happy to learn, no adjustments are needed, so we can just crank out the answer without breathing hard. I am sure most of you finished before I typed the last sentence, but if not, let’s step through it. Black leads 95-119, so R is 95, and L 24. So 900 + 95 + 400*24 = 10,595, and that must be divided by 95 + 168 + 25, or 288. It comes to about 37%, which, added to 50% brings us up to a cubeless winning probability of 87%. A rollout finds we have come within 1%, as the actual CPW is 88%, and, Paul informs us, confirms that a redouble is marginally correct.

It isn’t clear that Paul would have actually redoubled at the table. A chart much later in the book for “gammonless redoubles” puts the doubling window at 87-89%. It is usually wrong to double at the bottom of the doubling window except in last shake situations – not the case here. So, the formula’s triumph is not unequivocal, but to be fair, missing your redouble in this situation is not critical, and the formula does a good job of confirming the need to take, which is. Another bit of rain on Paul’s parade is that the problem comes directly beneath a table of race advantages keyed to length and lead, and should one use the table instead of the formula for this particular race one would put Black’s winning chances at 91%, a 3% error.

Those quibbles aside, the formula seems quite effective. The question is: who will use it? I’m afraid it might scare the socks off of most beginners, and old timers are often set in their ways. Still, for the advancing intermediate, or any others with ambition, a flair for math, and a good memory (egad, he wrote this book for me!), it is worth their while learning it. I am reminded of a conversation I overheard years ago. A player was talking with Bill Davis one Tuesday night, and said: “I realize I’m an intermediate, and in most situations I can’t compete with the experts. But I think there is one area I can outdo them. When I count a race, I use the Magriel, Ward, Robertie, Thorp, and Kleinman counts, and if they don’t all agree I go with the majority.” His math was questionable, but there was always a chance his opponents might expire while he was counting.

What else will you find here? A lot! The book examines the concept of equity, of how it is calculated, and how money and match equities differ – not just whether a decision changes in a match, but why it does. Among other things, Paul offers his own formula for computing match equities while at the table, and his own match equity charts. These are a lot of wheels to reinvent, especially in a book that runs only 120 pages, some of those devoted to a glossary and lists of books and internet sites. Wherever he does dip his toe in the water though, he dips it deep. It makes me wish that he had picked one area to focus on, and put in the same amount of work, as he might have produced a classic, instead of merely a good book. (He might also have caught two embarrassing goofs. His solutions to parts A and C of problem 13 are both wrong, due solely to poor proofreading would be my guess.)

Here’s an example of the sort of problem that is well handled.

It’s a money game – should Black redouble, and if he does, should White take? Now I could give you an answer to both parts of that question, but in some ways the answer is irrelevant. This position is worthless as a benchmark; there are too many variables. So an answer in a vacuum is also not worth much. By itself, all it tells you is what to do for a problem you will never actually face. However, the explanation of how to answer this question for yourself, at the table, that is valuable. It (the explanation) is also somewhat long, too long to reproduce here, and I am sure any paraphrase I gave would not do justice to Paul’s actual words. I’ll leave you to look it up for yourself. The book is a bargain at $14.95. You can afford it, and it is worth it.

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