Backgammon Lesson – 7
Free Series of 13 Backgammon Lessons for Beginners
Paul Money
Beginners Please – Lesson 7
by Paul Money – October 2006
A lot of what we have done so far has been theory that demands a lot of attention and effort on your part. I hope that you’ve put the effort in; believe me it will all be worthwhile if you can build your game on a solid understanding of the basics.
This time I’d like to bring you a ragbag of odds and ends, all slightly peripheral to the game, but all of interest, so we can relax for a change and have some fun.
Last time you will recall that I mentioned the Jacoby Rule for money play, which stipulates that neither player may win a gammon unless the cube has been turned. It is never used in match play. The late Oswald Jacoby was one of the first players to really examine the theory of the game and he wrote a nice book in partnership with John Crawford, The Backgammon Book, which is still worth picking up – it is often available at second-hand bookstores. He also introduced us to the Jacoby Paradox, a nice set of positions from the end game that illustrate a curious facet of cube ownership. The first position looks like this.
This position is quite easy to work out. Cubeless, Red 9on roll) will win 64.5%. He wins almost 53% immediately and wins almost a quarter of the rest of the games. Nevertheless, giving the cube away is a large error, because it permits White’s devastating recube to 8 in 47% of the games.
When White is on roll with one checker on the 6pt, she can win 75% of the time, a perfect position for a double. The paradox that Jacoby spotted is that if you make Red’s position weaker by moving White’s checker to the 1pt, then Red does have a redouble! White will never be able to recube, so all the games will end at the 4 level.
The other four positions where the Jacoby Paradox applies are where White has checkers (a) on the 5 and 1pts, (b) on the 4 and 2pts, (c) the 3 and 2pts and (d) two on the 2pt. Is it very useful to remember this? Probably not. I only recall seeing this position once in real life in the last 17 years!
When White redoubles with one checker on the 6pt, should Red take or pass? His equity is identical in both cases, always losing 4 points when he passes, and losing 4 pts per game on average when he takes.
Note the subtle difference between “always” and “on average” though. If you take you sometimes win 8 points and of course more often lose 8 points, but at least you sometimes win. Passing you always lose! Thus most players prefer to take.
In money play, it is often agreed to allow beavers. The beaver is a third option open to players who are doubled. As well as the take or the pass, they may also say “Beaver”, take the cube and immediately turn it to the next level. The cube stays on their side. Of course usually it only pays to do this if you are the favourite to win the position after taking the cube.
The position above is not a beaver if Red redoubles for example, because he is still a favourite even though his redouble is wrong. However, the Jacoby Rule for money play raises the possibility that a position may exist that is a correct double and also a correct beaver. The doubler may be an underdog to win the game, but have a strong gammon threat that he activates by doubling.
For years people constructed highly unlikely positions where the doubler won only 30-33% of the games but always won a gammon when he did win. This type of position is called a Kauder Paradox, named for Jimmy Kauder who is thought to have first proposed this shortly after the introduction of the Jacoby Rule. I have only seen one plausible example in real life and it cropped up in a chouette. UK expert Rick Janowski spotted this one and from memory, it looked like this.
Red is on the bar. If he doesn’t double, then he will win whenever he hits and that happens 14 times out of 36. He will win because after hitting, he doubles and White passes. He will lose all the rest because if he misses, White doubles him out. In 36 games he loses 8 points in total, 8/36 = 0.22.
If he doubles and White takes, now gammons will count. He will win enough gammons to almost become the favourite, but not quite. After he doubles and White takes, Red’s equity gets to about -0.039. White must beaver so that Red’s equity will drop to –0.078. This is still better than Red would get if he didn’t double, so hey presto, it is right for him to double and right for White to beaver! How useful is it to know this? Not very, but not everything beautiful is useful!
There are many paradoxes in backgammon and one of the best textbooks of the seventies was Barclay Cooke’s “Paradoxes and Probabilities”. This and his “Backgammon, The Cruelest Game” still come up on E-Bay in nice hardback editions with dust jackets for very little money. They make a nice addition to any gammoner’s bookshelf, although the answers to a lot of the problems can no longer be relied on.
Cooke was writing in an age when there were of course no computers to help and he was one of the very few people who were thinking about the game in an analytical way. Perhaps as a companion volume, you should also obtain “Classic Backgammon Revisited” by Jeremy Bagai. Jeremy is fresh from winning the Super Jackpot in Monte Carlo this year, $100,000 worth and is an excellent writer on the game. His book looks at several early textbooks, corrects their errors and explains how and why the giants of yesterday went wrong. I recommend it to you.
Of course the truly great volume from the seventies is “Backgammon” by Paul Magriel. It still can’t be bettered as a comprehensive guide to ways of thinking about the game, although again, some of the positions can now be shown to be incorrect. The original hardback editions are much sought after, but it has now been reprinted in soft covers with new diagrams, so you may think that to be better value.
Most of the rest of the text books from those days are, frankly, worthless, merely recycling the ideas of other writers, although one who did have some original ideas was Bruce Becker. Sadly, his volume “Backgammon For Blood” is arguably the worst book on the game ever written. Buy it for fun, but don’t copy his playing style! It is apparently, a very popular book in the Bush household, so if you ever get the chance to play Dubya, you might have some fun and win some money.
There can be few greater honours than to have a position named after you. The last one above should really be called Janowski’s Position, although he is probably better known in the game for his theoretical work on Live and Dead cubes. Snowie uses his theories for the algorithms that control its cube actions.
The following position is a fun one, known as Robertie’s Position. Bill Robertie has written many books on the game, most of which are excellent and was the first man in the modern era to win the World Championship twice, in 1981 and 1987. This is called Robertie’s Position, because he claims to have played it three times and lost 16 points on each occasion!
Now if you have followed all the articles so far, you will remember that we discussed two roll endings in Lesson 3. You will remember that a pure two-roll ending is a double and a pass. However, this is not a pure position. White has some extra ways to win. Red only has four winning doubles instead of the usual six and he will also lose if he rolls two consecutive ones. That’s worth 10% to White in itself and there is also the small sequence where Red rolls x-2 followed by 2-1. All this adds up to 24.7% for White, just fractionally short of a cubeless take, but there is a cherry on the cake as well.
Usually White has no recube to consider in a two roll ending, but here she can turn it back at 8 after Red rolls 2-1! Although she has three checkers left to Red’s two, she wins immediately with all her doubles and will still be a small favourite even if she rolls a singleton. You may not see this position very often, but the redouble with three or four checkers against two does come up, so it is well worth remembering.
The Robertie Position is the reference point for two-roll endings. As White you can of course take if Red’s position is worse in any way. It is much more common to see this with the spare on the 3pt moved to the 4pt for example.
Three-roll endings are very close to being takes for money anyway, so any weakening of the doubler’s position may well indicate the chance for a take. This next position however, is not even a double although it looks like a three-roll ending.
As a little thought will reveal, Red will lose immediately if he rolls any 2 (30%). Add this to White’s basic 21% in a three-roll position and add on several losing sequences next turn as well, to see that White is actually a small favourite here and should beaver an improper double. I call it my position because I have beavered this three times as White, losing two 8 cubes and a 16 cube in the process!
This position demonstrates the necessity of mastering a skill that Robertie calls “Reading the Numbers”. All this means is that you quickly scan through the rolls on your next turn to see what harms you and what is good. You can ignore the doublets for most practical purposes as they usually fall into the “good” category, so you just need to look at the 15 singleton numbers.
Doing that here as Red will save you from an expensive mistake, unless you are playing me of course, in which case history suggests that you should double and I will beaver and lose!
Earlier I mentioned the name of John Crawford, a writer and top player of the sixties and early seventies. He gave his name to the Crawford Rule, universally used in match play. It stipulates that the cube may not be turned in the next game after one player has reached one point away from victory. We haven’t looked at any match play, but next time we will and we will start with the 2-point match.
This format is popular on backgammon servers for quick tournaments and is also often seen as a sideline at many major live events. It is essential to know how to play these well, not least of course because many matches get to a stage where both players need two points to win, or 2-away, 2-away as it is known. For the first time we will get to look at those mysterious columns in the response charts called GammonGo and GammonSave!
That’s for the next lesson, for now let’s finish with this one with another response chart.
Note: The moves suggested above are the #1 results of rollouts, however, sometimes other possible moves for a dice roll may have been listed as very close or as a reasonable alternate.
24/15 isn’t often seen as an opening play, but perhaps it should be with 6-3 at least. It is very nearly as strong as 24/18, 13/10 and certainly less complex, so a nice play for beginners. If you try it, you will find people particularly reluctant to hit with 3-1! As before, don’t try to learn these charts, but do keep a record of your plays in the opening (and those of your opponents) so that you can check later to see if you are on the right track.
Homework for this time? Check out E-Bay for some classic books and search your local second-hand bookshops and charity shops. A nice little shelf of books will wile away the time in between backgammon sessions!
Until our next lesson, enjoy the game!