Introduction

When to Double By Phil Simborg (2005). One of the best players in the game, Kit Woolsey, taught me what I consider to be the single best tool for winning backgammon: Woolsey's Law. When you are trying to decide whether or not to double, put yourself in your opponent's shoes. Ask if you are sure he has a take or sure he has a pass. If you are not sure of either, then double! Introduction to the Doubling Cube By Paul Money (2006). It is never too soon to learn cube play. Why? Because if your cube action is good, you will be able to survive in quite strong company. The reverse is not true. Poor cube play will see you destroyed by clever cube handlers. Be a PRAT By Alan Webb (2000). In his book, Improve Your Backgammon, Paul Lamford uses the acronym PRAT as a guide to doubling and accepting. PRAT stands for Position, Race And Threats. Lamford recommends doubling when you have an advantage in two of three areas. And a clear advantage in all three means your opponent should pass. Doubling Cube Strategy By Hank Youngerman (2002). You will never be successful at backgammon if you don't know proper cube play. Many beginning and intermediate players play like they're afraid of the doubling cube. In my experience, beginning players tend to take much too often, and intermediate players tend to drop much too often. The Doubling Rule By Kit Woolsey (1991). Reprinted from Inside Backgammon in 1991, this is the original exposition of Woolsey's Law for Doubling. Woolsey's Law states: If you are not 100% sure that the position is a take, then it is always correct to double. Threading the Needle By Kit Woolsey (1999). How the doubling cube can affect checker play. Players sometimes say "I would have made a different move if the cube had been in the center" or something like that. Do they have any idea what they are talking about, or is it just rationalization?. Reference Positions By Kit Woolsey (2001). How do we make our cube decisions? Our judgment is mostly based on experience, from positions we have seen before. The best approach is to understand several common types of positions and know whether they are passes or takes and by how much. Then, when we see a position over the board, we can relate to a similar position which we know. We make adjustments for the differences, and come to a conclusion. Reference Positions By Hank Youngerman (2000). In this article we will discuss the general idea of reference positions. If you can find these break-even points and remember them, you will have taken a huge step toward improving your game. Cube Handling By Tom Keith. Articles on the strategy and theory of cube play. From the Backgammon Galore Forum Archive.

Cube Theory

Take-Points in Money Games By Rick Janowski (1993). This article was written in 1993. It presents a model of doubling in money games which allows for gammons, backgammons, the Jacoby rule, and beavers. Danny Kleinman checked through this work and acknowledged it as the most accurate model to date. (Warning: Heavy-duty math.) Doubling Strategy in Backgammon By Peter Max Friis Jensen (2003). An overview of the doubling decision. The rules of doubling, the value of holding the cube, when to double, when to drop, market losers, the doubling window, cube handling in match play, tables of take points. Doubling Theory By Kit Woolsey (1999). How can we determine if it is theoretically correct to double? Although this article is theoretically oriented, some understanding of the concepts involved will be of practical value for making doubling decisions at the table. Doubling Theory and Market Losers By Hank Youngerman (1999). The goal of doubling and taking or dropping is not to "cash" a game that is already all but won, or to show your resolve not give up easily. It is not to end the game, nor to prolong it. The goal of doubling is to increase your long-run expected number of points won per game. Cube Paradoxes By Paul Money (2006). A potpourri of cube-oriented positions with names: the Jacoby Paradox, the Kauder Paradox, and Robertie's position. Doubling the Stakes and Brownian Motion By Jochen Blath and Peter Mörters (2001). This article presents a mathematical model for the doubling cube and calculates the appropriate time to offer a double, and to accept or drop a double, based on the assumption that the players' chances in the game vary continuously.

Holding Games

Doubling in High Anchor Games By Bill Robertie (2006). A look at doubling strategy in high anchor games. In almost all normal 5-point holding games, you have a pretty easy take. Your opponent needs about a 15% lead in the pip count to have a good double. More About High Anchor Games By Bill Robertie (2006). Some exceptions to the rule that any 5-point holding game is a take. Low Anchors By Paul Money (2006). Cube handling when your opponent holds your ace-point, two-point, or three-point.

Blitzes

The Blitz By Kit Woolsey (2000). Cube decisions in blitzes can be difficult. It is easy overestimate the danger of the blitz and pass a clear take, or overestimate the defensive chances and take a clear pass. Unfortunately there is no yardstick one can use to measure a blitz position and decide whether or not it is a take. There are a lot of different variables which have to be put together properly to come up with the right answer. Evaluating Blitzes By Bill Robertie (2007). Blitzing positions as a group are probably less understood than any other group of positions. They don't have much of a theory connected with them. Instead, players learn benchmark doubling configurations and their associated values as a guide to assessing positions that arise in actual play.

Going for Gammon

Too Good? Too Bad! By Paul Lamford (2000). How to tell when you are too good to double. For it to be correct to play on rather than cash in a money game, the general rule is that you must have twice as many gammon wins as losses. Playing for the Gammon By Kit Woolsey (2000). The question of whether or not to play on for a gammon can be a very perplexing one. Here are some examples of positions where it is better to play on for another roll than double immediately.

Miscellaneous

What Lies Beyond the Board? By Frank Frigo (2006). Efficient doubles are all well and good, but having a strong suspicion that your opponent will pass a proper take, or take a proper pass can be a significant advantage. This suspicion can be derived from their body language or simply from a player's history. Michael's 432 Rule By Michael Bo Hansen (1998). A rule of thumb for estimating your equity after you've been hit and your opponent's board is closed. When "the player with one man on the bar" has 4 men left on his ace point, the probability of "the player with the closed board" winning the game is between 30% and 20%. Initial Doubles By Hal Heinrich (1999). Heinrich looks at positions in the first three rolls of the game where doubling is correct. These are positions we encounter frequently. An Unusual Back Game By Bill Robertie (2007). Some cubing decisions in a back game. Volatility By Kit Woolsey (2003). Volatility in backgammon is a difficult thing for us to measure. Yet it is so important when deciding whether or not to double that it must be considered, even if it has to be judged subjectively. Failing to turn the cube in those volatile positions where a lot is likely to happen on the next exchange is very costly. Inducing Cube Errors By Kit Woolsey (2003). If your opponent plays perfectly, you will never have an advantage. You will increase your winning chances if you create situations where your opponent makes errors. When you have a close decision as to whether or not to double, take the opponent's tendencies (to take late or drop early) into account in making your decision. Evaluating Early Doubles By Bill Robertie (2007). When contemplating a middle game double, don't just look at your own position. Remember to look at the weaknesses of your opponent. It may be his weakness, rather than your strength, that gives you a good double.

Pip Counting

The Half-Crossover Pipcount  By Douglas Zare (2000). Knowing whether one is ahead in the race or not is vital for correct playing strategies. Here is a new, simple method for establishing an approximate pip count, which often is good enough. Then when you need it, a few extra calculations gives you the exact pip count. Cluster Counting  By Jack Kissane (1992). Jack Kissane, backgammon master from Albany, New York, is known in many chouette circles as the fastest pip counter in the world. In a June 1989 Chicago Point interview, Kissane claimed that he can count almost any backgammon position within five seconds. Here, Jack Kissane shares his counting techniques with the backgammon community. Cluster Counting Revisited By Dean Gay (2007). As I started using Jack Kissane's Cluster Counting, I began to spot other clusters that are easy to count. Some are just variations on Jack's, but they come up so often I just added them to my "vocabulary." The Pip-Count of Monte Cristo By Michael Crane (2000). If pip counting is alien to you or you find the mathematics of counting all those checkers on all those points just too much work, then here are some handy shortcuts. Sho Sengoku's Five Count By Sho Sengoku (2003). Advantages of this system are that it takes advantage of the symmetric nature of the board, uses base 5 counting, minimizes arithmetic, requires no "mental shift", is easy and quick to master, and is reliable. Naccel: a[N accel]erated Pipcount By Nack Ballard (2001). In this pip-counting method, you count crossovers and residual pips separately. The crossovers are offset by 1 from the crossovers we are used to. By learning a series of common patterns, called "squads", you can convert multiples of six residual pips into crossovers. Dice Counting, an Alternative to Pip Counting By Grant Hoffman and Jackie Hoffman (2001). This article suggests keeping a running count of the difference between your own pip count and your opponent's. Every throw, adjust the difference by adding or subtracting the playable total of the dice thrown. General Form for Different Pip Counting Systems By Sho Sengoku (2003). There are some similarities among the three pip counting systems: Douglas Zare's "Half-Crossover Pipcount", Nack Ballard's "Naccel", and Sho Sengoku's "Five Count". Those three systems separate all 26 points of the board in a number of groups, assign a group center, and provide error numbers from the center point to locate any points in a group. The Northern Michigan Pip Count By Robert Townsend (2006). A summary of Douglas Zare's Half-Crossover Pip Count with some shortcuts and tricks to make it easier to use. Pip Counting By Tom Keith. Articles on different techniques of pip counting. From the Backgammon Galore Forum Archive.

Races

Cube Handling in Noncontact Positions  By Tom Keith (2004). A description and evaluation of several popular methods of making cube decisions in noncontact positions. This article introduces the "Keith" count, an accurate formula for making race cube decisions. Effective Pip Count  By Walter Trice (2000). Trice introduces the term "effective pip count" and describes some clever methods of calculating it in different types of positions. In positions where it applies, the effective pip count is a very accurate method of making cube decisions in noncontact positions. Cube Handling in Races By Paul Money (2006). Many of our games will end in a race, with no hitting possible and gammons out of the question, so it is important to know how to handle the cube, or we will give away large amounts of our precious equity. The Cube and the Race By Robert Townsend (2007). Introduction to equity and basic concepts of doubling, Woolsey's Rule of doubling, the Jacoby Paradox, and cube racing formulas. The Ultimate Pip Count By Kit Woolsey (1993). In this article reprinted from Inside Backgammon, May/June 1993, Woolsey introduces the concept of a "pipple." A pipple is a unit of measure equal to 1/100 of a roll. Knowing both players' pipple count gives a good idea of your racing chances. Unfortunately, the calculations are quite tedious and "almost certainly not worth the effort at the table". Count Your Pipples By Jean-Luc Seret (2000). "Pipples" is a term coined by Kit Woolsey to represent a time unit worth the 1/100 of a roll. Seret develops a method of estimating a pipple count based on: (1) the numbers of men on each of the 6 points home-board points, (2) the shape of the distribution, and (3) which points have no checkers on them. Estimating Winning Probabilities in Backgammon Races By Andrew M. Ross and Arthur Benjamin (2007). A simple model of backgammon is used to approximate the chances each player has of winning, and a computable and reasonably accurate approximation is developed. From there, the model is compared to simulated backgammon games, and the approximation is modified to fit the real data. Cube Handling in Races By Tom Keith. After contact has been broken and the game is a pure race, it is easier to estimate your winning chances and make accurate doubling decisions. Here are some articles on handling the cube in races. From the Backgammon Galore Forum Archive.

Match Play

Five Point Match  By Kit Woolsey (1999). Woolsey examines every possible score for the five point match and looks at cube and checker play strategy for each score. Introduction to Match Play By Tom Keith (2006). When you play a series of games to a specified number of points, this is called "match play." The article explains all the rules of match play, introduces match equity tables, and provides a brief explanation of how to use match equity tables. Calculating and Using Match Equities By Stephen Turner (1999). Turner presents several formulas for estimating match equities at different scores, including his own formula which is noted for its easy calculation. The article goes on to give an example of using match equities to influence doubling decisions. Match Equity For Idiots By Phil Simborg (2008). I have stopped drilling match equity and take points. It just doesn't work for any but highly skilled players. Instead, I teach the concepts to beginner and intermediate players. (And if you are a more experienced player, this is not a bad refresher course for you, too!) Double? Take? The Mathematical View By Roy Hollands (2001). A quick introduction to doubling decisions in match play. Hollands presents a match equity table and shows how to use it to estimate your take point in various match situations. Match Play Doubling Strategy By Tom Keith (1995). In tournament play, where matches are played to a specified number of points, proper doubling strategy is different than when games are played for money. This article presents a number of the considerations a player must make when handling the cube in match play. Staying Ahead By Lasse H. Madsen (2003). How do you protect a lead in a long match; e.g., when the match might still have a long way to go? The focus of this article is on taking doubles. When do you accept an initial double? When do you accept a redouble? Charts show the patterns of how the match score influences take points, and the article develops some general guidelines along with some examples. Crawford and Beyond By Kit Woolsey (1999). In match play, the value of a gammon varies according to the score of the match. This article looks at situations where the correct checker play depends on the score of the match. Post-Crawford Doubling By Matt Reklaitis (2000). How long you can hold off doubling with an odd number of points to go in post-Crawford games. Three-Point Match By Paul Money (2006). In a three-point match, life is a little more complex. Just as in the two-point match, the cube decisions at every score in the three-pointer are different to money play. Two Away Three Away By Kit Woolsey (2003). Many players have serious difficulties making the adjustments required to their cube decisions at 2-away/3-away. Here are some examples of how to adjust your normal cubing strategy at this match score. Adjusting for the Match Score By Bill Robertie (2007). In matches, optimal play is often affected by the score in the match. The leader becomes more defensive. He wants to reduce the likelihood that the cube will reach a high level, and he wants to avoid gammonish positions. The trailer, on the other hand, is willing to double a little earlier than normal, and perhaps take cubes more aggressively. Only a 4-Cube By Kit Woolsey (2001). You have to be extra cautious when you have a big lead in the match and the cube starts flying. Before sending it over, see how things will look at the level your opponent will be potentially turning it back to. If you do this, you can avoid giving away a match when you are a big favorite to win. The Frozen Cube By Kit Woolsey (2001). This article discusses the concept of a frozen cube, situations where because of the match score one player or both will not want to turn the cube. The concept comes up more often than you might imagine. It is important to look ahead and see what the future ramifications of doubling or not doubling are. Sometimes the conclusions can be very unintuitive. Memoriable Weekend By Kit Woolsey (2002). Some match-play cube decisions from the Chicago Memorial Day Tournament, 2002. Match Equity and Doubling Windows By Hank Youngerman (1999). How to use a match equity table to make correct doubling decisions in match play. The fourth in a series of articles for beginning and intermediate players. Match Play Postings By Tom Keith. Articles about doubling strategy in match play. From the Backgammon Galore Forum Archive. Two-Point Match By Paul Money (2006). In match play, the score is the all-important factor, so let's start by looking at the shortest match length possible, the two-point match. Two-Point Match Strategy By Phil Simborg (2005). Playing a two-point match is a fun and popular game, both on the net and as a side event at tournaments. The interesting thing about playing at this score is that both your cube strategy and checker strategy are different from playing at any other score. And the Trailer Doubles By Anthony Patz (2001). Anthony Patz shows that when both players are two points away from winning the match, it can pay to double even when you are the underdog in the current game. Doubling Windows and Special Doubling Situations By Hank Youngerman (1999). How your doubling varies depending on the score of the match. Special doubling situations: the free drop, 2-away/2-away. This is the fifth in a series of articles for beginning and intermediate players. Match Play at 2-away/2-away By Tom Keith. Articles about the correct doubling strategy when both players both have two points to go in the match. From the Backgammon Galore Forum Archive.

Match Equities

Woolsey's Match Equity Table By Kit Woolsey (1999). A match equity table tells you your chance of winning the match at each different score. It is a useful tool in making doubling decisions. Woolsey's table is derived from a combination of empirical data and assumptions about gammon probability and the value of doubling potential. While it may not be totally accurate, the figures should be correct to within a percent or two. The table has proven to be of practical value, and is used by most experts today. Mec26: A New Match Equity Table By Albert Silver (2003). Mec26 is based on a small program published in 1996 by Claes Thornberg. The post-crawford equities were calculated by Joseph Heled using GnuBG rollouts. This table performed very well in comparisons with Kit Woolsey's table, Jacobs's/Trice's table, and the Snowie table. Snowie Equities By Kit Woolsey (2001). Kit compares Snowie's match equity table with his own match equity table and wonders why there is a big discrepancy at the score 4-away/3-away. How to Compute a Match Equity Table By Tom Keith (1995). This article describes how a match equity table can be derived mathematically if you assume a constant gammon rate and efficient cube usage. Lots of diagrams show the process step by step. Match Equities: Simplification vs. Accuracy By Nigel Merrigan (2000). Nigel Merrigan compares three formulas for estimating match equity: Janowski's formula, Turner's formula, and his own Merrigan formula. He finds that his own formula does the best overall while still being reasonably easy to compute. Graphical Match Equity Charts By Sho Sengoku (2003). Using these charts, you can visualize important characteristics of the possible scores in a match. Match Equity Posts By Tom Keith. Various match equity tables and formulas for estimating match equities. From the Backgammon Galore Forum Archive.

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Last updated: 11 Jul 2008