"Keep the Skeer on 'Em"
—Nathan Bedford Forrest

Part 3:  The Cube and the Race

Backgammon in its most basic form is simply a race to bear off your 15 men before your opponent does. How you do it is up to you, so long as you accomplish the holy grail of backgammon—you win the race in the end. In this article we will discuss strategy and doubling situations in long/medium races and in short races during the end game. We will also discuss the cube in general (the cube decision is the game within the game of backgammon) and the concept of equity and how to use it to guide our decisions in all aspects of the game. One final thing to remember is that the point of these articles is not to give the answers to problems, but to teach the reader to ask the right questions and use a proper thought process to come up with the answers themselves across the board. In the American Civil War (or the War of Yankee Aggression as my SC friends call it), Confederate General Forrest had a unique way of taking enemy positions. He would show some strength (or imply it without actually showing his hand) and then send a message to the opposing commander. The message was simple: "Surrender now without a fight and we will take you prisoner, make us storm your works and we will kill every one of you". This turned out to be remarkably effective. The doubling cube uses a similar threat. Unlike a money game—which you actually have to win, with the cube a significant threat to win is all you really need; you don't need to necessarily to do it, you just need to threaten to from a position of force. How much strength do you need? Let's discuss the concept of equity. Say you and a regular opponent are playing your weekly game. You know from past experience with this player he is exactly at your level, neither stronger nor weaker. At the start of the game, he is just as likely to win as you are, if you are playing for a dollar, you can set a value on your game. The calculation is simple, in two games you will win one and lose one (you are equally matched and both deal with the same dice right?). If you win, you win a buck, if you lose, you win nothing. Since the end result of 2 games is that you win one dollar, one dollar divided by 2 is 50 cents a game. 50 cents is the value of your game. If I came by and wanted to buy your game from you, I would have to give you 50 cents. Obviously, as the game goes on and one side gets advantage over the other, the price to buy each side adjusts accordingly. You play along, and he doubles you. You assess the position and decide, based on a number of factors, you still have an equal chance to win. In other words, your chances of winning are 50:50. Now you have a choice to make. If you drop, you lose 1 point. If you take, you will either lose another point or you will gain 3 points and win a total of 2 points. Let's look at the value or equity of your two choices. Choice one is to drop, this will cost you −1 point in one game, which is −1/1 game or −1 points per game. Choice two is to take. Since your chance to win is 50:50, if you play two games you will either win 2 points or lose 2 points. So (2 − 2)/2 games is 0 points per game. Clearly since your choice is between losing a point and breaking even, breaking even is the better choice, so you take. What if instead you evaluate the game when the double is made and decide your opponent is twice as likely to win as you are? If you play three games from the current position, you can expect to win one and you expect him to win the other two. Again, you have the same two choices: Choice one is to drop, this will cost you −1 point in one game, which is −1/1 game or −1 points per game. This is just the same as the last example. Choice two is to take, with the cube now on 2, you will win one game for 2 points and lose 2 which will cost you 4 points. So now we have (2 − 4)/3 games or −2/3 point per game. Since your choice is now between losing 1 point (passing) and losing 2/3 point (taking), you are still better to take. Another way of looking at equity is the total points you stand to gain with one choice vs the total you gain with the other. In this case you risk one point (the difference between passing and losing 1 point vs taking and losing to lose 2 points) against the possibility of gaining 3 points (the difference between passing and losing 1 point and taking and winning 2 points). In the first article of this series, we talked about the 75/25 rule of doubling. Now we can see why you can still take with your opponent having a 3-1 advantage over you. Again, if you pass, you lose one point. If you take, you win once for 2 points and lose 3 times for −6 points. This is (2 − 6)/4 games or −1 point per game, exactly the same as passing. So if you estimate your winning chances are 25%, you can take or pass; it makes no difference. 26% you take, 24% you pass. We may need to modify these numbers in match play, but that is more than we are discussing in this article. (For information on cube handling in match play, see Kit Woolsey—How to Play Tournament Backgammon.) But this is the basic strategy and the thought process behind it. One term to know is that if you double with a 75% advantage you are giving a maximally efficient cube. The 75% advantage is a target point to get the most from the cube. So we now see how to estimate what is called the take point—the lowest winning percentage we can have and still accept the cube. It should be noted, for accuracy, that this take point does vary with the match score. In some cases at certain match scores it can be as low as 3%, but those considerations are beyond the scope of this article and we will use 25% as a good rule of thumb. The next major consideration is: When do we offer a double? Technically it is correct to offer a double when you have an advantage over your opponent. So if your winning chances are 50% or more, you can double. This brings up a new term. If you can double at 50% and your opponent can take with 25%, then we say that 50% to 75% is the ‘doubling window’ Again, this varies with match score, we discussed this earlier with looking at takes. The same considerations affect the other end of the window. A good example is the case in match play where you offer a double, but your opponent can redouble you for enough points to win the match. This would be the case, for instance, when you needed 2 points to win a match and your opponent needed 3. By doubling, you give him the chance to turn the cube to 4 (What does he lose? If you win, you win the match, if he wins he might as well win the match too). Clearly in this case, you want more than a 50-50 chance for the amount of risk you are taking. How do we note or evaluate cube decisions? In this and other articles on backgammon, cube action is referred to by a code. If a double is technically correct, and a take is technically correct, a cube decision is referred to as double/take (sometimes further defined as easy take, borderline take, strong double, or weak double etc.). If a double is not correct but is made, it is no double/take (+/− beaver). This means that if a double was offered, your opponent would take or even beaver you. Beavers are not always allowed, but if they are, you may, if given an improper double, turn the cube again and retain ownership of the cube. This is designed to punish a foolish double. So we know when we can offer a double—with an advantage. When should we? Ideally, you want to stack the deck in your favor. The closer you are to the maximum advantage your opponent can accept the better. In other words, you would like a 74-75% advantage. If you have an advantage but you have a number of potential rolls that will increase it to such a point that your opponent will not be able to accept the cube, you may want to consider turning the cube on the chance you will roll and crush him. Another factor you can use to guide your cube action is whether or not your opponent will have a take next roll even if your roll your best roll and he rolls his worse. If the answer is yes, you lose nothing by waiting another shake to turn the cube, as you can turn it next roll when perhaps you have more advantage. This is handy if there is a sudden turn in the game—you reduce your risk if you hold off on the cube until things are clearer and more decisive. The final bit of advice on cube action is known as Woolsey’s Rule, after Kit Woolsey—one of the best players in the world. Woolsey’s Rule:  "If you are not sure your opponent has a take or a pass, turn the cube." The explanation of this rule is simple, if you are not sure your opponent has a take or a pass, your opponent probably doesn’t either. Put the stress of that decision on him. If the position is a pass, he might incorrectly take the cube. If the position is a take, he might pass it. Either way, your equity goes up. Remember that casinos have only a 0.5% edge in blackjack, yet make 10%. This is because people make bad decisions and lose games needlessly. The same thing happens in backgammon. Another aspect of the cube we need to consider is cube ownership. Both players can use the cube in the middle, but if a double is made, the side doubled owns the cube. If one side owns the cube and his opponent rolls a very good roll (a joker) or otherwise turns the game around, he can't turn the cube on you and double you out. Even if you only have 5 winning rolls, you still get to keep the equity of those 5 rolls if you own the cube. This is important in our last example from the first article—something called the Jacoby Paradox. The Jacoby Paradox:  A position which is an initial double/take (but not a redouble) that results in a redouble/take if it is not won on the initial roll. In a money game, here is the breakdown of the position: Red rolls and wins 19/36 games outright. White then rolls on the 17 remaining games and wins 26/36 of those or 12 games. Red then rolls on the last 5 and wins them. (Ok 2-1 twice in a row would not be good, but what are the chances?) Here we will look at the equity in a third form, we will compare two series of 36 games, see how many points we win overall with one choice and compare it to how many we win overall with the other choice. With the cube in the middle: Variation 1: Red does not double Red wins 19 games on this roll @ 1 pt each +19 points White doubles and wins 26/36 of the 17 games @ 2 pt each −24 points Red redoubles and white passes @ 2 pt each +10 points ———— +5 points Variation 2: Red doubles Red wins 19 games on this roll +38 points White redoubles and wins 26/36 of the 17 games @4 pt each −48 points Red redoubles and white passes @4 pt each +20 points ———— +10 points With the cube in the middle, red doubles and wins 5 more points overall. With the cube on 2 and owned by red: Variation 1: Red does not redouble Red wins 19 games on this roll @ 2 pt each +38 points White wins 26/36 of the 17 games @ 2 pt each −24 points Red redoubles and white passes @ 2 pt each +10 points ———— +24 points Variation 2: Red redoubles Red wins 19 games on this roll @ 4 pt each +76 points White redoubles and wins 26/36 of the 17 games @ 8 pt each −96 points Red redoubles and white passes @8 pt each +40 points ———— +20 points With the cube owned by red, red does not redouble and gains a net of 4 points over 36 games. The ability to control the cube (cube ownership) prevents your opponent from capitalizing on a turn of the game in his favor. This is why you should not be as anxious to redouble as you are to give an initial double: you are giving up control of the cube and it may come back to bite you. Recently, an expert player and I were talking about the cube in races and he made the statement: "Races are very complicated when it comes to the cube". I am not sure about this; I find them pretty easy if you look at them in an organized fashion. First a definition: A pure race is when you find yourself in a situation where you no longer have contact with enemy checkers. You cannot be hit or blocked, and you are running home for the bearoff. A second definition is a long race vs a short race. Races have been long been defined in terms of pip counts as long/medium/short, I prefer a simpler classification. If all your men are in your home board, it is a short race. If some are in the outfield, it is a long race. Let's deal with short races first, they are more fun, more volatile and the cube action is more precise. Proper use of the cube in the final bearoff accounts for at least as much cube action as all the game play leading up to it. It was the first aspect of cube usage I studied because I got the biggest bang for the buck for my time. Back games, prime battles and other interesting and complicated games occur, but most games do end up in a short race, so we need to understand them and use the cube efficiently. There are several key features to short races. The most important is the smoothness of your bearoff compared to your opponents—gaps are very important, if you have none and your opponent has 3 empty points, you will have the advantage because he will not bearoff if he rolls a gap number. Another factor is pip wastage—are there many checkers stacked on the one and two points that will be taken off with higher number rolls? Finally, do all doubles take off 4 men and all non-doubles take off 2? Further modifications need to be made if one side has more checkers off than the other. Based on these factors, the pip count is modified and cube decisions are made. Several formulas to adjust the pip count in short races exist—the classic is the Thorp Count. These formulas attempt to adjust for gaps, pip wastage and average dice rolls to give you a number you can compare to your opponent and make a cube decision. An excellent discussion of these formulas is given in an article titled Cube Decisions in Non-contact Positions by my friend Tom Keith. In honor of his contributions, I am going to use the Keith Count rather than the Thorp, I also prefer the Keith Count over the board because of its simplicity and power. First, do a quick pip count—with all the checkers in the home board, this is easier than you think. Second, divide your count by 7 and add it to your total (forget the decimals and fractions) Third, look at the 4/5/6 points and add 1 point for each empty point. Fourth, look at the 3 point, add 1 point for every checker there more than 3 (3 for 3 I always say) Fifth, look at the 1 and 2 points, you should have no more than one checker on each. If you have any more, add 2 points for each extra on the 1 point, and 1 point for each extra on the 2. Finally, do the same for your opponent, but skip step 2 (this step adjusts for the fact you are on the roll). Ok, cube decision time. Assuming that both players have the same number of checkers on the board, if you have a Keith count not more than 4 more than your opponent, you can double. If your count is not more than 3 higher, you can redouble. If your count is at least 2, your opponent can take. Note that if you are on the receiving end of a cube, you need to do the above calculations from the viewpoint of the doubler (you add 1/7 of his pip count to his count and don't add it to yours). One special situation in the short race is the n-roll position. This special type of race has two conditions: All doubles bear off 4 checkers. All non-doubles bear off 2 checkers. Thus in a perfect bearoff position with 15 checkers there are 7 rolls remaining, this is 6 non-doubles and 1 double (1/6 of all rolls are doubles). Before deciding you are in an n-roll position, you must carefully play devil's advocate to be sure all your doubles and all your non-doubles work, with the same going for your opponent's position. Even one double or non-double that does not meet the conditions invalidates the position. A 1-roll vs. 1-roll position. A 2-roll vs. 2-roll position. A 3-roll vs. 3-roll position. A 1-roll position is a 100% win, just roll the dice. A 2-roll position is a double/pass (84:16). A 3-roll position is also a double/pass (77:23). A 4-roll position is unique in that it is an initial double/take but it is not a redouble (remember the value of cube ownership). 5/6/7-roll positions are all double/takes, but it is best to simply watch them and act after a couple of rolls bring you to one of the above positions. You can exert some degree of control as you are bringing your checkers to the home board to set up a bearoff position that can work to your advantage The key to this is to set up as smooth a bear off as you can. Another feature to pay attention to is the minimum number of rolls to bear off your last man. As an example, a position with 7 men remaining and one with 8 men both will require 4 rolls (excluding doubles) to bear off. There is no advantage to taking a risk to bear off an additional man if you have 8 checkers to go. If there is a question of gammoning your opponent, you might consider taking an extra risk with 7 men to take an additional man off to get to 6. Now that we have a handle on the short race, let's extend that to the long race. The long race has all the same considerations as the short race, with a few twists thrown in. Since it does have many of the features of a short race, adjustment of the raw pip count with the Keith Count is certainly appropriate and will bring you closer to the truth. Another factor that must be considered is the number of crossovers needed to bring your (and your opponent's) checkers home to start the bearoff. An example of this concept is seen in this short race. How would you handle this position? Red to play a 4. Determine which checker to move and why, then see my discussion here. Resulting pip counts from any move are: 25 for white, 17 for red. With white on roll, Keith Count will be 25 (pip count) + 3 (25/7) + 1 (2pt extra) + 1 (4pt gap) = 30. Option A:  5/1 Red Keith Count (as above, white on roll): 17 (pip count) + 2 (1 pt extra) + 2 (gaps on 4 and 5 pt) = 21 Option B:  6/2 Red Keith Count (as above white on roll): 17 (pip count) + 2 (gaps on 4 and 6 pt) = 19 White does not have a double, red an easy take in both cases. Now, give white a prime roll, 6-5. White's Keith count is 15 (pip count) + 1 (2 pt extra) + 3 (gaps 4, 5 and 6) = 19. If red played 5/1 we have Option A: Red's Keith Count is 17 (pip count) + 2 (17/7) + 2 (1 pt extra) + 2 (gaps 4 and 5) = 23. If red played 6/2 we have Option B: Red's Keith Count is 17 (pip count) + 2 (17/7) + 2 (gaps 4 and 5) = 21. In Option A, red's Keith Count is 4 greater than white’s. Cube decision is borderline initial double/clear take, no redouble/clear take. In Option B, red’s Keith Count is 2 greater than white’s. Cube decision is clear initial double/borderline take, clear redouble/borderline take. Discussion Note that we gave white his best non-double roll. A good understanding of the game makes the choice between Option A and Option B relatively easy—Option B avoids large stacks and has better flexibility with fewer miss numbers. (Option A does not bear off with a roll containing 2, 4 or 5 whereas Option B misses only with 4—27 bad rolls vs 11 not accounting for doubles.) However familiarity with the Keith Count can also help us to determine the optimal bear off distribution, especially with more complicated positions. Rather than go through the in depth calculations we can just look at white's major winning variant (red rolls a lone 2) and realize that white will double us out of those 10 games, so in reality we are looking at being 26:10 favorites, so it is a double/take. That is the quick and easy calculation over the board and very close to the truth.

A final piece of advice comes from Douglas Zare, mathematician and backgammon theorist. It has come to me (one roll too late in many cases) far too frequently. "The reason you lost to a joker roll is that you gave your opponent the chance to roll it when you should have doubled him out." Pick your ground, do your math, and keep the Skeer on 'Em. Happy Journey. Continue to Part 4

Article © 2006 by Robert Townsend.

Bob Townsend is the director of the Northern Michigan Backgammon Club.
You can contact him at DrRTownsend@msn.com.

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