This is probably the most popular way to play this roll right now, so you do need to be clear as to how to reply. This chart should help to avoid some of the most common mistakes. Get your board out, set out the pieces and see how many you can get right. Don’t worry about columns three and four for the moment, we will only need those later when we are looking at match play.
Responses to 64 played 24/18, 13/9 
Roll 
$ 
DMP 
GammonGo 
GammonSave 
66 
24/18(2),
3/7*(2) 
24/18(2), 13/7*(2) 
13/7*(2)/1*(2) 
24/18(2),
13/7*(2) 
65 
13/7*,
6/1* 
13/7*, 6/1* 
13/7*,
6/1* 
13/7*, 6/1* 
64 
24/20, 13/7* 
24/20, 13/7* 
24/20,
13/7* 
24/20,
13/7* 
63 
24/21,
13/7* 
24/21,
13/7* 
24/21,
13/7* 
24/21, 13/7* 
62 
24/16* 
24/16* 
24/16* 
24/16* 
61 
13/7*, 8/7 
13/7*,
8/7 
13/7*,
8/7 
13/7*,
8/7 
55 
8/3(2),
6/1*(2) 
8/3(2),
6/1*(2) 
8/3(2),
6/1*(2) 
8/3(2),
6/1*(2) 
54 
24/20, 6/1* 
24/20,
6/1* 
24/20,
6/1* 
24/20,
6/1* 
53 
24/16* 
24/16* 
24/16* 
24/16* 
52 
24/22,
6/1* 
24/22,
6/1* 
24/22, 6/1* 
24/22,
6/1* 
51 
13/7* 
13/7* 
13/7* 
13/7* 
44 
24/16*, 13/9(2) 
24/16*(2) 
24/16*,
13/9(2) 
24/16*(2) 
43 
24/21,
13/9 
24/20,
13/10 
24/21,
13/9 
24/20,
13/10 
42 
13/7* 
13/7* 
13/7* 
13/7* 
41 
24/20,
8/7* 
24/20,
8/7* 
13/9,
8/7* 
24/20,
8/7* 
33 
13/7*(2) 
13/7*(2) 
13/7*,
8/5(2) 
24/21(2), 13/10(2) 
32 
24/21,
13/11 
24/21,
13/11 
13/11,
8/5 
24/21,
13/11 
31 
8/5,
6/5 
8/5,
6/5 
8/5,
6/5 
8/5,
6/5 
22 
24/16* 
24/16* 
13/11(2),
6/4(2) 
24/20(2) 
21 
24/22,
8/7* 
24/22,
8/7* 
13/11,
8/7* 
24/22,
8/7* 
11 
8/7*(2),
6/5(2) 
8/7*(2),
6/5(2) 
8/7*(2),
6/5(2) 
8/7*(2),
6/5(2) 
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.
Last time, we looked at a tough onepoint match and we will certainly do that again, but in this lesson I want to talk about the cube. If you are very much a beginner, still not sure how to set up the board and counting the points when you move around the board, then you are probably not ready to play with the cube, but bear with me. This isn’t very technical and it will come in useful later.
For everybody else, it’s never too soon to learn cube play. Why? That’s because if your cube action is good, you will be able to survive in quite strong company, even if your checker technique is still poor. The reverse is not true. Even if your checker play is expert, poor cube play will see you destroyed by clever cube handlers.
Why is this? Because the very great majority of games are decided by overwhelming luck for one side or the other and all the skill in the world with the checkers won’t change the result. Most of the skill is in ensuring that the cube is in the right place to maximise your return from the game.
The rules covering the doubling cube are very simple. Before rolling, you have the option to offer a double, by turning the cube to 2 and saying “I double”. Your opponent can pass, paying one point or she can take the cube and play on for double stakes. She is said to own the cube. On a later turn, she can redouble to 4, and you can either say “I drop” (refuse the cube) and pay two points or say “I take” and play on for four times the stake. Now you own the cube and can later redouble to 8! This process can in theory be repeated up to any number that you can think of, but in practice 16 cubes and above are extremely rare.
What is the point of doubling? When should one take or pass? This bit is very important...
All good cube actions, doubling or deciding not to double, taking or passing, increase your equity. All bad cube actions reduce your equity.
What is equity? It is just what the position is worth to you on average. Every position is worth something, depending on how likely you are to win and how many gammons and backgammons are likely on average. Your equity may be positive in a position that you are favourite to win, or negative where your opponent is favourite.
Your equity may be as high as +3, in a position where you are certain to win a backgammon and of course as low as 3 when you are certain to lose one. Usually of course it is somewhere in between and almost always some fraction of a point or 1. Sometimes it may be possible to calculate your equity exactly, but much more often it is an educated guess. Sometimes the equity depends on the differing skill of the players, sometimes not. Is that clear? Don’t worry if it isn’t, you’ll soon get the hang of it. Look at the position below. Red is on roll, the cube is in the middle and it is a money game.
Position 1
This game will be decided on the next turn. No skill is required, so the result can be calculated exactly. From your probability chart (Lesson 1, print it off for easy reference) you can see that Red has 26 winning rolls and 10 losers. He will lose every time that he rolls a single 1 and win all the rest.
Twentysix winners minus 10 losers means that in 36 games he can expect to come out 16 points to the good. Of course he might win them all, or even lose them all, but on average, he will win 16 points in 36 games. Putting it another way, his average gain per game is 16/36 points. Usually when discussing equity we like to do it in decimals and 16/36 = 4/9 = 0.44 points per game (ppg). In this position Red’s equity is 0.44 ppg and of course White’s equity is 0.44 ppg, because she must lose what Red wins.
Does this mean that Red should double? If he does, should White take? Let’s see. We know what the equity is if Red doesn’t double, so will he increase his equity if he doubles? If he doubles and White takes, all the games will now finish at the 2 level, so Red’s equity will rise to 0.88 ppg. If he doubles and White passes, he will win a whole point, so his equity will be exactly 1 ppg. Clearly for Red doubling is a very good idea, as his equity will rise to either 0.88 or 1, depending on what White decides to do. Note in passing the difference between these two figures though; the 0.88 is an average figure, whereas the 1 is a real point on the score sheet. However, his equity will go up if he cubes, so it satisfies the criteria for good cube action.
What should White do? If she passes, her equity is 1, whereas if she takes, it is only 0.88. Taking will save her 0.12 ppg on average, so clearly she should take and limit her average loss. Here we have calculated the effect of a 2 cube, but the size of the cube doesn’t matter. Even if it is a 64 cube, double and take is still the right answer.
Do we have to calculate the equity every time that we consider our cube action? Fortunately, you will be pleased to hear that we don’t but I think it is important to know the theory behind the decision. In practice, all that you will need to do is have an idea of how often you can expect to win. Here Red wins just a fraction over 72% of the time and this sort of advantage is usually enough for a strong double.
Does this mean that whenever Red has positive equity and he doubles that he will double his equity? In every case where he is the favourite to win and the game will finish at the 2 level, yes, that is true. This can happen in a number of cases. Here it is true because White will never have the chance to recube to 4. It would also be true if they were playing for $1 a point but had agreed that the maximum to be won or lost in a single game would be $2. This is often the case in online money play on servers and although rarely done in live play, it is a good idea to have a limit for players with a limited bank roll.
It is also true that in 2pt matches, where of course the first cube turn is the only one possible, but we will come to those another time. However, in games where White may get the chance to redouble later, then doubling doesn’t always increase Red’s equity. If he is only a small favourite and the game still has some way to go, then doubling can reduce his equity. This is because owning the cube has some value. Once you own it, you can’t be doubled out. Not only that, you can sometimes return the cube at 4 in an advantageous position when things go your way. In a game a long way from the end, cube ownership is worth a lot. We will look at this in much greater detail on many occasions in the future. For now, let’s look at some practical examples of cube action.
First, let’s look at some very simple end game positions that occur frequently. These are called roll races because all the checkers are piled up on the low points and only the likely number of rolls left in the game are a factor.
Position 2
This very common set up is known as a tworoll ending, with both sides certain to be off in a maximum of two rolls. This is quite simple to calculate. Red has two ways to win. He can roll a doublet immediately and the chances of that are 1/6. He will also win if he rolls a singleton and White fails to roll a double on her turn. The chances of that are 5/6 x 5/6 or 25/36. And 25/36 + 1/6 = 31/36. In a cross section of 36 games, Red wins 31 and loses 5, so his average gain will be 315 or 26 points in 36 games which is 26/36= 0.72 ppg.
As before, we can see what effect doubling will have on the equity for both sides. If Red doubles and White takes, his equity will rise to 1.44 ppg.
If he doubles and White passes, he gets 1 ppg. Clearly Red should double and White should limit her losses to –1 and pass.
By adding two more checkers to each side, we get a threeroll ending:
Position 3
As in all roll races, there is no skill required except for the ability to roll doublets! It is possible to calculate the result exactly and sparing you the details, Red will win this position 78.8% of the time if he doesn’t use the cube. Deducting White’s 21.2% wins from this we can see that in 100 games, Red has an average profit of 56.6 points, so his equity is 0.56 ppg. If he doubles, his equity will rise to 1.12 ppg if White takes, or 1 ppg if White passes. Red should double and White should pass.
Adding two more checkers to either side brings us to the fourroll ending:
Position 4
This is a fourroll ending. Red can expect to win this position 74.5% of the time. 74.5% less the 25.5% games that White will win gives him an equity of 0.49 ppg. If he doubles, that will rise to 0.98 ppg if White takes or 1ppg if White passes. However, there are two things different about this position. The first is that White should take, losing only 0.98 ppg rather than the 1 that she gives up by passing. The second is that for the first time, White may get the opportunity to redouble to 4 later on. This will happen when Red rolls a singleton, White rolls a doublet and Red then rolls another singleton.
Then, we would be down to a tworoll ending with White on roll. We know that this is a double and a pass, so White can turn the cube to 4 and win two points! If she can get down to the tworoll ending, she will win it every time, because she owns the cube. This is our first example of the value of owning the cube.
If White chooses never to redouble, Red’s equity is 0.98 ppg, but if White redoubles correctly when she can, Red’s equity comes down to about 0.91 ppg. This is the cubeful equity, the equity taking into account future cube use, as opposed to cubeless equity, where all the games are played to the end without use of the cube.
One last one of these roll races, the fiveroll ending, with both sides having 10 checkers left. Incidentally, you will have noticed that all the checkers are on the 1pt, but the differences if some of them are on the 2pt are very small and can usually be ignored.
Position 5
As the race gets longer, there are more rolls for White to catch up. Red is still a strong favourite, winning 71.7% of the time cubeless, so he should still double and White should take.
However, there is one subtle difference to this position. In the two, three and fourroll endings, the correct cube action would still have been the same for Red if he had owned the cube. It would have been correct for him to redouble, but in the fiveroll ending, if he owns the cube, he does better to hold on to it and wait a turn. This is because the cube has much more life in it for White. She now has two separate paths that lead to a cube turn, sometimes being able to double a threeroll ending and sometimes in the tworoll ending as before.
The more value it has for White, the more reluctant Red must be to give it away and if he redoubles now, his equity will drop very slightly. Compare this position with Position 1 that we looked at, where Red had two checkers on the two point. In both cases, Red has cubeless winning chances of about 72%, but in the first case Red could redouble because the cube had no value for White. We call this a dead cube. Here the cube is live and Red does better to wait for a slightly stronger position.
So, to summarise:
2roll ending  
Red wins 86% cubeless  
Redouble/pass 
3roll ending  
Red wins 79% cubeless  
Redouble/pass 
4roll ending  
Red wins 74.5% cubeless  
Redouble/take 
5roll ending  
Red wins 72% cubeless  
Double/take
(but not a redouble) 
These figures are well worth memorising as these positions come up a lot. Usually of course, they will be slightly different, as it is unlikely that all the checkers will be piled on the ace point, but these are the basic reference positions from which you can start to calculate your chances.
Up until now, we have concentrated on the doubler’s chances. Can we say what winning chances White needs to have a correct take?
Indeed we can. In all cubeless games, where there are no chances of a gammon, you can take if your chances of winning are 25% or greater. We can demonstrate this very elegantly by doing what investors call a Risk/Gain analysis.
Our base line for a take/pass decision is –1, i.e. what we will lose if we pass. Taking a 2 cube risks an extra point over and above the point that we agreed to play for at the start of the game  if we can win the game, we will be three points better off than if we had passed. Instead of being 1 we will be +2. So, we are risking one point to try and gain three. If we can win one time in four (25%), we will break even. If we can win more than that, we will gain. That’s cubeless.
If the cube has some value for us, that is if we will sometimes be able to redouble later, then we can often take with slightly less than 25%. We will look at that in more detail another time. For now, don’t worry about it, because only the very best players will ever be able to work out their chances with this degree of accuracy. Focus on winning one out of four and make that your bench mark.
Last Week’s Homework:
Your chart for your chances of entering from the bar on a board with x points open should look like this.
Points Open 
Rolls that enter 
Chances to enter 



5 
35/36 rolls enter 
(97%) 
4 
32/36 rolls enter 
(89%) 
3 
27/36 rolls enter 
(75%) 
2 
20/36 rolls enter 
(56%) 
1 
11/36 rolls enter 
(31%) 
Did you get that? If not, go back to your probability chart to find out where you went wrong. These are essential figures to remember, or you can’t gauge the dangers of being hit. Try to assemble a chart for your chances of entering two checkers from the bar.
Now that’s quite enough for this lesson and if you are new to all this, your head is probably spinning! Don’t worry, gradually the mists will clear. Reread this as many times as you need to as some may not absorb it all on the first go; believe me!
Until next time, enjoy the game!
