This article originally appeared in the March 2003 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
Backgammon is, in essence, a race.
I believe it was Bill Robertie who first uttered these immortal words. He was trying to illustrate the importance of the race as opposed to the positional considerations which had been over-emphasized in the late 70's and 80's.
Every backgammon game which doesn't end with a cube turn and a pass must, by definition, end with a race of some sort. It might be a very lopsided race, it might be a race to scramble off a gammon or a backgammon, but it must be some kind of race. At some point contact will be broken, and only racing considerations will apply.
Since all backgammon games played to a conclusion must end with a race, the race is the most common position type we see except for the opening position. Since it is so common, it is worth knowing how to play it correctly. Of course, most of the time there is very little equity difference in racing plays. If you just move the pips you are dealt that is usually sufficient. It definitely isn't worth worrying much about how to play some ace in the outfield in a straight race. Just shove a checker one pip and get on with the game. Thinking a lot about that sort of play isn't going to help. It will just annoy your opponent. Backgammon is a fast-paced game. While there are positions worth spending some time on, races aren't one of them. What we want to do is to establish a few good guidelines which we can follow at the table without much thought, so we can play races quickly without giving up any substantial equity.
Our goal is to get our men off as quickly as possible, of course. We are limited by the dice which are rolled. We can't move any more pips than are on the dice. However, sometimes we are forced to move fewer pips than are on the dice in a bearoff. When this happens, we have wasted pips. Our goal in the bearoff and the preparation for the bearoff is to position our checkers so as to minimize the likely wastage.
How does wastage occur, and what does it mean? let's look at a simple example:
Blue is on roll, with White having an 11 pip lead. That 11 pip lead is not nearly as great as the pip count indicates -- in face, this is a very close race. Blue is unlikely to waste anything, while White is almost certain to waste a fair amount on his next roll. As an illustration, suppose both player's roll 6-5. They have rolled the same number of pips, but White's once mighty 11 pip lead has shrunk to 2 pips and it is Blue who wins the game. What happened was that White was only able to play 2 of the 11 pips he rolled, so he wasted 9 pips.
Note: In my examples in this article I am ignoring the cube. Of course there are special situations where the cube may come into play, but for now we are just looking at general racing principles.
How does wastage occur? Obviously the only condition under which you can waste pips is when you roll a number which is larger than the point of your farthest back checker. We can see how having a bunch of checkers on the low points may lead to wastage, as in the above example. Once a checker is on the ace point, it cannot be moved except by rolling an ace in the bearoff or until all the other points have been cleared. Hence, there is potential for eventual wastage with a lot of checkers on the ace point, so that is something to avoid.
Another way we can waste is to have the high points cleared and roll a big number. If we roll as six with the six point empty, we can only move five of the six pips and have thus wasted a pip. This argues for keeping checkers on the six point as long as possible. I have often seen this type of position misplayed:
It turns out that 6/0, 2/1 is slightly the best play, followed closely by 6/0, 3/2, then 6/0, 5/4, and 6/0, 4/3. These plays are all fairly close, and it would not be a terrible error if you chose any of them. However, 6/5, 6/0 is far worse than any of the other plays. The reason is that if you roll a six, that means immediate wastage of a pip. Yet, it is not uncommon to see an average player make this play in order to be able to take four checkers off with 5-5 next turn. A player who understands what wastage is all about would never consider 6/5, 6/0.
The object is to take all the checkers off. When you can take a checker off, it is usually correct to do so. A very simple rule which will save you a lot of thinking and fumbling is as follows:
If you have rolled an ace, 5, or 6 on one of your dice and you have a checker on the ace, 5, or 6 point, then it is always correct to used the number to take a checker off. No exceptions! How often have I seen a player in a position such as:
play 6/5, 6/0 with the argument that he wanted to keep a smooth position. That is just plain wrong. It doesn't do any good. Look what happens. Suppose we take two checkers off and then roll the dreaded 5. The position will just transpose. We will have the same single checker on the ace point and no checkers on the five point that we would have had after playing 6/5, 6/0 and rolling that five. The gain from taking the checker off occurs when we don't roll any fives or aces for the next few rolls. We may be able to clear the six point without having that annoying checker sitting on the ace point. Trust me on this one -- it is always correct to rip that checker off.
Note that I said it was always correct to use the number to take a checker off. That doesn't mean it is always correct to take the checker off that point. We are all familiar with the following sort of situation:
If Blue snatches the checker off with the five and plays 5/0, 4/1, he will miss with any 2 or 3 next roll. Better, of course, is to play 5/2, 4/0, so he can take a checker off next turn with a 2. Note that the 5 is still being used to take a checker off -- only the checker isn't being taken off the five point.
There are special positions where it is correct to not use a 2, 3, or 4 to take a checker off the 2, 3, or 4 point. The best known of these is the following:
If Blue plays the obvious-looking 4/0, 2/0 and then rolls two twos while clearing the three point he will miss twice and come down to an odd number of checkers, costing a full roll. After the proper 4/2, 3/0, Blue won't miss unless he rolls three twos.
For completeness, here is a position where it isn't right to take a checker off with a three:
For similar reasons, it is correct (by a small margin this time) to play 6/3, 4/0 rather than 6/0, 3/0.
Finally, a rare position where it is wrong to take a checker off the four point with a four:
It is correct (by a fair amount) to play 6/2, 5/0 rather than 6/0, 4/0. If Blue takes two checkers off, every four and two are misses for quite a while, so it is better to have a checker on each of these points now even if it means one fewer checker off immediately.
While these positions do come up, they are rare enough so they aren't worth worrying about for the most part. When I play on Gamesgrid I always have the automatic bearoff toggled, which means that the program will misplay these positions. My overall equity loss is tiny. Simply following the rule of taking a checker off with one die when you can is almost always right.
When a combination of both dice is needed to take a checker off, that can be another story. For example:
Blue could take a checker off with 5/0, but that would be a serious error. If he makes this play, subsequent fours and fives will force him to dive down to the two and ace points, and these checkers on the ace and two points will lead to eventual wastage. It is much better to have a checker on each of the four and five points while clearing the heavy six point so Blue won't be forced to go deep.
When you can't take a checker off, what should you be doing with the roll? As we have seen, we want to avoid having checkers on low points. We also want to keep the most men on the highest points to avoid clearing these points too soon and then wasting. Roughly speaking, we would ideally like to have a triangle which is tallest for the highest points and slowly slopes down.
Back in the 70's before we had our huge computer data bases available, I tried
writing a simple program for bearoffs. I wasn't aiming for perfection --
I just wanted something decent for simulation purposes. The simple logic
my program followed was:
Of course my program made mistakes. For example, in the previous position with the 3-2 to play my program would have blindly taken the checker off. Still, it probably played the bearoff better than 90% of the players at that time. Quite frankly, that is pretty much the way I think at the table during a bearoff, with an eye toward the exceptions when they jump up.
Here are some guidelines when having an ace to shuffle in the inner board:
1) Make the bad numbers play well. Suppose that on your next roll you roll a number on one of your die which doesn't take a checker off. You want to have that number play conveniently, smoothing the position out and filling up gaps. For example, let's reexamine a position we looked at earlier:
If you remember, I said that 6/0, 2/1 was the best play. Why is this best? Suppose you make the play and then roll a 2 (your worst). The number plays comfortably, slotting the empty two point and unstacking the heavy four point. If you make one of the other plays and then roll an ace, that doesn't play quite so comfortably.
2) Watch the fives and fours. In particular, watch the fours. These are the numbers which will force you to play deep from the six point if these points get empty. It is important to look at the status of the lower points to see how serious it will be if you are forced to go deep. If the ace and two points are empty, then being forced to play a five or a four to the ace point won't be too bad. In addition, an ace or a two can be used to unstack the heavy six point onto the five or four points. However, if there are already checkers on the low points, then being forced to put more checkers there can be very bad. In addition, if you roll an ace or a two and have a checker on the ace or two point you will have to use the number to take the checker off, so you won't have the opportunity to unstack the six point. For example:
Both 6/0, 6/4 and 6/0, 5/3 help smooth out the position. 6/0, 5/3 gets a checker onto the three point so Blue can take a man off if he rolls a three. 6/0, 6/4 keeps extra men on the five and four points, but may be starting to deplete the six point too early. It turns out that 6/0, 5/3 is slightly better. Having the extra checkers on the five and four points isn't too vital, since the two and ace points are open. However, change the position slightly to:
Now 6/0, 6/4 is slightly better. Those two checkers on the ace point represent potential wastage, and if Blue is forced to dump more checkers there the wastage problem will become severe. Blue needs to keep his five and four points well-fortified in order to avoid this.
The three point is a different animal. When you have a lot of checkers on the six point, there is little need for men on the three point. The idea is that if you roll a three you can play 6/3 which is perfectly comfortable. For example:
All plays are close, but it is sligtly best to play 6/0, 3/2. Note that this follows the principle of making the bad numbers play well -- now if Blue rolls a three and misses it plays comfortably. If Blue instead plays 6/0, 4/3, his twos aren't as comfortable. Also, thinning out the four point may lead to wastage if Blue suddenly starts rolling a bunch of fours. Note that the best play looks more like a triangle than any of the other plays.
This time, 6/0, 4/3 is best. Blue will have only one checker left on the six point after this move, and when that checker is moved threes will become a potential problem. If Blue doesn't fill in the three point quickly, he will find that a bunch of threes will force him to dive to the ace and two points for potential wastage.
When bearing in, you should naturally be preparing for the bearoff, and trying to bear in with a structure which will lead to the least wastage. As we have seen, this means as few checkers on the lower points as possible, and a triangle on the higher points with the most men on the six point, then on the five point, etc. Not surprisingly, the optimal bearoff structure for 15 checkers (that is, the structure which will lead to the least wastage on balance) is 7 on the six point, 5 on the five point, and 3 on the four point. This is roughly what you should be aiming for. Of course you seldom are going to hit this exact formation. It requires perfect cooperation of the dice for that to happen. However, if your structure looks anything close to this you are doing fine and probably played the bearin perfectly or very close to perfectly.
It has been said that you can't take them off before you bring them in. There is some degree of truth to this, but it shouldn't be overdone. We already know to avoid putting checkers on low points if we can avoid it. However, getting a smooth structure has the highest priority. Crossovers are completely meaningless. They would matter only if you are so far behind in the race that you need the miracle of rolling 6-6 every turn. Otherwise, there is no particular gain at all from bringing a checker home. There is a temptation to just grab checkers and bring them home. Don't fall for this. You aren't going to lose any pips if you start bearing off checkers one roll later, but if you don't have adequate fortification on your five and four points you may lose a roll later due to wastage. For example:
Playing 8/6, 7/6 would be an error. Blue might never roll the necessary numbers to fill in his five and four points as he brings in his remaining outfield checkers, and if these points remain thin that could result in later wastage. Instead, Blue should take this opportunity to unstack with 6/5, 6/4. His good diversification in the outfield will let him put the rest of his checkers where they belong pretty much whatever the dice say.
This concept can be overdone. For example, make the position:
This time I believe Blue should bring them home with 8/6, 7/6. His five and four points already have some fortification, so unstacking the six point to fill them in more isn't too vital. More important is that if leaves the outfield checkers where they are and then rolls big numbers next turn, he will be forced to bring these checkers deep before he can start to bear off, which is exactly what he is trying to avoid. In the previous position Blue had other checkers farther back in the outfield to handle large numbers, so the danger of being forced to move the checkers on the seven and eight points in too deep was minimal.
Diversification in the outfield is vital. You want to avoid having extra checkers on the same outfield point, as this restricts your ability to bring them where they belong. In particular, having extra checkers on the seven and eight points is very dangerous, since if these are your only outfield checkers then you are forced to bring them deep if you roll fives or sixes. For example:
Blue must bring two checkers home with 8/6, 7/6, even though this overstacks the six point and leaves the five and four points thin. It is too dangerous to leave the extra checkers on the seven and eight points.
In general, it can never be wrong to diversify in the outfield. If you have your outfield checkers all scattered on different points, you will be able to bring them in where you want regardless of what the dice say. This can even have priority over bringing men into perfect places. For example:
It might seem natural to play 9/4, 7/5, since play puts these checkers where we want them. However, that would leave no diversification in the outfield. Blue would be forced to move from the midpoint next turn, and if the dice aren't cooperative these checkers may not go where they belong. Instead, Blue should diversify completely in the outfield with 13/11, 13/8. There will be plenty of time and opportunity to fill in the five points with Blue being this well diversified. We can see the danger of play 9/4, 7/5 if we imagine what things will look like if Blue rolls 6-6 on his next turn. Which play would you rather have made? The diversification in the outfield gives Blue the opportunity to recover from anything.
Understanding the importance of proper bearoff structure can help one's middle game decisions. For example, consider the following very common sort of position:
Blue can save a six by playing 6/2, 6/1, but is it worth it? That puts a fourth checker on the ace point, which as we know will lead to much wastage if get to a race. In addition, if Blue does roll that six he will have to put yet another checker on the ace point in order to play safe. 7/2, 6/2 is a much smoother play.
Let's suppose that Blue does play 7/2, 6/2, White rolls 6-4 (13/3), and now Blue rolls the dreaded 6-4 and is forced to play 18/8. Why did I ever learn about smoothness, Blue is thinking. The position is now:
If White hits, it is all over of course. But suppose White rolls something like a 5-3. Now Blue is all smiles. White is forced to do some wasting of his own. Blue is a favorite to get his back checker to safety, and if he does so he will have a moderate lead in the race.
Contrast with what the position would look like if Blue had saved a six:
Ugly! Blue has piled six checkers onto his six point, so even if he does turn the position into a race his racing chances will be considerably less than the pip count would indicate. Also, if White is forced to break off the midpoint by rolling a six, Blue may not even be able to afford to hit! The saving of the six comes at quite a cost. It is true that if Blue knew for certain he would roll a six next turn he would rather have saved the six, since giving White 17 game-winning shots is very bad. However, the downside of saving that six can be quite large. In any variation where Blue doesn't roll a six next turn he would rather have not messed up his position, particularly if he rolls small doubles. The bottom line is that the correct play here is 7/2, 6/2 because the potential wastage from saving that six is too great. Now that we know what this wastage means, we can look ahead and see how our bearoff structure should affect our middle-game play.
Following the principles outlined in this article will enable you to play races quickly and efficiently. There will be no need to ponder over every play. The checkers will automatically go where they belong, and you can save your mental enrgy for more important decisions. You will be a more popular opponent since you aren't wasting everybody's time with these decisions. Also, you just might win a couple of races you would have otherwise lost.