Counting the Position
Oswald Jacoby and John R. Crawford, 1970
From The Backgammon Book, Chapter 6 (pp 107–110)

The count represents the total number of points you would have to move to bear off all your men assuming no waste motion. This of course is hypothetical, since there is always going to be wasted motion due to the fact that you tend to bear off men from the higher-numbered points first, and invariably come down to a position in which first your six point, then your five point, and then the four and three points have no men left on them, so that you don’t get full value out of any numbers on the dice except 1 and 2.

However, it stands to reason that in most running-game positions you can measure your relative standing by this optimum count.

Calculating the Count

The count of each man in your own board is that of the point he is on. Thus a man on your six point is counted at 6, while three men on that point would be counted at 18; a man on your bar point is counted at 7, and so on. The count for men in your opponent’s board starts with 13 for each man on his twelve point (since he is 13 moves away from bearing him off), 14 for men on his eleven point, and so on up to 24 for each of your men on his one point. One way to determine the exact count for these men is to count the number of pips (points) necessary to bring them to the black twelve point and then to add 13 for each such man.

Thus, in the Position 39a your count is 109. This is made up of 30 (two men on the black ten point); 26 (the two men on the black twelve point); 9 the man on your nine point); 18 (the three men on your six point); 15 (the three men on your five point); 4 (the man on your four point); 3 (the man on your three point); and 4 for the two men on your two point.

The black count is 118, represented by 52 for his four men on your twelve point; 18 for the two men on his nine point, and 8, 7, 12, 10, 8, and 3 for the rest of his men.

Notice that this isn’t entirely a running-game position yet. There is still contact between your men in black’s outer board and his men in yours; there is some slight chance that one of you will have to leave a shot, but this chance is small enough to be ignored. Thus, this may be considered a straight running-game position.

Taking Differences

There is an alternate and quicker way to count the position: you work on differences. Your four men on the black ten and twelve points are a total of 4 points worse than your opponent’s four men on your twelve point. You begin by counting 4 against yourself. Your man on the nine point cancels out one of his two men on his nine point. This leaves him with one other man on the nine point and single men on the eight and seven points. The three count 24 (9 + 8 + 7) against him, and after taking out the 4 you were behind, you come into the home boards with a 20-point advantage.

You lose 11 of this 20 here. The extra man on your six point counts 6 against you; the extra man on your five point counts 5 against you; your man on the four point balances one of his two men on the four point so you gain 4 here for a new so far of 7 behind in the home boards; the single men on the three points cancel each other out, while the two men on your two point cost you 4 more to bring the home board net to minus 11. Subtracting that from the 20 you were ahead outside the home boards leaves your final position 9 points better than (i.e., less than) your opponent’s.

Cube Decisions

In using the count to decide whether or not to double, bear in mind that the average roll in backgammon has been computed to be 81/6 , allowing for the extra plays with doubles; so if it is your roll, you have a decided advantage in this case: 9 points plus whatever your roll will be. If it is your opponent’s roll, you still have an advantage, but it is a very small one. Assuming it is your roll, is your position good enough to double? The answer is — almost! But with such a long way to go it is better to wait a few rolls before losing control of the doubling cube.

While the exact count is important, you must add some common sense to it. Some running-game positions are better than others. It is a matter of just where your men are located. In general, it is better to have men scattered around than grouped on a few points. For example, you almost never get full advantage of your rolls if all your men are grouped on your one, two, and three points. The count for those men is small indeed, but the count is misleading since (unless you roll a double) you can’t bear off more than two men in any one roll.

As an extreme example, suppose that you have borne off ten men and have five left on your one point. Your count is 5, but barring doubles you need three rolls to get off. Even with a double, you still need two rolls.

But let’s say that your opponent has borne off thirteen men and has two men left on his six point. His count is 12, but he can get off in one roll with double 6, double 5, double 4, or double 3, and his chances of getting off in two rolls (see Table 4) is 78 per cent. In terms of odds, his chance of getting off in two rolls are one hundred sixty-nine to forty-seven, or slightly better than three and a half to one. Thus, even though your count is only 5, with your five men on the one point you are a decided underdog.

Counting the Number of Turns Left

At a certain stage of the running game you should forget about points and just count the number of rolls each of you should need to end the game.

Except for the position in which all your men are on your one and two points with at least half of them on the one point, there is always a chance that repeated bad rolls will cause you to miss, but if your men are all well forward in your board, you should just treat this as an unacknowledged possibility and not really allow for it.

Take this example: you have two men each on your one, two, and three points.

How many rolls to get off?

You expect to get off in three rolls, but there is a 2½ per cent chance that you will need four. Don’t worry about this; estimate three rolls.

Suppose that you are down to twelve men, placed two each on your one to six points.

How many rolls to get off?

Should you expect to get off in six rolls? Decidedly not. The odds are that you will miss at least once. Can you afford to estimate seven rolls here? Yes, you can! You can miss twice and still get off in seven rolls. The first miss will leave you with an odd number of men and cost you a roll, but the second miss will cost you nothing. It will merely give you an even number again.

This is most important: When you have an odd number of men, you can afford to miss bearing off two men on one roll without costing yourself a turn; when you have an even number of men, just one miss will cost you a full roll. It may seem obvious, but think about it and remember it!

Doubling and Redoubling after Counting the Number of Turns Left

When it is your roll and both you and your opponent need the same number of rolls to get off, you have an obvious advantage. In any such position, if the doubling cube is still in the middle, you should make the first double — provided that your chance to get off on schedule is at least slightly better than your opponent’s.

Suppose that you and your opponent each have fourteen men placed forward in your inner boards. It is a seven-roll situation for each of you.

Seven roll position

If your men are placed just the least bit farther forward than his, you should double. If they are placed the same or worse, don’t double. Wait a roll or two!

Furthermore, don’t be as quick to redouble as you may be to offer the first double. Remember that a redouble moves the doubling cube from your side of the table to his. This consideration is important enough so that you definitely should not redouble when there is a six-turn-against-six-turn position (that is, twelve men against twelve men) or a five-turn position (ten men against ten men) in which your opponent has all of his men down on his one and two points.

Four roll position
(eight men each)

In a four-turn position (eight men against eight men) you have a good situation in which to redouble. He should accept if all his men are on his one and two points, but he should refuse if two of his men are on his three point (or higher, of course). The reason for this is that he won’t bear off four men with double 1 or double 2, so that those two doubles won’t save him a roll.

In a three-turn position, he should refuse your double. To win he needs to roll a doublet on one of two turns, and even if he rolls one at his first turn, he gets no chance to redouble you since you will either win or lose the game on your second roll.

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