Backgammon Articles
 Burying vs. Bearing In by Douglas Zare

 This article originally appeared in GammonVillage in 2001. Thank you to Douglas Zare for his kind permission to reproduce it here.

The 7-5-3 Bearoff Position
In a large fraction of backgammon games, at some point the players break contact and race. Some people think racing is pure luck, but there is still some skill left: handling the cube and positioning one's checkers properly. Since it comes up so frequently, one should know the basics of bearing in correctly.

Experts rarely make large errors in non-contact checker plays, but beginners often cost themselves a noticeable amount in each game. This is easy to avoid once one knows what the correct goals are.

What is the fastest way to bear off 15 checkers?

A novice will think of the top position, and an expert will think of the bottom, the 7-5-3 position. Certainly no matter who is on roll, the top player is a big favorite. However, one never has a choice between these two positions. While bearing in, you have no control over the pipcount. The top player has a pipcount of 15, and the bottom player has a pipcount of 79. A more reasonable comparison is between the following sides:

 Which side would you choose?

Now, the pip count is 15 for both players, and the bottom player will be off in 2-3 rolls, whereas the top player will probably take 6-7 rolls. Clearly the bottom position is better, even though not all rolls take at least two checkers off.

Of course, one almost never sees someone try to put 15 checkers on the ace point. However, one probably would not stumble upon the 7-5-3 position (at bottom in the first diagram) by chance. It was checked by a computer, and you can simply rely on the fact that among all bearoff positions with 79 pips, the 7-5-3 position is in some sense the most efficient, bearing off fastest on average. Whether you are ahead or behind in the race, the 7-5-3 position is a good goal.

A Contest
The following is the record of an exercise, not an actual backgammon game. Both sides receive the same rolls and move simultaneously. I'll take the bottom checkers and aim for 7-5-3, but my opponent, normally a strong player, will not. I can't promise my moves are right, but this will illustrate why bearing in high is better than spreading out the checkers evenly in one's home board.

 White and red each startwith 15 checkers on theirown twelve point.
 Roll #1: White: Red: 5-4 12/7, 12/8 12/7, 12/8 Roll #2: White: Red: 2-1 7/4 12/11, 12/10 Roll #3: White: Red: 6-3 12/6, 4/1 12/6, 12/9

Moving a checker to the ace point is already an error. The best white can hope to do is break even. At worst, white will have to move other checkers to the ace point, perhaps because of a gap on the 4 or 5 point in the bear off, or too many checkers left on the bar point at the end of the bear in. Extra checkers on the ace point are bad.

Meanwhile, I'm happy to unstack rather than get an extra crossover. This will help me make the most efficient use of doubles.

 Roll #4: White: Red: 6-5 8/2, 6/1 12/6, 10/5

Moving another checker to the ace point is terrible, as is burying a checker on the 8 point rather than unstacking the 12 point.

 Roll #5: White: Red: 4-3 12/5 9/6, 8/4

My position is so flexible that it probably doesn't matter what I do. Still, it might have been better to keep unstacking. Playing 12/9 would give me another useful 5, as I would rather not pile checkers up on the bar point.

 Roll #6: White: Red: 5-2 12/7, 12/10 12/5 Roll #7: White: Red: 1-1 10/8, 7/6, 5/4 12/11, 12/10, 7/6 Roll #8: White: Red: 6-6 12/6(3), 8/2 12/6(2), 11/5, 10/4 Roll #9: White: Red: 5-2 6/1, 6/4 12/5

I don't know whether it is the largest error in absolute terms, but sending a third checker to the ace point is hideous. It was in the right place on the 6 point; one would be happy to have 7 checkers on the 6 point in the 7-5-3 position. Also, on the ace point that checker can't be used to fill the gap on the 5 point, which might become a serious problem. Any 5 played to the ace point in the bearoff will be a disaster because there are already too many checkers on the ace point.

 Roll #10: White: Red: 6-5 12/6, 12/7 12/6, 12/7 Roll #11: White: Red: 3-1 12/9, 6/5 7/4, 6/5 Roll #12: White: Red: 4-1 9/5, 7/6 11/6 Roll #13: White: Red: 6-2 12/6, 12/10 6/off, 5/3 Roll #14: White: Red: 6-3 12/3 6/off, 3/off

I disagree with plugging the gap on the 3. Gaps on the 3 point are not a problem if you have extra checkers on the 6 point, since a 3 can be played 6/3 in a useful fashion, and even 3-3 plays well. A gap on the 3 point is a problem if you run out of checkers on the 6 point. A gap on the 4 point is bad immediately, since a 4 will miss and still leave a gap.

 Roll #15: White: Red: 6-1 10/4, 1/off 6/off, 4/3 Roll #16: White: Red: 5-2 5/off, 2/off 6/4, 5/off

White has rolled a couple of small numbers, clearing off two of the checkers on low points that shouldn't have been there. However, White still has problems because of the two checkers on the ace point and the thinness of the 5 point. If White rolls 2 5's before clearing the 6 point, another checker will probably have to go to the ace point.

 Roll #17: White: Red: 5-1 5/off, 1/off 6/off

It would also be reasonable for me to play 5/off, 3/2 or 5/off, 4/3.

 Roll #18: White: Red: 4-4 6/2, 4/off(3) 5/1, 4/off(3) Roll #19: White: Red: 5-4 6/2, 6/1 6/2, 5/off

Oops. Two misses for white. Meanwhile, my position is still smooth, i.e., I am prepared to make good use of almost every roll.

 Roll #20: White: Red: 5-1 6/off 5/off, 1/off

The pipcounts are equal: 11 pips each. Not having enough checkers on the 6 point could easily have lead to falling behind in the race due to the wastage from using large numbers to bear off from low points. However, that's not the only consideration. White has more checkers, and no checkers on high points, so white expects to take longer to bear off for the same pipcount.

 Roll #21: White: Red: 5-5 3/off, 2/off(3) 6/1, 3/off, 2/off, 1/off

So, this time, white fell behind by a roll by bearing in too deeply. White's bearoff was correct, but my guess is that white wasted the equivalent of about 5 pips by bearing in badly. White had more than a 50% chance to require an extra roll. Of course, it might not have mattered, and fluke final rolls might have allowed White to bear off more quickly, but in the long run White will lose a lot of close races by making these errors. If the race should be even, white becomes about a 2:1 underdog by playing like this.

If one tries to close one's board, e.g., 3-3-3-2-2-2 instead of 7-5-3, this wastes several pips. I would guess that one requires an extra roll just under half of the time if one aims for a closed board instead of 7-5-3. There is more information about the 7-5-3 position in the rec.games.backgamon archives in these two articles.

Tips
 Don't bury checkers deep in your board. If you aren't worried about saving/winning the gammon, don't worry about maximizing crossovers. Stay flexible. Having multiple checkers on your bar point is not flexible. Checkers on your 6 point are usually good. Put many there in the bear in. Gaps on your 1, 2, and 3 points are ok, but gaps on your 4 and 5 are bad. In the bearoff, if you can take two checkers off, it is almost always right to do so. These principles are consistent with the statement that the 7-5-3 position is the most efficient 79 pip bearoff position. There are a lot of extra checkers on the 6, yet not so few checkers on the 4 and 5 that those points are likely to turn into gaps. The lack of checkers on the lower 3 points is not a problem, as small numbers can still be played to redistribute the checkers on the high points, fill lower points, or bear off. By the way, who was my clueless opponent in this exercise? Snowie 3.2, 3-ply, 100%, with racing database.