The Bear-Off

The bear-off is the last stage of a backgammon game, when all of a player's checkers are in his home board and he starts removing them. Beginners are often surprised to learn that many games never reach the final checker: a double is offered and passed long before. If the game does get to the bear-off, there is a good chance it will end with a double or redouble there. So it pays to know both how to remove your checkers and how to handle the cube in typical endgames.

Removing checkers: the basics

A few simple rules cover the vast majority of bear-off positions without contact. There are exceptions, but these will rarely steer you wrong:

1. Keep your distribution smooth. Checkers spread over several points let more numbers bear off. Stacks leave gaps, and every gap means some rolls bear off nothing. Each wasted roll is a gift to your opponent.
2. Bear off a checker whenever you can.
3. If you cannot bear off, fill the highest gap, as long as doing so does not open another gap.
4. Otherwise, move a checker from your tallest stack.

When your opponent still has a checker in your home board or on the bar, the priorities change: safety comes first, and you should bear off in a way that leaves as few shots as possible.

Endgame doubling positions

Many bear-off cube decisions can be judged by counting the rolls each side needs. A player who will clear his checkers in two turns needs two rolls, and so on. The guidelines below assume a non-contact position and a money game.

1. One roll ahead

If the player on roll needs at least one roll fewer than his opponent, he should double (or redouble), and the opponent should pass. For example, you are on roll and need two rolls, while your opponent needs three. To win, your opponent would need doubles that save him a roll while you roll none, and in a position like this one even that is usually not enough.

2. Equal number of rolls

When both sides need the same number of rolls, the player on roll is the favorite because he gets to go first. The table shows his approximate cubeless winning chances in a simplified model where any doubles save exactly one roll and nothing else is wasted. Real positions with misses will differ a little.

The 2-roll figure is easy to check. The player on roll loses only if he fails to roll doubles and his opponent then rolls them: 5/6 times 1/6, or 5/36, about 14 percent. A take requires roughly 25 percent winning chances, so the 2-roll and 3-roll cases are passes, while the 4-roll case is just about a take.

3. Last roll

If your opponent is certain to finish on his next turn, you should double whenever you are a favorite to bear off all your remaining checkers this turn, since the cube can no longer be used against you. Your opponent should take if he has at least 25 percent, that is, if you will miss at least 9 of the 36 rolls. With two checkers on the 1- and 2-points you miss nothing; with two checkers on the 6- and 5-points you succeed with only 6 rolls, so you should not double at all.

4. Three checkers against two

Suppose you have three checkers left low in your board, enough to finish in two turns, and your opponent has two. You should not double if he is a favorite to bear off both of his checkers on his next turn. But if his two checkers are on high points, for example the 6- and 5-points, only 6 of 36 rolls (6-5, 5-6, 6-6, 5-5, 4-4, 3-3) take both off. You win at least 30 times out of 36, about 83 percent, so you should double and he should pass.

Conclusion

These rules cover most bear-offs you will meet. Count the rolls each side needs, keep your checkers spread, and remember that a take needs about one chance in four. For tricky cases, bear-off databases built into GNU Backgammon and eXtreme Gammon give exact answers, and our probability tables and the race article cover the steps that come before.