A cute position: Black can win 1/6 of the time with a double. Although he will be redoubled when he misses, he will then be only a very slight underdog (19–17). The chance of winning immediately makes this a correct double. White has far better than a 25% chances of winning, so he would take.

Black must double; this is effectively the last roll of the game and Black is a favorite. Evaluate White's equity as consisting of three parts:

1. White can roll a double after Black doesn't (approx 14%).
2. Black rolls 3-1 or 3-2 and loses (approx 0.5%).
3. Black rolls 3-1 and subsequently misses (approx (0.5%).

In a pure three-roll position (6 men each on the one-points), Black would double and White would pass. (White's equity would then be 22%). Here White has the extra possibility that Black can roll three ones in a row (probability approx 3%). In fact, after the sequence Black 2-1, White nondouble, Black 2-1, White can redouble! Therefore, take.

Obviously Black has a double. What's not so obvious is that White has just enough equity to take. This type of position is particularly difficult to evaluate wince White has no constructive game plan. His best strategy is to wait and hope to get a lucky shot as Black brings his men around the board, relying on innate randomness of backgammon. In 100 trials of this position, White lost a total of 90 units owning the cube at 2 (as opposed to the 100 units he would have lost by dropping).

Again, an easy double. Curously, this is an easier take than in Position 4. Although Black's home position is stronger, White still has the possibility of building a prime of his own. This problem illustrates the defensive value of owning your own 5-point. Without that point, White would have to pass.

Black's double here is overly optimistic in view of White's 4-point prime. However, Black is still a slight favorite, so White should only take.

Too good to double! Black's distribution of men makes him a favorite to win a gammon. Don't be deceived by White's semi-4-point prime; once Black closes his board he'll have no trouble escaping his back men.

White still has a clear take in Position 8 even after Black closes the one-point. Black should therefore wait to avoid the variation where Black rolls 2-any and White subsequently makes the the one-point. Black should not double until he sees the distribution of his three spare checkers for the bearoff.

Black must wait, since he will still have a double and take even after closing the 4 or bar-points. White should not beaver; Black is still a slight favorite.

An interesting position which violates the usually reliable principle that you should wait until the last possible moment, when your opponent still has a take, before doubling. White has a clear take now, and will have a clear drop after Black rolls a single six. Nonetheless, Black almost doubles his equity in the position by waiting until he rolls a six, then cashing the game.

A good example of the general principle mentioned in Problem 10. Black should double now. After next turn, he ay have three or four builders aiming at the 4-point, and White would have to pass if still on the bar. White has a 15-pip lead in the race and just enough equity to take now.

Time to put this problem to rest once and for all. It is true that with the Jacoby Rule in effect, there theoretically exist positions which are both initial doubles and beavers. For such a position to occur, three conditions must hold:

1. Black must win between 30% and 33% of the time.
2. Black must win a gammon whenever he wins.
3. Black must lose a single game whenever he loses.