An easy take, but a somewhat surprising redouble. An extensive calculation revealed that Black's raw probability of winning is 63%. Normally with two or three rolls left in the game this would not be enough to justify a redouble. Notice, however, that even if Black throws his worst number (2-1), White will not have a strong enough position to redouble to 8. Since the cube has no value to White on his next turn, Black has no reason to hold onto it this turn. By redoubling now, Black insures that he will not lose his market if he bears two men off and White bears off only one man.

The easiest way to see that this is not a double is to consider the most likely case: Black and White both bear two men off next turn. In that case, Black can double and White can still take. (White's winning chances would then be 28%.) It is very rarely correct to double if your opponent will still have a take in most cases on the next turn. Owning the cube, White is actually a slight favorite in this position. Black can lose in three ways: by throwing a one next turn, by White's throwing a double, or by throwing a one on his second turn. The cumulative sum of these probabilities is slightly greater than 50%, so White should beaver.

The basic 3-roll position (6 men each on the one-points) gives White a winning probability of 22%. Here White has an additional winning chance: Black can throw an immediate 2-1 and be redoubled out of the game. The extra probability of a 2-1 is 5½%, so White has more than enough equity to take.

A position that can be evaluated using the pip count; Black's count is 126, White's is 135. Black's lead of 9 pips means that White would just have a take in a noncontact position. Here Black has the additional possibility of pointing on White next turn (with 10 rolls: 5-1, 3-1, 5-3, 1-1, 3-3, 4-4, or 5-5). Even if Black misses, White may not be able to escape his straggler and could get attacked on a subsequent turn. So drop.

Black has too many winning options in this position for White to risk a take. Black can win with an attack followed by a closeout, or by simply escaping his back checkers. White's distribution is too awkward for him to improve his position easily. White's most reasonable hope is to establish a 3-point anchor behind Black's prime, not much of a reason to justify a take.

This is nothing like a double, and actually closer to a beaver than a drop. Black has a long way to go to complete a closeout in this position, even if he hits the checker on the 10-point. White will have a significant timing advantage as soon as he can establish an anchor anywhere. A healthy dose of caution is reasonable, but a drop in this position is pure cowardice.

Black must double now; he loses his market entirely if he rolls a six next turn, and he has an advantage even if he doesn't. White, at the same time, has a very clear take. He is a long shot to be gammoned, his back men can't be completely trapped, and he has a chance to put Black behind a 5- or 6-prime should Black not escape his last checker soon. White even retains some equity in the position should Black escape this turn. Easy double, easy take.

A widely misunderstood position. Four-point anchor positions need to be evaluated in terms of racing equity, in much the same way as noncontact positions. In the time it takes Black to dismantle his outside blocking points, White has many chances to escape his back checkers and join the race. Black has some extra equity in his ability to point on White's last checker (if White is forced to run with just one man) but most players overrate this factor in their doubling decisions. Here the pip count is: Black 88, White 69. No double.

Like Problem 6, this is almost a beaver. Black could double this sort of position if he had a reasonable number of throws to make the fifth point in his board. Here he has only 5: 2-1, 3-2, 1-1. Black will, of course, hit loose if he can. However, White's possibility of a return hit, coupled with his 4-point prime, give him excellent winning chances. Black should save the cube for next turn, where he will have chances to use it effectively.

A solid double. White's take might look risky, but his ownership of the 15-point give him a flexible structure, and once he enters his checker from the bar he can work on improving his own position. Of course, Black doesn't even cover the 5-point with 5 rolls (5-2, 5-4, and 5-5) after which it's anyone's game.

In contrast to Problem 10, Black's position here is stronger in two ways: he has only one man back, and he is one roll away from establishing a 5-point prime. White should definitely pass. Incidentally, many players overlook the strength of having just a single man back. One man back vs. two or three for the opponent is almost a doubling advantage itself.

If you tend to take positions like this you need to seriously reassess your evaluation of back games. White's position is hopeless whether or not he ever succeeds in making the 4-point. Black's two men on the 21-point give him plenty of time to fiddle while White's game burns to the ground. Remember that playing a back game usually doesn't allow for any prolonged maneuvering. Unless the back game player is in position to recirculate checkers to the outer boards, he is continually threatened with disaster on every roll. The longer the game drags on, the more certain he is to roll the large double that spells finis.