Bill Robertie, 1980
Las Vegas Backgammon Magazine, October 1980
Here are the answers to the Doubling Quiz (presented here). Give yourself one point for each correct answer (two points per problem). A grading scale appears at the bottom of the page.

Problem 1
Should Black double?
Should White drop, take, or beaver?
Double, take.

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.

Problem 2
Should Black double?
Should White drop, take, or beaver?
Double, 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%).
Since the sum of all these equities is greater than 25%, White should take.

Problem 3
Should Black double?
Should White drop, take, or beaver?
Double, take.

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.

Problem 4
Should Black double?
Should White drop, take, or beaver?
Double, 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).

Problem 5
Should Black double?
Should White drop, take, or beaver?
Double, take.

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.

Problem 6
Should Black double?
Should White drop, take, or beaver?
No double, take.

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.

Problem 7
Should Black double?
Should White drop, take, or beaver?
No double, drop.

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.

Problem 8
Should Black double?
Should White drop, take, or beaver?
No double, take.

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.

Problem 9
Should Black double?
Should White drop, take, or beaver?
No double, take.

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.

Problem 10
Should Black double?
Should White drop, take, or beaver?
No double, take.

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.

Problem 11
Should Black double?
Should White drop, take, or beaver?
Double, take.

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.

Problem 12
Should Black double?
Should White drop, take, or beaver?
No double, beaver.

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.
In Position 12 (from an article in the March 1980 Las Vegas Backgammon Magazine conditions (1) and (3) hold, but condition (2) fails. If Black hits, he will win a gammon about 80% of the time, win a single game about 15%, and lose about 5%. Unfortunately, the more blots Black hits, the greater his chances of leaving a blot in the bearoff and losing. The net effect is that Black's initial double is incorrect. White, of course, should still beaver. (In fact, White must beaver to make Black's double incorrect.)