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Old 12-06-2009, 02:48 PM   #1
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Join Date: May 2004
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Problem of the Week #39: Solution

Problem of the Week #39: Solution

Cash game. Black owns the cube. Black on move.

(a) Black to play 5-1.

(b) Black to play 5-1.

Problem 39 is our first look at the fun world of post-ace-point games.

“Post-ace-point” is a little bit of a misnomer. These positions can be reached from ace-point games, but also from deuce-point games, or back games, or even games where someone was on the bar and closed out. The main idea is that you held on and finally hit a shot, then contained the hit checker or two, then completed a closeout, and finally started to bear off. Mostly your problem is figuring out exactly when to redouble, but sometime the problem lies in how safely you should play your checkers.

Problem 39 shows two examples of the most common checker play quandary. In each case, Black has a choice between bearing off one checker and playing completely safe (5/off 5/4), or bearing off two checkers while leaving a shot (5/off 1/off). What’s right, and how do we make the decision?

The first metric we want to calculate is the crossover count. A crossover is simply a move of a checker from one quadrant to another, or from the bar to the opponent’s inner board, or from the inner board to the bearoff. Let’s start with Problem 39a. Black has 15 checkers in his inner board to be borne off, so his crossover count is easy: it’s just 15. White’s is a little more difficult. His six checkers in his inner board represent six crossovers obviously. His checker on the bar represents another five crossovers: one to enter, three more to get from Black’s inner board to White’s inner board, and one more to bear off. White’s total crossover count is 11.

So in 39a, Black trails in the crossover count by four, 15 to 11. In 39b, he also trails by the same four crossovers, 14 to 10.

Next we employ the following rule of thumb:

>If you trail by two or less in the crossover count, play safe. You’re doing well enough in the race that there’s no need to take additional risks.

>If you trail by five or more in the crossover count, take two checkers off and leave a blot. You’re a big underdog in the race, and you need the extra checker speed.

>If you trail by three or four, you’re in a grey area.

Well, that’s nice. We’re in the grey area in both positions. What next?

In the grey area, decisions depend very much on the exact arrangement and count of checkers in the inner board. You next want to look at all of the following considerations and see if they point toward one play or another.

(1) If you trail by three crossovers, tend to play safe. If you trail by four, tend to bear off.

(2) If White has a blot in his board, tend to bear off. If no blot, tend to play safe.

(3) If taking two checkers off brings you to an even number of checkers, tend to bear off, otherwise tend to play safe.

(4) If you have a speed board, tend to play safe, otherwise tend to bear off. A speed board is one where Black’s home board spares are heavily concentrated on the one and two points, which implies that small doubles are more likely to bear off four checkers through the bearoff. With a slower board, where the checkers are spread evenly across points, small doubles often won’t save a roll.

Now let’s see how Positions 39a and 39b compare across these four criteria.

(1) Crossover count? Black trails by four in each position.
Problem 39a – favors bearing off.
Problem 39b – favors bearing off.

(2) White blot? White doesn’t have a blot.
Problem 39a – favors playing safe.
Problem 39b – favors playing safe.

(3) Getting to even? Taking two off in 39a brings Black to 13 checkers, an odd number. But in 39b, taking two off brings him to 12, potentially saving a roll.
Problem 39a – favors playing safe.
Problem 39b – favors bearing off.

(4) Speed board? Black has a slow board in both positions.
Problem 39a – favors bearing off.
Problem 39b – favors bearing off.

For Position 39a, our four criteria split two and two. Rollouts show the position is actually a tossup, with a minute edge for playing safe.

In Position 39b, three of our four criteria favor bearing two off, and rollouts show that’s the correct play by a wide margin.


Position 39a: Tossup

Position 39b: 5/off 1/off
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