Problem of the Week #32: Solution


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




Black to play 5-1.


When bearing in against an ace-point or other low anchor game, you generally have two goals in mind. The first is safety; you want to create formations that are less likely to leave blots as you bear off. The second is winning a gammon; you’d like to maximize your chances of winning a gammon if you can.

It’s usually good practice to look at the gammon chances first. If your gammon chances are either very large or very small, then it’s not likely that any play you make will affect them enough to matter. In those cases you just make the long-run safe play, whatever that might be. But if the gammon is up for grabs (which in practice means gammon chances in the 15% to 40% range) then you may need to look for riskier plays which win more gammons at the cost of some extra losing chances. (Typically these plays involve piling checkers on the 6-point and 5-point, hoping to hold the prime as long as possible.)

So what’s happening in Problem 32? Here it’s pretty obvious that Black is going to win a gammon unless he gets hit. White has one checker to enter off the bar, then four checkers to extract from the ace-point, and several checkers in the outfield as well. All told, he needs 20 crossovers to get his checkers home, even after he manages to enter. Any reasonable play leaves Black with gammon chances in the 58% to 60% range, which is so large that Black can forget about the gammon as an issue and just concentrate on safety. So now we can ask the interesting question: What’s the long-run safest play?

When playing strictly for safety, Black has four goals. Here they are, arranged in order of obviousness:

(1) Don’t volunteer shots. Here this simply means that Black won’t slot the 7-point with his ace. Giving White even a single indirect from the bar means giving him an extra 5.5% to hit, which could easily be a game-winner.

(2) Try to ensure that 6-6 and 5-5 don’t leave shots. This won’t always be possible, and even when it is possible you may decide to leave one of these numbers in order to secure better distribution.

(3) Strip the 6-point for quick clearance later. In general, you want to place your spare checkers on the 3, 4, and 5-points, preparing to clear the 6-point when you get all your men in.

(4) Avoid phantom interior gaps. A real interior gap occurs when an interior point like the 3-point or 4-point is actually open when the bearoff starts. Interior gaps are very weak and those formations will leave many more shots than a completely made board. Less weak but still a serious flaw are what I call “phantom gaps”, where you have all the points made but some points lack spares. For example, if you have spares on the 6, 5, and 3-points, but no spare on the 4-point, you have a phantom gap. Most players ignore this situation, but it’s actually more serious than it appears, leading to awkward formations later and more shots.

Note that some phantom gaps are worse then others. The 2-point and the 5-point are the least serious. The 2-point is so deep that it generally gets filled later, and the 5-point isn’t too important because it’s sometimes an advantage to clear the 5-point before clearing the 6-point. The serious phantom gaps occur on the 3-point and the 4-point.

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If we now put all these ideas to work, we can find a clear winner with the 5-1, namely 8/3 6/5! All criteria are met: no shot volunteered, 6-6 and 5-5 are both safe, the 6-point is stripped, and no phantom gaps.

Other plays don’t work as well. 10/5 4/3 blots on both 6-6 and 5-5 and leaves a phantom gap on the 4-point. 10/5 6/5 doesn’t blot but also doesn’t put a spare on the 3-point, which may prove useful.


Solution: 8/3 6/5

Two notes:
1) If gammon chances is 5% or 95% - it does not matter. It is still important to increase them. I think in this position gammon chances is high. GNU evaluates them as 80%. But here all moves that increase wins increase gammons.
2) Are you sure that 10/5 6/5 that gives extra good five in future is really worse than 8/3 6/5?

I disagree. You have to watch the exchange rate between gammons and losses. If your gammon chances increase by 5% by making a risky play, but this also causes your losing chances to increase by 25%, then you're making a poor trade. This ratio is more important than the goal of simply increasing gammons.

Here's a question for Bill (and anyone):
If you play the safer 8/3, 6/5, and have concluded that White poses some threat if he hits, would you continue to play safe during the bear off? For instance, if your next roll were 41, would you clear the six point or bear a checker from the five point?

A Note about Gammon Risks
I think aoleg1 may be confusing the aggressive play recommended against an oppenent who is closed out with the much more cautious play required against an ace-point game. Of couse, he is certainly right that anytime you can win two or more gammons for every victory you sacrifice, then you should take the risk.

On the sequence
Black 51, played 8/3 6/5
White fans
Black 41
here's how I would evaluate Black's choices:

6/2 6/5: no good, allows White to hit my outfield blots with 62 and 64.

10/5: OK distribution, no interior gaps, but blots next turn on 55.

8/4 10/9: good distribution, no interior gaps, doesn't blot on either 66, 55, or 65.

8/3: blots on 55.

I probably wouldn't go any further, and I'd just play 8/4 10/9.

If I thought a play was aggressive but won more than two gammons for each extra loss, I'd make it. My original point (perhaps badly expressed) was that when gammon chances are already either very low or very high, there don't tend to be any such plays. Those plays most often occur when gammon chances are in the middle somewhere. Here they are high (over 60% mostly) so good plays are safety oriented.

I'm sorry, Bill, I made a mistake in my question. I was trying to ask about overall strategy: safe or bold. I forget for the moment that the bear in was not complete.

Assuming that it was, is it better to clear from the rear as soon as possible? Or, would you bear off checkers with rolls such as 4-1? This is predicated on White's position holding stable, without joker doubles, and so on.

Thanks for you help.

If playing safe was the goal, I would almost always clear the 6-point with a 4-1, rather than bear somebody off.