Black – Pips 89 (+22)
Black to Play 5-3
One-way holding games are, for the most part, pretty easy to play. You hold onto your anchor, watch your opponent try to clear points, wait for a shot, and then hopefully hit it. Alternatively, you roll a 6-6 or 5-5 somewhere in this process and move into a race where you’re close to even or maybe a little ahead.
The tough decisions come when you hold onto your anchor but start to run out of moves that improve your home board. You’re still behind in the race, but not so far behind that racing is completely hopeless. Now you throw a number that give you some options: you can stay for a last-ditch shot, break contact and run with both men, or run with one man. How do you decide?
There are no hard and fast rules for these positions (that I know of, at least) but there are several features that you’ll want to consider before making your play. Put them all together and you should be able to come to a good answer. Let’s take a look at the key features and how they apply to this position.
(1) The race. Obviously, the further behind you are in the race, the more incentive you have to stay and get a shot. Evaluating the race, however, can be tricky. In “normal” racing positions, we often resort to the 8-9-12 rule for a quick approximation: double if you’re up 8%, redouble if you’re up 9%, and pass if you’re behind more than 12%. Here however, the pip counts after Black plays will be 67 for White and 81 for Black, with White on roll. Can we easily convert that pip count into winning chances?
Actually, we can. Let’s start with a useful table. It shows the winning chances in a straight race where the leader has about 70 pips (a so-called “medium-length” race) and the trailer is behind by 15% or more, with the leader on roll. The table assumes that both sides have well-structured home boards (no gaps or big stacks) and the trailer has a couple of more crossovers than the leader.
Trailer is behind by 15%, leader on roll with about 70 pips:
Trailer’s chances are in the 19% to 20% range.
Trailer is behind by 20%, leader on roll with about 70 pips:
Trailer’s chances are in the 14% to 15% range.
Trailer is behind by 25%, leader on roll with about 70 pips:
Trailer’s chances are in the 10% to 11% range.
Trailer is behind by 30%, leader on roll with about 70 pips:
Trailer’s chances are in the 8% to 9% range.
(This is a good table to commit to memory. While it helps in this position, it’s especially useful in tournament matches where you’ve entered a race with a lopsided score. When the player winning the race also has a big lead in the match, he’ll often be able to use his match equity tables to calculate a doubling point. The table above will then let him see what racing edge corresponds to the advantage he needs.)
Here White has a pip count of 67, and he’ll lead by 14 pips (21% of 67) if Black runs. Since his pip count is a little less than 70, and his lead is a little bigger than 20%, we can look at the table and guesstimate Black’s chances in a pure race at about 12%.
We need to massage that number a bit more however, because the table assumes a race with well-structured home board and no gaps or big stacks. In this case, White has two gaps, one on the 3-point and one on the 5-point. The severity of a gap is related to the number of checkers White still has to bring in and whether he has time to use small numbers to fill them. Here White has only three outside checkers, so as he brings them in he won’t have a lot of spare time to be gap-filling. A good general rule is to add a couple of percent to the trailer’s chances for each gap. Here that raises Black’s chances from 12% to about 16%. That number actually squares well with the rollout results after 20/17 20/15, which put Black’s chances at 16.5%.
(2) Pointing numbers and pick and pass numbers. If Black decides to run with just one checker, he has to pay attention to how pointing numbers and pick and pass numbers play. Are there a lot of these numbers? How would they play if Black ran with both men? How would they play if Black stayed with both men?
In this case White has four pointing numbers (2-2, 1-1, and 2-1) as well as eight pick and pass numbers (1-3, 1-4, 2-3, 2-4). Note that with the exception of 2-2, these are all White’s bad racing numbers. If Black runs with both men, these are the numbers that start to let Black back in the race. If Black stays with both men, these numbers don’t play badly.
(3) Hitting chances. Suppose Black thinks his racing chances are just too small and stays with both men. What are his chances of getting a shot, hitting a shot, and then winning?
Next turn White has no chances of leaving a shot. If White moves one checker off his 7-point, but keeps four checkers on his 6-point, then he’ll leave a shot on his second turn with just two numbers (6-2, 5.5%). Black then hits about 30% of the time, or 1.5% overall.
If on his second turn White has two checkers on his 7-point and three on his 6-point, he has a total of five shot numbers (6-6, 5-5, 4-4, and 6-2, 13.8%). Black will again hit about 30% of the time, or about 4% overall. But White only reaches this situation after rolling 6-5, 6-4, or 5-4 on his first turn.
All told, Black’s chances of hitting a shot after hanging around for two turns are a little over 2%. Does hitting a shot win? Not really. Consider the following fabulous sequence for Black:
Black 5-3: plays 6/1 5/2
White 5-2: plays 7/2 4/2
Black 4-3: plays 5/1 4/1
White 6-2: Plays 7/1 6/4, leaving a shot
Black 3-2: plays 20/18*/15
If White now flunks, it’s a double and a marginal take/pass for White. But if White enters, Black is only even money or a little worse.
We can now put the whole picture together pretty clearly. Running one checker is out, both because the pointing numbers and the pick-and-pass numbers turn White’s worst throws into great throws, and because Black’s hitting chances are so small anyway. But staying with both checkers is out because you’ll need to stay for two turns to get a small number of shots, while your racing chances, which are currently pretty reasonable at 16.5%, go down the drain. So go with both.
