Problem of the Week #80: Solution
Cash game, center cube. Black on roll.
Should Black double? Should White take if doubled?
Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.
Black’s off to a good start in Problem 80. He’s made the best possible anchor, built his 4-point, and also has the start of a prime. White’s fallen behind early, and now has a third checker back. He has managed to build his 10-point, which has some value. Clearly Black has to consider a double, and White might be facing a tough decision whether to take or drop.
We’ll start with the double first. Sylvester’s Law has helped us in the past in early game positions, so let’s use that first. We’ll look at the race, the position, and the threats, and see if Black has an edge in two out of three. If he does, we’re usually happy doubling.
The race: Black leads 143 to 175. Big edge to Black.
The position: Solid edge to Black, with his anchor, better home board, and the start of a strong blocking position.
Threats: Solid edge to Black, with the chance of making his 5-point quickly. Black’s anchor prevents White from doing anything soon.
Case closed. Black has a good edge in all three areas. This must be a double.
Now let’s look at the harder question: Should White take or drop?
Before doing any analysis on a take/drop question, I like to step back a little, look at the position as a whole, and see what my gut impression is. If any analysis I do turns out to be inconclusive, I’ll just go with my gut. For Problem 80, my first impression is “take”. Black only has a two-point board, White has an anchor, and there aren’t a lot of White blots floating around. Most drops in the early stage of a game are predicated on serious gammon chances (in the 30%+ range), and that’s not the case here. Combine this with White’s reasonable structure (10-point, generally good distribution, and no buried checkers) and it’s hard to believe that White should pass this.
Clearly White’s game goes downhill quickly if Black is able to make his 5-point, so we can do some simple calculations to see how often that actually happens. On the first turn, Black can make the 5-point with 10 shots: 1-1, 2-2, 3-3, 4-4, 3-1, 2-1, and 3-2. Of these shots, only half (1-1, 3-1, 3-3 and 4-4) are really crushing. Four of the other pointing numbers (2-1 and 3-2) leave a blot on the bar-point, giving White 10 quick return hits. The last number, 2-2, leaves the bar-point open, but it’s still a good shot.
Black can also make the 5-point after a loose hit, followed by a miss, followed by a cover. Assuming Black won’t hit loose with a roll like 5-2, Black has a total of 14 loose hits: 4-1, 5-1, 6-1, 4-3, 5-3, 6-3, and 6-2. Of those, White hits back a little over half the time. When he doesn’t hit back, Black is a pretty big favorite to cover. Of the 14 loose hits, Black probably ends up covering next turn in something like 4 or 5 games.
If Black makes the 5-point and White doesn’t respond with something good, Black’s moved well into pass territory; we can probably assume he’s something like an 80% favorite or better in these games. When Black doesn’t make the 5-point, White gets close to an even game if he can then make the point. If neither side makes the point for awhile, Black’s chances probably stay in the 60% range.
We can sum all this up in a short table, rounding categories off to even numbers:
(1) Black makes the 5-point, White doesn’t do much: 13 games.
(2) Black makes the 5-point but White hits loose on the bar: 1 game.
(3) Black hits loose, White hits back: 8 games.
(4) Black hits loose, White misses, Black doesn’t cover: 2 games.
(5) Black shoots a blank, White makes the 5-point: 4 games.
(6) Both sides make no progress: 8 games.
Our last job is to group these into a couple of narrow categories and make a good estimate as to just how many games Black and White can win in each category.
In Group (1), Black’s doing very well, and the position is now a clear pass with some gammon chances. I’d make Black an 80-85% favorite here in 13 games.
Groups (3) and (5) fit together well. In these 12 games, Black is a slight favorite.
Groups (2), (4), and (6) also fit well. In these 11 games, Black is probably a little better than 60%.
Summing up these numbers from White’s point of view gives us a quick estimate of his winning chances.
In Group (1), White wins 15% of 13, or about 2 wins.
In Groups (3) and (5), White’s a slight dog in 12 games, so we’ll give him 5 wins here.
In Groups (2), (4), and (6), we think White wins a little less than 40% of 11 games, so let’s give him 4 more wins.
Our total is about 11 wins for White out of a 36 game sample, just under one-third. We’ve probably been a little pessimistic for White, and we haven’t factored in the value of cube ownership. We also haven’t factored in gammons, but this doesn’t rate to be a hugely gammonish position. However, the analysis backs up our initial impression of a take, so over the board you should feel pretty comfortable taking.
In general, this is the approach I like to take when deciding take/drop decisions over the board. Break the position down into a few categories based on your opponent’s next roll, make some common-sense estimates of wins in each category, and sum up the results. If gammons play a big role, try to get an overall sense of gammon probability, and remember that the defender needs one extra win to balance out every two gammons. Looking one roll ahead has the practical value of extracting some of the emotion from a take/drop problem, and simplifies the process of making a well-reasoned decision.
Solution: Black should double, and White should take.