Problem of the Week #146: Solution
(a) Cash game, Black owns a 2-cube.
Should Black double? Should White take if doubled?
(b) Same position with the score tied 1-1 in a 5-point match. Cube in the middle. Should Black double? Should White take if doubled?
(c) Same position with the score tied 7-7 in a 15-point match. Black owns a 2-cube. 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
(a) We’ll look at the cash game question first.
At first glance, Problem 146 looks pretty straightforward. Black played a 2-3 back game and everything went well. He got a shot, he hit the shot, then he got another shot and hit that too. To make things even better, all this happened while his board was still intact. Now White has one man back on his 24-point and another in the air, while Black is moving in for the kill. Must be double and easy pass. Maybe too good? Nah, probably not. Let’s cash the game and get on with the next.
Hold on though. Things aren’t quite that simple. This position isn’t your basic ‘I-hit-two-men-and-now-I’m-cashing’ position. Let’s take note of these unusual features:
> Black is still holding on to both his back game points. That’s nice if he wanted to continue playing a back game, but right now he has to extricate those men, and quickly. That might be easy to do, and then again it might not.
> White’s position is still intact, with no dead checkers yet. Usually by the time the back game player hits two checkers, his opponent’s game has collapsed. That’s not the case here. Threes and fours are awkward numbers for White, so he may start killing checkers soon. Then again, he may not. Black will be running his back checkers soon, and if White enters and hits, things could get tricky.
> White’s still ahead in the race by 33 pips. That’s going to matter in some variations.
> Black’s rolls aren’t trivially easy to play. A lot of rolls will involve difficult choices between playing quietly, hitting loose, or running off one of the back points. How should Black play a 3-1? Hit loose, slot the 2-point, or just play 8/4? How about 2-2? Hit loose or switch? How about 5-3? Hit loose, or run off the 23-point, or run off the 22-point? Then there’s 6-5. Hit loose or run 23/12 or maybe the juicy duplication with 23/18 23/17?
Black does have a real advantage with some crushing threats. Almost any roll that makes significant progress, followed by White’s dancing, will be double-pass next turn. ‘Progress’ simply requires making the 2-point, or getting one of his back checkers to the outfield. A few sequences actually leave Black too good to redouble: any of the three big doubles (6-6, 5-5, 4-4) followed by a fan, or 5-4 making the ace-point followed by a fan. So Black can lose his market a significant part of the time, and some of those market-losers are real crushers. Hence Black must double.
We’ve already listed some of the concrete reasons White should take. He’s ahead in the race substantially, with a good board and no checkers out of play yet. Black has a lot of checkers to extract, and he doesn’t as yet have any outfield presence. If White enters quickly, Black finds himself squeezed, with a lot of work to do on both sides of the board. Solid take for White in a cash game.
(b) Same position, score tied 1-1 tin a match to 5.
As we’ve seen in the past, the initial double at the 4-away 4-away score strongly favors the side with some gammon chances. A solid edge with gammon chances greater than 20% usually translates into double and pass. The defender does better to play on from 2-1 down than to take the chance of losing the match on a gammon. That’s exactly the case here. Black’s gammon chances in a cubeless game are about 25%, making this a monster pass at this match score.
(c) Same position, score tied 7-7 in a 15-point match, Black owns a 2-cube.
At first this looks very similar to (b). Black doubles, and White loses the match if he loses a gammon. Must be a drop, right? Not quite. The key is that dropping costs White more in (c) than in (b).
In (b), if White drops he trails 2-1 to 5 and has about 43% winning chances. In this situation, a drop leaves him trailing 9-7 to 15, with only about 38.5% chances.
At the same time, turning the game around and winning is better for White in (c). When White turns the game around and cashes in (b), he leads 3-1 to 5, for match chances of about 67%. In (c), winning the game gives White an 11-7 lead to 15, with about 74% chances.
Although in each case he goes to zero when he loses a gammon, his increased chances in these other two variations allow him to take more easily. Instead of the huge pass he had in (b), he now has a marginal take.
(a) Cash game. Double and take.
(b) 4-away, 4-away, cube at 1. Double and big pass.
(c) 8-away, 8-away, cube at 2. Double and marginal take/pass.
Oh, and here are the answers to the checker plays I posed earlier. Assume a cash game and Black has doubled to 4, with White taking.
Black to play 6-5
: Two out with 23/18 23/17. The nice duplication of entering and hitting numbers makes this a little better than running one checker off either point. Note that loose hitting (with 6/1*) is a bad play compared to running. The loose hit creates a self-squeeze, where Black now has more work to do on both sides of the board.
Black to play 5-3
: A very unusual position where running off the front point (22/14) actually trumps running off the back point (23/15). Running off the front point duplicates White’s entering and hitting numbers, while giving White a very bad 3. Running off the back point turns 1-3 and 2-3 into good numbers.
Black to play 3-1
: It’s a photo finish between 8/4 and 5/1*. When Black rolls two small numbers and can’t move his back men, hitting loose becomes a reasonable plan.
Black to play 2-2
: Switching is bad, as the game is likely to go on a long time and Black needs the best board he can get. The top choice is attacking with 8/4 and 5/1*.