Problem of the Week #114: 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.


Problem 114 shows White stuck in a 3-point holding game, while Black leads by 43 pips and is ready to start bringing his men home. Black’s structure is pretty good; his only weakness is the lack of his bar-point, which will cause him a few extra problems as he later tries to clear his 8-point. White’s position is a little unusual in that he has a third man back, on his 23-point. The extra checker may give White a few more indirect shots in the future, but it’s also a target for Black to attack. With a 43-pip lead, it’s hard to believe Black doesn’t have a big enough edge to double. The real question is whether or not White can take.

To see what’s going on here, let’s take a look at the cube action in a few typical 3-point holding games. We’ll leave Black’s structure alone, and play with White’s position a bit. Here’s our first example:




Position 114a: Black on roll, cube action?

Here Black’s lead is only 13 pips, 115-128, and White doesn’t have an extra checker back. Black has a marginal double/no double, and White has an easy take. Black’s cubeless winning chances are about 74%, with some tiny gammon chances. Black’s racing lead of a little over 10% represents the minimum he needs to turn the cube with his bar-point open. White has some racing chances and some hitting chances, which together give him enough equity to take comfortably.




Position 114b: Black on roll, cube action?

With a 20-pip lead, 115-135, Black’s cubeless chances have risen to a bit over 76%. His double is now solid rather than marginal. White has a take, but it’s no longer a blunder to drop, just a mistake.




Position 114c: Black on roll, cube action?

With a 27-pip lead, Black’s double is strong and White’s take is now marginal. These three problems illustrate one of the key differences between 3-point holding games and 5-point holding games. In 5-point holding games, falling further behind in the race makes your chances of holding on and hitting a shot go up even as your racing chances go down. As a result, 5-point holding games are a take over an extremely wide range of racing leads. On the other hand, 3-point games are carefully bounded for any given structure. There will be a point where the doubling window opens and another point where it closes. With Black having the 4, 5, 6, and 8-points, and White’s structure having no particular flaws, the doubling window opens around a 13-pip lead, and closes with Black having about a 30-pip lead.




Position 114d: Black on roll, cube action?

With Black’s lead at 33 pips (115-148), it’s now a double and a clear pass.




Position 114e: Black on roll, cube action?

With White having a third checker back, his race deficit grows to 42 pips (115-157). His winning chances are almost the same as in position 114d (about 22%) but now his chances of losing a gammon have started to rise. Black now wins about 10% gammons, whereas Black’s gammon chances in 114c and 114d were in the 5-6% range. With a third checker back, there will be some variations where the third checker never gets moving until late in the game, and White won’t quite be able to run off the gammon. As you might expect, Position 114e is an easy double and a solid pass.


With these examples in mind, evaluating our original problem is pretty simple. Black has a double based solely on his lead in the pip count. White is so far behind he would have a pass even if his third checker back were safe on the 22-point. As a blot on the 23-point, White’s checker allows Black to pick up even more gammon equity when rolls like 6-2, 4-2, and 6-4 put him on the bar, perhaps for a while. As a result White is considerably worse off than in position 114e, and he has a big pass.


Solution: Black should double, and White should pass.

I'm such a fish!

Does it show my age that I still say drop instead of pass?

No comment.

I got suckered by this one. I thought Bill might be trying to "stick" it to anyone who says the three point is no good!

I've got to remember to play the position, not the QF (quiz factor).

In any event, it is nice to see a cube problem where the answer is not "Double, take."

Gotta think on the next level, like a poker player.

This makes me want to change my answer to problem #115