Problem of the Week #142: Solution


Money game, center cube, Black on roll.




Should Black double? What should White do if Black doubles?



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 142 is an interesting problem with a few non-standard elements. On the one hand, it’s a prime versus prime problem; both sides have 5-point primes, and both sides have some escaping to do. On the other hand, it’s an action play problem. Black is on the bar shooting at a couple of blots, and a lot hinges on whether he hits or not next turn.

So we have a blend of priming elements and action elements. Which elements are more important? That’s actually an easy question to answer. Take a look at Position 142a:




Position 142a: Black on roll. Cube action?

Position 142a is just our original position with one White checker moved from his 2-point back to his 14-point. Black now has 27 shots from the bar at three blots, so the action aspect of the position is even stronger. (Look at how 1-1 now plays!) But – the checker on White’s 14-point now gives White more timing, so the priming aspect of the position is worse for Black.

So in 142a, Black has more action but less timing. Which is more important? If you said “timing”, good for you. White’s extra timing dominates Black’s extra action, and in fact Black’s winning chances drop by almost 7% in going from 142 to 142a. Fundamentally, this is a prime versus prime position. The action element is secondary, although if Black’s advantage is big enough, the action element will push Black to double now rather than later, as a bigger than usual swing could happen on the next roll.


Now let’s take a closer look at the position itself. Clearly Black has the edge. His threat is to enter and hit a checker – then White enters and breaks his prime – then Black hops out and hits another checker – then White enters and cracks some more – then the inevitable gammon. A grim scenario. Black should double and White should pass, right?

Well, let’s hold up a bit on the pass. In evaluating positions, we need to avoid a certain trap which is easy to walk into, especially (but not exclusively) in priming games. I call it the Trap of the Mainline, and it’s simply the tendency to find what appears to be the main line of a position and assume that it happens much more often than it actually does. In positions where neither side has an especially strong inner board, the game can slide off into a lot of different variations, most of which offer the defender more resources than the main line. Players who train themselves to supplement computer rollouts with the occasional manual rollout are more inoculated against this trap, but we all fall victim to it from time to time.

(If you’re a hold’em poker player, the Trap of the Mainline has an exact analogy in the fallacy of “putting your opponent on a hand”, where you assume your opponent has a single hand and you play against that hand, instead of putting him on a range of hands and making the play that works the best against that whole range.)

While the main line in Position 142 is a disaster for White, there are plenty of other variations where he does all right. In prime against prime games, I think it’s useful to look at the hard cold facts of the position as it now stands: Black has three men behind White’s 5-point prime. That’s a lot of men behind a big prime. Black is only a slight favorite to hit a shot, and whether he hits it or not, he still has three checkers to extract. When White’s prime does break, it breaks from a 5-prime down to a 4-prime, but plenty of games are won by a 4-point prime.

On the bright side, Black has some timing left and White has almost none, so there will be plenty of variations where Black hits a checker and wiggles free. Those variations won’t necessarily lead to a gammon, because Black only has a 3-point board right now. But there will be plenty of games where White falls into a low-anchor game with a busted board, and those are easy for Black to win.

Conclusion: Black should double, and White has an easy take.

How much timing would make this a pass for White? Take a look at Position 142b:




Position 142b: Black on roll, cube action?

Compared to our original position, Black has seven more pips of timing. Each pip gave him about an extra half-percent in winning chances, so 142b is a strong double and just barely a pass. There’s no formula that I’m aware of that “solves” these priming games, but notice that in 142b, Black can play at least two full rolls with his outside checkers (barring an awkward double) before he has to think about breaking his prime. That idea – how many rolls can I play without rolling a six and without breaking my prime – seems to be a useful parameter to keep in mind when judging these tricky situations.


Solution: Black should double, and White should take.

Sigh missed this as a problem but a fascinating solution exceptionally well told. Thanks as ever Mr Robertie.