Problem of the Week #136: Solution


Money game, White owns the cube, Black on roll.




(a) Black to play 6-5, 6-3, 5-4, 5-3, 5-2, and 3-2.




(b) Slightly different position, Black to play all the same numbers as in (a).


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 136a is an interesting holding game situation. White’s has the 20-point, but now he’s got a man stuck on the bar. Black has a nice 5-point board, and a couple of blots not in direct range of White’s anchor. He also has the 9-point, which was an asset for a while, but is now a liability to be cleared. How much risk should he be willing to take while White is on the bar?

In approaching these technical positions in the later part of the game, I like to use a technique I call “bracketing” (from the old naval gunnery term). Instead of trying to solve one play for one position, I pick a bunch of non-trivial rolls, then make one or two modifications to the position, then solve all the rolls for all the positions, then go back and look for patterns. The patterns will tell you much more about the positions than could any single problem.

So how should Black plan to come home? Here we’ve got six rolls for Black, and two positions, one where White has a closed board, and one where White has a high point open with the possibility of covering it later. Let’s look at the cross-section of plays and problems and see what we can discover about our starting position.

The first big point to notice is the difference between the two White positions is pretty much an optical illusion. Rollouts showed that moving from (a) to (b) only changes the right play in the case of 6-5, and then only by a tiny margin.

Why wouldn’t a closed board make Black play safer? Three reasons:

(1) The closed board doesn’t always hold up. Suppose Black’s first roll is 5-2, played 13/6. Of White’s 11 entering numbers, 5-1 and 5-4 (4 shots) break his board by force. But on 5-2 and 5-3 (4 more shots), White should break his board voluntarily, rather than moving into the outfield and giving Black a double shot, which will win a few extra gammons when he hits. Only 5-5 and 5-6 actually keep White’s board intact.

(2) White doesn’t hit very often, so whatever difference there may be doesn’t come into play much.

(3) When White does hit, the 5-point board is mostly good enough to win anyway.

The second big point is that if Black can clear his 9-point safely, he should do so. Clearing with 6-3 and 5-3 is much better than bringing in a checker. Clearing with 6-5 is tied with making the 7-point, but that’s only because the 7-point makes clearing the 9-point much easier later.

Clearing is such a big deal that’s it’s worth doing with the 3-2 roll (9/6 9/7), despite leaving an indirect shot. (It’s a closer but still correct play if White has a closed board, because playing 13/10 12/10 makes 5-5 an awful shot for White.)

If Black can’t clear, but he can bring a blot home, then he should bring home the checker on the 13-point; with 5-2 he plays 13/6, and with 5-4 he plays 13/4. He safeties the checker on the 13-point because the checker on the 12-point can get home with more numbers.

Note that with 3-2 Black doesn’t have the option of bringing a blot home, so clearing the 9-point becomes best despite the risk.


Solutions:

6-5: (a) 9/3 9/4 (b) 13/7 12/7. Both are very close.
6-3: 9/3 9/6 in both.
5-4: 13/4 in both.
5-3: 9/6 9/4 in both.
5-2: 13/6 in both.
5-1: 9/6 9/7 in both. In (a), 13/10 12/10 is very close.