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.

This position shows the tail end of a long prime versus prime struggle. It was an unusual game in that both sides had a lot of men back. The primes have advanced about as far as they can, and now each side is starting to face cracking danger.

White’s 3-1 roll presents him with a choice: he can play safe for now, with either 6/3 4/3 or 6/3 5/4, or he can slot the 2-point with 6/2 or 5/2 4/3, leaving a blot but giving himself a much better chance to still have a 4-point prime (or even a 5-point prime) next turn. What’s the right plan?

We’ll attack the problem in three parts. First we’ll compare a slotting play with a safe play to see which plan looks best. Then we’ll pick the best play of that type. Finally, we’ll look at some more typical priming games to see whether we’ve latched onto a general concept, or whether the tactics are too specific to be generalizable.

PART 1: Slot play versus safe play.
We’ll actually choose 6/2 as our sample slotting play and 6/3 4/3 as our sample safe play. (Actually 6/3 4/3 clearly dominates the other safe play, 6/3 5/4. The first play leaves 3-1 and 2-1 as good non-double shots which keep White’s board, while the second play has only 2-1 as a good non-double shot.)

We’ll then break Black’s 36 replies into categories and see how each category does against the slot and the safe play, respectively. The equities I’m quoting here will all come from XG’s evaluations. (Using rollout results take a lot more time and yield only trivially different numbers.)

Group 1: Black fans (16 rolls)
White’s equity after slotting = + 0.014
White’s equity after playing safe = -0.277
Equity gain from slotting = 0.291*16 = +4.656 points.
This is by far the largest group and yields a huge profit from slotting, which shouldn’t be a surprise. If you knew Black was fanning, wouldn’t you want to slot?

Group 2: Black enters but doesn’t hit, escape, or crack (2 rolls: 1-5 and 5-1)
White’s equity after slotting = +0.176
White’s equity after playing safe = -0.105
Equity gain from slotting = 0.281 * 2 = +0.562 points.
Just two rolls in this group, but they’re almost as favorable as the fanning numbers.

Group 3: Black enters without hitting and cracks (4 rolls: 1-3 and 1-4)
White’s equity after slotting = +1.00 (double/pass)
White’s equity after playing safe = +0.750 (average of 1-3 equity and 1-4 equity)
Equity gain from slotting = 0.250 * 4 = +1.00 points.
White can cash after slotting for another solid equity gain.

Group 4: Black enters and escapes without hitting (2 rolls: 1-6)
White’s equity after slotting = -0.642
White’s equity after playing safe = -0.832
Equity gain from slotting = 0.190 * 2 = +0.38 points.
White’s in bad shape either way, but slotting gives him a chance at a 5-point prime.

Group 5: Black enters, (hits), and cracks (5 rolls: 2-2, 2-3, 2-4)
White’s equity after slotting and 2-2 = +0.545
White’s equity after playing safe and 2-2 = +0.644
White’s equity after slotting and 2-3 = +0.547
White’s equity after playing safe and 2-3 = +0.694
White’s equity after slotting and 2-4 = +0.584
White’s equity after playing safe and 2-4 = +0.767
Equity loss from slotting = (-0.099*1) + (-0.147*2) + (-0.183*2) = -0.759
White’s doing well either way, but he’s doing better if he’s not on the bar.

Group 6: Black enters (hitting), and escapes (4 rolls: 2-5 and 2-6)
White’s equity after slotting/2-5 = -0.993
White’s equity after playing safe/2-5 = -0.832
White’s equity after slotting/2-6 = -1.146
White’s equity after playing safe/2-6 = -0.892
Equity loss from slotting = (-0.161*2) + (-0.254*2) = -0.830
White’s crushed either way, but getting hit is somewhat worse.

Group 7: Black enters, (hits), and makes the 2-point (3 rolls: 1-1 and 1-2)
White’s equity after slotting/1-1 = -0.626
White’s equity after playing safe/1-1 = -0.678
White’s equity after slotting/1-2 = -0.575
White’s equity after playing safe/1-2 = -0.514
Equity loss from slotting = (+0.052*1) + (-0.061*2) = -0.070
After Black plays 1-1 by Bar/24 24/23(3), White’s actually better off on the bar because he won’t have to play many of his fives.

Taking all this data into account, we get a pretty clear picture. Slotting leaves White better off after the 24 rolls in Groups 1-4, plus 1-1 in Group 7. Playing safe leaves White better off after the 9 rolls in Groups 5 and 6, plus 1-2 in Group 7. To compensate, Black has to do much better than White after his good sequences, but he doesn’t. In the rolls in groups 1-4, White is gaining between 0.2 and 0.3 point per roll, while he’s losing less than that with the bad rolls in Groups 5 and 6.

This result makes complete sense. If Black throws 2-5 or 2-6 next, White was doing poorly in any event, so it doesn’t cost him that much to get hit. When Black throws a bad number like a dance or a cracking number, White’s game becomes much stronger if he has a threat to make a five-point board. Since these sequences are much more common than Black’s good numbers, the slot easily dominates the safe play.

PART 2: Which slot?
Once White decides to slot, he has to decide how to arrange his spares. Should he play 6/2, leaving spares on the 5-point and 4-point, or 5/2 4/3, leaving spares on the 6-point and 3-point?

We can solve this by counting rolls after Black fans, keeping in mind what we’re trying to do: maximize our chance of making a 5-prime, and minimizing our chance of leaving a blot. Let’s look at the chance of making a 5-prime first.

Rolls that make a 5-prime after 6/2: 11, 63, 62, 53, 52, 32, 31, and 21. (15 rolls)

Rolls that make a 5-prime after 5/2 4/3: 11, 64, 61, 54, 51, 41, 31, and 21 (15 rolls)

No difference! Both play give us about a 40% chance of making a 5-prime, assuming we’re not hit. Now let’s count the rolls that leave a shot.

Rolls that blot after 6/2: 66, 55, 33, 65, 64, 61, 54, and 51. (13 rolls).

Rolls that blot after 5/2 4/3: 66, 55, 33, 65, 63, 62, 53, 52, 43, and 42. (17 rolls)

The Solution: So 6/2 is much less likely to leave a blot. That’s our play. (We could also have just noticed that the position after 6/2 looks prettier than the position after 5/2 4/3. In backgammon, that’s usually a quick guide to the right play. But not always.)