Problem of the Week #103: Solution


Cash game, Black owns the cube. Black on roll.




Black to play 3-1.



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 103 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.

Black’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 this problem in three parts. First we’ll compare a slotting play with a safe play, to see which plan is 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 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 Black’s board, while the second play has only 2-1 as a good non-double shot.)

We’ll then break White’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 Snowie’s 3-ply evaluation. (Using rollout results or using XG as the evaluation source yield only trivially different numbers.)


Group 1 – White fans (16 rolls)

Black’s equity after slotting = + 0.014
Black’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 White was fanning, wouldn’t you want to slot?


Group 2 – White enters but doesn’t hit, escape, or crack (2 rolls: 1-5)

Black’s equity after slotting = + 0.176
Black’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 – White enters without hitting and cracks (4 rolls: 1-3, 1-4)

Black’s equity after slotting = + 1.000 (double/pass)
Black’s equity after playing safe = +0.750 (average of 1-3 equity and 1-4 equity)

Equity gain from slotting = 0.250 * 4 = +1.000 points.

Black can cash after slotting for another solid equity gain.


Group 4 – White enters and escapes without hitting (2 rolls: 1-6)

Black’s equity after slotting = - 0.642
Black’s equity after playing safe = - 0.832

Equity gain from slotting = 0.190 * 2 = +0.38 points.

Black’s in bad shape either way, but slotting gives him a shot at a 5-point prime.


Group 5 – White enters, (hits), and cracks (5 rolls: 2-2, 2-3, 2-4)

Black’s equity after slotting/2-2 = + 0.545
Black’s equity after playing safe/2-2 = + 0.644

Black’s equity after slotting/2-3 = + 0.547
Black’s equity after playing safe/2-3 = + 0.694

Black’s equity after slotting/2-4 = + 0.584
Black’s equity after playing safe/2-4 = + 0.767

Equity loss from slotting = (-0.099*1) + (-0.147*2) + (-0.183*2) = -0.759

Black’s doing well either way, but he’s doing even better if he’s not on the bar.


Group 6 – White enters, (hits), and escapes (4 rolls: 2-5 and 2-6)

Black’s equity after slotting/2-5 = -0.993
Black’s equity after playing safe/2-5 = -0.832

Black’s equity after slotting/2-6 = -1.146
Black’s equity after playing safe/2-6 = -0.892

Equity loss from slotting = (-0.161*2) + (-0.254*2) = -0.830

Black’s crushed either way, but getting hit is somewhat worse.


Group 7 – White enters, (hits), and makes the 2-point (3 rolls: 1-1 and 1-2)

Black’s equity after slotting/1-1 = -0.626
Black’s equity after playing safe/1-1 = -0.678

Black’s equity after slotting/1-2 = -0.575
Black’s equity after playing safe/1-2 = -0.514

Equity loss from slotting = (+0.052*1) + (-0.061*2) = -0.070

After White plays 1-1 by Bar/24 24/23(3), Black’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 Black better off after the 24 rolls in Groups 1-4, plus 1-1 in Group 7. Playing safe leaves Black better off after the 9 rolls in Groups 5 and 6, plus 1-2 in Group 7. To compensate, White has to do much better than Black after his good sequences, but he doesn’t. In the rolls in Groups 1-4, Black is gaining between 0.2 and 0.3 points per roll, while he’s losing less than that with the bad rolls in Groups 5 and 6.

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


Part II: Which slot?

Once Black 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 White fans, keeping in mind what we’re trying to do: maximize our chance of making a 5-prime, and minimize 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 plays 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)

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.)


Part III: How strong is slotting generally in prime versus prime games?

In general, slotting is a key tool in prime against prime games, one that many players tend to overlook. Take a look at this position:




Position 103a: Black to play 6-3.

Here we have a pretty symmetrical prime versus prime position with both players having two men back. The obvious play is 10/4 7/4. Black keeps a 5-prime, leaves all his builders in play, and takes no risks. I’d wager most players would make this move without thinking much and pick up their dice. But playing safe here isn’t just a mistake, it’s actually a full-fledged blunder!

Here’s the problem: once you’re working with only 13 checkers, it’s actually fairly difficult to keep rolling your prime forward smoothly. In the position after 10/4 7/4, Black can make the 3-point with combinations of 1s, 2s, 3s, and 5s, but any 4 or 6 will cause a problem. Instead, Black should slot with 10/4 6/3! Slotting is a low-risk way of keeping the prime moving. You’re unlikely to be hit, and getting hit still won’t necessarily lose the game for you.

Does duplication matter here? Note that after playing 6/3 in Position 103a, White’s deuces are partly duplicated: he needs twos to hit but many of his twos also make his 3-point. Let’s change the position a little bit and see what happens:




Position 103b: Black to play 6-3.

We’ve removed the direct duplication, but 10/4 6/3 is still the best play by a wide margin. (There is still some duplication involved, since White’s deuces have value on his side of the board – they make the 2-point instead of the 3-point.)

What if the positions aren’t so symmetrical?




Position 103c: Black to play 6-3.

Here Black has a few more pips than in 103a but White has moved 10 pips forward, giving Black a distinct timing advantage. Slotting with 10/4 6/3 is still correct, but by a much smaller margin than before.




Position 103d: Black to play 6-3.

White has moved a little further forward compared to the last position, and now he’ll have to break his bar-point with fives. Black’s overall edge is substantial, and now 10/4 6/3 is just marginally better than 10/4 8/5.

The moral here is that slotting is a powerful tool in relatively even prime against prime struggles, but becomes less necessary as one side develops a timing advantage.


Solution: 6/2