Problem of the Week #69: Solution


Cash game, White owns the cube.




Black to play 1-1.


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.


In Problem 69, Black has escaped his back checkers, while White has a single checker remaining in Black’s board, behind a 5-prime. How these positions are conducted depends on whether White has one man back or two men anchored.

If White is anchored, we name the position by the quality of the anchor: ace-point game, three-point game, and so forth. Those positions are fairly simple to conduct. Black keeps his prime, brings his outside checkers home, clears the outside points of the prime, and finally starts to bear off. Keeping White contained isn’t a big deal, because White is usually so far behind in the race that he doesn’t want to run; keeping the anchor and trying for a shot is his best game plan.

If White has only one man back, as here, the position has a different character. Now the race is often close, so keeping that one checker trapped is important. I call these positions “One-Way Primes”, to distinguish them from prime versus prime games, where both sides have checkers trapped behind primes.

Black’s plan in a one-way priming game breaks down into three stages:

(1) Build a 6-prime, to keep the checker trapped.

(2) Roll the prime home and close out White’s checker.

(3) Bear off safely.

Building a full 6-prime is Black’s first job. If White doesn’t have a prime of his own, this is a pretty straightforward process:

(1) If White is at the edge of the 5-prime, hit him with any spares on top of the prime.

(2) If White isn’t at the edge of the 5-prime, slot the prime from the back.

(3) Spread builders in the outfield to cover the rear of the prime.

For a typical example, take a look at Position 69a.




Position 69a: Black to play 3-1.

Here Black has a 5-prime with the back of the prime already slotted. White is at the front of the prime, ready to hop into the outfield, and White is also 6 pips ahead in the race (94 to 100). White’s home board is decent, but his 5-point is open, and he can’t build a prime of his own anytime soon.

White’s racing lead combined with his home board gaps means that Black should pursue a prime in the most direct way. The right play is simply 3/2* 13/10! Black hits White off the front of the prime while keeping the back of the prime slotted and giving himself a couple of cover numbers for the 8-point. If White doesn’t roll a 2, Black will complete a 6-prime with every number but 4-3 or 3-1. If White does roll a 2, Black needs to enter before White rolls a 6, and he’s a big favorite to do that.

Why is making a full 6-prime so important? Why not just play safe, keep a 5-prime, and tip-toe home?

The power of a 6-prime is twofold: it renders the race moot, since White can now never escape, and it creates a position that’s easy to roll home in more or less complete safety. Once you have a 6-prime, you just slot the front of the prime, cover the slot, then slot the new front of the prime, and repeat until you’ve rolled the prime completely home. Without a 6-prime, the process is more awkward and dangerous.

Suppose White’s board is better than in 69a? What then? Take a look at 69b:




Position 69b: Black to play 3-1.

With White having both a 5-point board and a 5-prime, the position is now too dangerous to hit loose; the right play is 8/5, 13/12. White is still an underdog to leap the prime, and Black has double shots after 6-1, 6-2, 6-3, or 6-5. Only 6-6 and 6-4 are outright winners for White. Study positions 69a and 69b carefully, because they represent the demarcation point between two separate plans of attack for Black.

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Now let’s return to our original problem. Although White’s position is the same as in 69b, Black has a tactical trick available with his 1-1 roll that’s not possible with other rolls: He can hit without remaining exposed on the edge of the prime! The right idea is 3/2*/1, 13/11. Now White’s only great shots are 1-1 and 2-6. Otherwise, hits on the 1-point leave White needing a 1 and a 6 to escape, while Black only needs to roll a 2 and then close his prime from the back. If White rolls a 2 without hitting, Black has all 5s and 3s (25 numbers) to complete a full prime.

The second-best alternative is the ugly-looking 3/2*(3) 8/7. While it gives up on the idea of making a prime, it does catch up in the race, and White may take several moves to get to the 3-point and then escape, during which time he may fall further behind. Although it looks awful, it’s surprising close to the purer play in equity.


Solution: 3/2*/1 13/11

Thanks for the really interesting problem Bill!

I'm surprised by this part, and can't help wondering if the computer rollouts were really squeezing all the equity from the position in the scenarios where White is collapsing.

I know the computers basically play perfectly now, but if there is any type of position where they still might miss something, I think this would be it.

Or, far more likely is that I'm just confused and White has a much better position than I thought.

I have been reviewing the problem of the week back from the start. I rolled this out using XG2 and got 3/2*/1* 13/12(2) as the better solution as there are more covers than 3/2*/1 13/11. Having 6's and 5's (after 13/12(2) ) gave 30 covers (51 is counted twice) but having 6's and 4's (after 13/11) gave 29 (42 and 22 are counted twice).

Maybe because the worst case scenarii are devastating for black (in the case of 3/2*/1 13/11), 13/36 times white hits and black fans a lot, and cube is at white' side: what is the equity after white rolls 2-6 or 1-1?

On the other hand 3/2*(3) 8/7 is quiet. Comparing moves with a big variance difference is not easy.

I guess you meant that having 5's and 4's (26 shots) to cover was better than 5's and 3's (25 shots).

Yes I wrote that in a hurry (too many things on my mind at present).

I have learnt so much from these problems of the week and hope that they will return in the future.

they are great!!!!!