Problem of the Week #123: October 16


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




Black to play 4-4.



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.

Don't think extra gammons trump safety here. I take one off, 6/4(2),9/5

9/1 5/1 4/off. Opening our board when the gammon is not guaranteed is not an option for me.

This play is not supersafe but it's still reasonably safe.

Yeah, when the gammons are competitive, I'm not sure how you can give him a chance of entering and getting a head start when you can play an alternative move that forces a fan and leaves no 1-turn shots and only a couple of 2-turn shots.

9/1 5/1 4/off

My first thought was "auto" 9/5 6/2(2) 4/0 but then I read mutes reply and realised I missed something! Now I like the non-opening play.

9/1 5/1 4/off

This guarantees an extra roll before white gets in, whereas any other play gives white a chance to enter quickly and race off the gammon. I don't know what the downside is.

It's between open or not to open. 9/5 6/2(2) 4/0 locks the win. The DMP play, no doubt. 9/1 5/1 4/off creates a "stripped" formation against the sniper on the roof. An ace or a deuce is then not so good to handle, but gammons are going up. 12 crossover against 14 crossover. A roll more on the bar is a lot. 1 crossover from 14 is a little bit more then 7%, plus dancing plus high doubles, let's say 8 - 9% more gammons. Do we loose more then 4% games compared to the DMP play? I can't calculate this exactly, but the stripped position has to blot at minimum 12% more. That's a lot. I don't know if i would choose the greedy play in the box or as captain, but for sure head to head.

9/1 5/1 4/off.

I agree with the gammon-seeking play. Regarding your chouette comment, I would only make the safer play if i were captain and the next guy in the order is very weak. There is too much edge in getting to be box vs a weak player vs having him as my captain.

Aggressive for the gammon or playing safe?

White needs 13 crossovers to save the gammon but will probably dance a roll or two. Black needs 16 crossovers before playing his roll. I'm discarding any move that leaves a blot, because White has a 5-prime on his side.

The candidates:

(a) 9/5 4/0 (3) : 3 checkers off
(b) 9/1 5/1 4/0 : 1 checker off
(c) 9/5 6/2 (2) 4/0: 1 checker off
(d) 9/1 6/2 (2) : 0 checker off

Between (b) and (c), (b) seems better because it keeps the 6-prime while bearing off the same number of checkers.

(d) leaves a nice structure but doesn't bear off any checker.
(a) bears off 3 checkers, but leaves a gap that could be problematic.

The gammon race is close, so I discard (d). I go with the aggressive way and hope for the best!

9/5 4/0 (3)

9/5 4/0 (3). The problem with this play is, that it isn't really aggressive, because after taking 3 men out, it slows the bear off considerably down. The gap on four isn't self-filling and when the sniper is on the roof, safety is a major issue. But 13 crossover are right, so i would estimate 45% for the safe play and 53% gammons for the stripped play. 9/5 4/0 (3) will win less games and less gammons.

9/1 5/1 4 off

And I am almost positive this play has a much higher gammon % than 9/5 4/off(3) and a significantly higher gammon % than 9/5 6/2(2) 4/off (this one is "safest" though)

At least I'm pretty sure that 9/5 6/2 (2) 4/0 isn't the winner, but once again I could be wrong!

As things stand, White needs 12 crossovers before he can bear off. But White is not favored to enter on the first try. About half the time, White will fan on his first two tries. When he does, Black should win quite a few gammons. Depending how Black plays this 44, he will need only 14 or 15 crossovers after this turn. Only when White enters on the first or second try, will the gammon will be close.

There are two approaches to slowing White down, and thereby winning more gammons:

1. Rip 3 checkers now, leaving a gap on the 4pt. White will enter immediately 30% of the time.
2. Keep the full prime, ripping only 1 checker. White will be closed out.

My solution: 9/1, 5/1, 4/off

For the Record
I am so often wrong that I like to post my record in these messages. It's kind of a truth-in-advertising thing. Grunch: I have been answering these problems without the use of a bot, and before checking the excellent solutions of others, since Problem 28. My record at this writing is 50%.

Keep your board closed and take one off.

Ultimately, the 2-to-1 gammon price determines whether to risk a "won" game in order to convert a single win into a gammon. In practice, however, the tough choices come when the chance of winning a gammon is in the broad middle range, say 25% to 75% or even 30% to 70%. When the gammon is a lock, for instance, 90%, it is foolish to take any voluntary risks. Conversely, when the gammon is an impossibility, say 10%, then there is also no advantage to taking a voluntary risk. Usually it is in the middle, when a small risk can add 5% or 10% to the gammon win rate, does it make sense to risk losing a "won" game.

I am curious then, if you do not think this position merits taking a risk, why is that? Is the chance of winning a gammon too low? Or is it too high? What ballpark figure do you put on the odds?

First of all what is the crossovergain from 9/5 4o(3)?

You get 3 off instead of 1 after 9/5 5/1(2) 4o
but since it gives white 11/36 x 2 crossovers
the real gain is about 1,4 crossovers.

How much gammons is this worth.
I know one man closed out is 5% gammon
2 men closed out is 45% gammon

so 4 crossovers = 40% gammons
So i guess the difference is about 14% gammons.

Now the risk, is it above or below 7%?

First we have (62) with a hit risk of 11/36 x 2/36 = below 2%
Then when white doesn't enter on the 6 (27/36 !!) we have
(61,51) here the hitchance is 27/36 x 11/36 x 4/36 = below 3%

Of course there will be some plusses (a quick black double throw)
and minuses (61 followed by a high double throw)
and another plus: black will get probably 5 or more men off

but it seems to me all in all that the risk will be below 5%

So my solution 9/5 4o(3)

One thing you're ignoring is what higonefive mentioned: After gapping you are gonna be bearing of slower (on average) because you have to mind the gap.

I don't think this math works. A lot of the extra gammons you win is because of your opponent failing to enter the second checker in time, not just the crossovers. Try setting up a position with one on the bar against a closed board and then two or three checkers in the outfield needing four crossovers to get home. That will give a better picture.

Hi mute,

thanks for your reactions.

I have thought about your remarks.
The first one seemed logical when i read it,
but then I have looked at several moves.
I didn't find a lot of moves where bearing off was slower because of the gap.

The only moves with a difference between a) 9/1 5/1 4o and b) 9/5 4o(3)
are: 52 (quicker on b) 51 (quicker on a, but creating a gap) 32 (quicker on b).

So b) seems to be quicker.

Now your second remark,

I have already taken into account that variation b) makes the opponent enter quicker, that's why i didn't count 2 crossovers extra for variation b) but only 1,4 crossovers. That's why i count 14% gammons instead of 20% gammons.

greetings k.

You're neglecting that you always get a free additional roll in the first case. You can always use it to rip at least one off (if you want to), and sometimes more, so there's pretty much no tempo advantage and you just have a hole in your board by force.

Hi,

let's compare:
a) 9/1 5/1 4o
b) 9/5 4o(3)

How do the next moves play?

Moves containing a 6: a) and b) both 1 checker off
D6: both 4 off
Moves containing a 5: (54,53,52) a) and b) both 1 checker off
(51) only a) one checker off
D5: both 4 off
Moves containing a 4: (43,42,41) only a) bears one checker off
But: The pip wastage on the 1-point that was compensated by the wastage of the hole on the 4-point is now coming to live in variation a).
D4: only a) bears 2 off
Moves containing a 3 or 2: (32,31,21) both could bear a checker off.
D3, D2, D1 no difference.

The difference lies in the moves (51,43,42,41,D4), they count for a total of 10/36 crossovers = 0,3 crossovers = 3% gammons.

This taken into account the difference will be 11% gammons instead of 14 %, so the risk should be below 5,5%.

So it's becoming closer, but the risk still pays off, since it's below 5%.

I'm completely unconvinced of the estimations of gammon based on crossovers. Is this a standard sort of thing that's backed up by either simulations or mathematics?

Also, the gap is a long-term problem. You can already see that in this roll it's slowing you down. Not much will change in the next roll, or probably for another two or three more rolls. This is a compounding problem.

You also introduce some long-term risks. For example 62 will leave a blot if you pull 4 off. And if you avoid that one, later on a 61/51 will possibly leave a blot. Until you get rid of that gap, you're going to have extra types of positions in which you can lose huge chunks of equity.

Hi Aaron,

thanks for your remarks.

On your first remark i have a rather good answer.
The gammon estimates are based on the following:

In Paul Lamford's book "100 backgammon puzzles"
there is a small table of chances of being gammoned in certain positions.
One man closed out and your other men in your board (5%)
Two men closed out and your other men in your board (45%)
Three men closed out and your other men in your board (90%)

Since the main difference between one and two men closed out is the extra 4 crossovers,
i estimated the gammonchance of 1 crossover equal to 10% gammons.

Now for your second remark:

you are right there will be a compound effect, but will it be big?
Most effects on the next turns tend to be fractions of those of this turn.
For example take the 62 move,
you're right but the fraction in which white enters with a 4 has already been taking into account
the fraction in which white enters with a 6 is not relevant.
After that only the D3 to D6 give a problem.
So the problem-fraction one move later is only 16/36 x 4/36 = about 5% of the original risk.
So i don't think later problems will raise the original risk of below 5% with more than let's take it high (10% of 5% = 0,5%),
making the risk still below 5,5%.

You're last remark about "huge chunks of equity" is therefore not right, i think, an extra risk of 0,5% will cost
0,5% x 3 (the difference in equity between winning a gammon and losing) = 0,015 equity.

But what about the longterm effect of the wasted checkers (pips) on the ace-point ?
And what about the pips that white will waste by staying on the 4-gap after entering with for example 45 or 46?
the other checkers in the outfield will have a wasty crossover,
and with a D1,D2,12,13,14 black can hit this checker off the 4-gap and further increase his gammonchance.

So conluding there will be some longterm plusses and minuses in both ! plays, but they will only be fractional.

I still think that the risky play pays off.

This is the part that makes me skeptical. I have my doubts that this is a reasonable way to estimate things. You win a ton of gammons with two men closed out because of the amount of time it takes for your opponent to get both checkers in, which isn't really related to a crossover count.

The risk is not of him coming in on another point. The risk is you leaving a blot that gets hit, which swings you from being a favorite in the high 90s% with a large number of gammons to being something like a 50-50 coinflip (or worse) with no gammon changes. That's a huge swing, and even if it's rare, it's large enough to be a consideration.

Edit: He has the cube, which means he will possibly be able to cube you out of your scrambling chances if he hits you.

Wastage is a small issue relative to the risks of being hit. In this case, you're talking about wasting perhaps 7-10 pips (bearing off two extra checkers from the ace point instead of from a more ideal position). That's one roll. This one roll can be made up by not opening your board and forcing white to waste an entire roll.

This reminds me of a friend telling me that duplication isn't a consideration that's large enough to trump any other decision. It's something that can be used when comparing two rolls that are basically the same in all other respects. Wastage in bearoff is basically irrelevant relative to making sure you don't open up long term liabilities because white's outfield control is a strong as it is.

I think you're still doing something wrong. You need to be comparing the position after 9/1 5/1(2) 4/off PLUS ANOTHER ROLL to the position after 9/5 4/off (3). So, let's say you only rip one on 6s and doubles and play everything else 6/x 6/y. He has one entering point, you have 1 man off, and, neglecting doubles, you get off in 7 rolls most of the time and 8 almost every time. You also can't leave a blot in 1 roll (playing off the 6) and it takes effort to leave one in two rolls. And once he enters, you can play efficiently.

Now, compare that position to ripping 3 immediately. He has one entering point, and while you have the theoretical chance of getting off in 6 non-double rolls, you have 2 on the 6, 3 on the 5, a gap on the 4, and the only non-double you can play safely and efficiently is 5-1, so you effectively never get off in 6. And given how badly you have to play your first roll, you're probably actually less likely to even get off in 7. And you blot with 7 numbers (66-44 62 52), and when he enters, you still have to tiptoe sometimes.

Hi Aaron,

you are right while white is on the bar, black will have some crossovers.
I really should adjust this figure.

But how?

I tried the following:

Let's say black has a closed board, and white has only outfield crossovers.
With a closed board and well spread spares (and none off) black will have between 48 and 57 pips.
Let's say 52,5 pips.

To be equal in the gammon(avoiding)race white needs 48,5 pips. With black to move.

So 48,5 pips equals 50% gammons.

Hey, that's a nice rule: Let's make it 1 pip is 1% gammon
And since a crossover is somewhat below 6 pips there is another rule (of thumb)
1 crossover = 5% gammons.

Since the risky play gains 1,4 crossovers, it gives us 7% gammons.
So now the risk of (somewhat below) 5% is too high.

I now vote for the safe side.

But you still didn't convince me on the risk side of the story.
We agree on the initial risk. But is still think the longterm risk is small.

You say:

Wastage is a small issue relative to the risks of being hit

and i agree, but what i meant was:

The wastage issue is comparable to the LONGTERM risks of being hit

BOTH issues are fractional to the main risk issue

I have already taken into account that white has the cube, since i see being hit as losing.