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Problem of the Week #105: Solution
05-12-2011
, 10:05 AM
Problem of the Week #105: Solution
Cash game, White owns the cube. Black on roll.
(a) Black to play 1-1.
(b) Black to play 1-1.
(c) Black to play 1-1.
(d) 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 105 (a) through (d), Black is bearing off against a White anchor on his 2-point. In each position, he has a decision to make: should he (a) clear a back point, minimizing his chance of ever being hit, or (b) rip off checkers, maximizing his chance of winning a gammon, or (c) look for a play which is some combination of the two?
During the 1970s and 1980s, the convention wisdom in these positions was to play safe. Players knew that you needed to win two extra gammons to compensate for any extra losses incurred, and it wasn’t thought that saving a roll in the bearoff would get you that many extra gammons. “Clear from the back and don’t ask questions!” was the operant phrase of the day.
By the 1990s, however, players were starting to ask questions about these positions. Their curiosity was caused partly by the emerging evidence of the bots, and partly because of the gradual accretion of knowledge and experience. What was thought to be an easy question was actually pretty complicated, and it turns out that there are lots of positions where bearing off checkers is right, and lots of other position where safety is right. Let’s summarize what we now know, and then take a look at what’s going on in these four positions.
Counting Pips and Crossovers
Our first job is to get a handle on the race and see just how likely it is that’s we’ll win a gammon in this position, assuming average rolls from this point on. There are two approaches here.
Method 1 is to compare Black’s pip count to White’s gammon count (the minimum number of pips White needs to throw to bear off exactly one checker). In 105a, for instance, Black’s pip count is 35 and White’s gammon count is 38. White’s count assumes that his four outside checkers get to his 6-point and he then bears a checker off his 2-point. Both of these counts contain a lot of wastage.
Method 2 (which I prefer because it’s simpler and quicker) is to just count the crossovers required for each side. Black needs 13 crossovers to bear off his 13 men, and White needs nine crossovers to bear off one man. Black’s on roll but he trails by four crossovers, which means he’s a solid underdog to win a gammon. His gammon chances are probably in the 25% to 40% range, depending on how aggressively he chooses to bear off. That may seem imprecise, but as we’ll soon see, great precision isn’t required here.
The Zones
Our next job is to realize when taking off checkers is likely to be right, and when it’s not. Here the positions tend to follow the same rules as do other “going for the gammon” situations. In general, the positions will break down into three zones:
Zone 1: The gammon is very unlikely either way, so taking extra risks can’t boost the gammon chances enough to matter. Play safe.
Zone 2: The gammon is very likely either way, and again the extra risks don’t boost your gammon chances much because they’re already so high. Play safe.
Zone 3: The sweet spot, where gammon chances are high enough but not too high, and the extra risks can buy you a significant bump in gammon chances.
Roughly speaking, Zone 1 consists of positions where the gammon chances are lower than 10% to 15%. In Zone 2, we’re looking at positions with gammon chances above about 70%. Zone 3 is everything in the middle. These are very rough numbers with plenty of exceptions, but you get the idea. Zone 3 is a big zone, and it includes lots of positions where the gammon is somewhat up in the air.
Here’s an example of a Zone 1 position:
Position 105e: Black to play 2-1.
Before rolling, Black is about 8% to win a gammon. He needs 15 crossovers to bearoff all his men, while White needs only 7 crossovers to get off the gammon. Black’s emphasis should be safety, and with a 2-1 to play the right move is clearing the 5-point with 5/4 5/3. Playing 3/off instead gains only 2.2% more gammons at the cost of 2.5% fewer wins – a particularly bad trade.
Now let’s look at a Zone 3 position:
Position 105f: Black to play 4-1.
Before rolling, Black is about 72% to win a gammon. He needs 14 crossovers to bear off all his checkers, while White needs 18 crossovers to get a man off. Once again, safety is Black’s main concern. The right play with 4-1 is just 5/1 5/4 instead of 4/off 1/off. Interestingly, playing safe is better because it wins more gammons! When Black gets hit on the 5-point, he may lose the game, but he’ll almost certainly lose his gammon chances.
Safety and Risk: Not a Clear-Cut Choice
This last example raises an important point about clearing points and bearing off checkers. Clearing points isn’t purely a safety play: it can also win gammons when you avoid getting hit and then roll well to score a gammon you would otherwise have missed. At the same time, bearing extra men off can help you win games you would otherwise have lost when you got hit. As a result of these two factors, the safe play and the risky play can have equities that are more similar than you might expect.
The Power of the Small Double
One last theoretical point to make: small doubles are especially powerful rolls when playing for a gammon. Not only do they take off up to four checkers, but they’re bearing off checkers that would otherwise come off later with fours, fives, and sixes. Small doubles guarantee minimum wastage, so bearing off checkers with these numbers should be your default choice in unclear positions.
Back to Our Original Problems
With the theory out of the way, let’s look again at our four original problems.
Problem 105a is clear-cut: bearing off four checkers is hugely correct. Black has a general safe position, he’s squarely in Zone 3, and double-aces gain a full roll compared to clearing the 5-point. Rollouts show that playing safe with 5/4(2) 1/off(2) increases winning chances by about 2% (93.5% to 95.5%), but playing 1/off(4) bumps the gammon chances by 9% (31.4% to 40.4%).
Problem 105b is more intriguing. Black has four different plays – he can take off either one, two, three, or four checkers, using the rest of his roll to make his position incrementally safer. Playing 1/off(4) leaves three points to clear plus three checkers on the 4-point. Playing 1/off(3) and 4/3 leaves him even-ended on the top two points. Playing 1/off(2) and 5/4(2) actually clears a point. Finally, playing 1/off 5/4 5/3 clears a point and gets even-ended.
The right play is still 1/off(4), but now it’s a little closer. If we compare it to 1/off(3) and 4/3, the safer play wins only 1% more games, but taking the fourth checker off gets 4.5% more gammons. Clearing the 5-point but taking two checkers off is worse still – it doesn’t win any more games than just playing 4/3 (an interesting point) but it wins 10% fewer gammons than ripping four men off! Cleaning up the position entirely with 5/3 5/4 1/off is the worst choice of all – it wins 2% more games than taking four men off, at the cost of 14% fewer gammons.
Problem 105c is the first position where we are seriously odd-ended. Now pausing to patch our position with 5/4 1/off(3) does beat 1/off(4), but just barely – 1.6% more wins, 2.4% fewer gammons. (At a match score where gammons were more important for the trailer, Black would be correct to stay odd-ended and rip off four men.) The plays that are safer than this, however, still trail by a lot.
Problem 105d is the first position where taking four men off is actually a serious mistake. Now 1/off(2) 5/3 and 1/off(3) 5/4 are essentially a dead heat, which is itself a little surprising given the awkwardness of the lone spare on the 4-point.
Conclusions
The lesson you want to take away from this problem is that when you’re in the zone, playing safe and clearing points is rarely a foregone conclusion. This is especially true when you have small doubles to play. Note that in this problem we gave White an excellent position, with a perfect board, no gaps or weaknesses, and outer checkers that could get home with aces. As you can imagine, any weakness in White’s position would make ripping checkers off even more correct.
Solutions:
Problem 105a: 1/off(4)
Problem 105b: 1/off(4)
Problem 105c: 1/off(3) 5/4
Problem 105d: 1/off(2) 5/3 or 1/off(3) 5/4
Cash game, White owns the cube. Black on roll.
(a) Black to play 1-1.
(b) Black to play 1-1.
(c) Black to play 1-1.
(d) 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 105 (a) through (d), Black is bearing off against a White anchor on his 2-point. In each position, he has a decision to make: should he (a) clear a back point, minimizing his chance of ever being hit, or (b) rip off checkers, maximizing his chance of winning a gammon, or (c) look for a play which is some combination of the two?
During the 1970s and 1980s, the convention wisdom in these positions was to play safe. Players knew that you needed to win two extra gammons to compensate for any extra losses incurred, and it wasn’t thought that saving a roll in the bearoff would get you that many extra gammons. “Clear from the back and don’t ask questions!” was the operant phrase of the day.
By the 1990s, however, players were starting to ask questions about these positions. Their curiosity was caused partly by the emerging evidence of the bots, and partly because of the gradual accretion of knowledge and experience. What was thought to be an easy question was actually pretty complicated, and it turns out that there are lots of positions where bearing off checkers is right, and lots of other position where safety is right. Let’s summarize what we now know, and then take a look at what’s going on in these four positions.
Counting Pips and Crossovers
Our first job is to get a handle on the race and see just how likely it is that’s we’ll win a gammon in this position, assuming average rolls from this point on. There are two approaches here.
Method 1 is to compare Black’s pip count to White’s gammon count (the minimum number of pips White needs to throw to bear off exactly one checker). In 105a, for instance, Black’s pip count is 35 and White’s gammon count is 38. White’s count assumes that his four outside checkers get to his 6-point and he then bears a checker off his 2-point. Both of these counts contain a lot of wastage.
Method 2 (which I prefer because it’s simpler and quicker) is to just count the crossovers required for each side. Black needs 13 crossovers to bear off his 13 men, and White needs nine crossovers to bear off one man. Black’s on roll but he trails by four crossovers, which means he’s a solid underdog to win a gammon. His gammon chances are probably in the 25% to 40% range, depending on how aggressively he chooses to bear off. That may seem imprecise, but as we’ll soon see, great precision isn’t required here.
The Zones
Our next job is to realize when taking off checkers is likely to be right, and when it’s not. Here the positions tend to follow the same rules as do other “going for the gammon” situations. In general, the positions will break down into three zones:
Zone 1: The gammon is very unlikely either way, so taking extra risks can’t boost the gammon chances enough to matter. Play safe.
Zone 2: The gammon is very likely either way, and again the extra risks don’t boost your gammon chances much because they’re already so high. Play safe.
Zone 3: The sweet spot, where gammon chances are high enough but not too high, and the extra risks can buy you a significant bump in gammon chances.
Roughly speaking, Zone 1 consists of positions where the gammon chances are lower than 10% to 15%. In Zone 2, we’re looking at positions with gammon chances above about 70%. Zone 3 is everything in the middle. These are very rough numbers with plenty of exceptions, but you get the idea. Zone 3 is a big zone, and it includes lots of positions where the gammon is somewhat up in the air.
Here’s an example of a Zone 1 position:
Position 105e: Black to play 2-1.
Before rolling, Black is about 8% to win a gammon. He needs 15 crossovers to bearoff all his men, while White needs only 7 crossovers to get off the gammon. Black’s emphasis should be safety, and with a 2-1 to play the right move is clearing the 5-point with 5/4 5/3. Playing 3/off instead gains only 2.2% more gammons at the cost of 2.5% fewer wins – a particularly bad trade.
Now let’s look at a Zone 3 position:
Position 105f: Black to play 4-1.
Before rolling, Black is about 72% to win a gammon. He needs 14 crossovers to bear off all his checkers, while White needs 18 crossovers to get a man off. Once again, safety is Black’s main concern. The right play with 4-1 is just 5/1 5/4 instead of 4/off 1/off. Interestingly, playing safe is better because it wins more gammons! When Black gets hit on the 5-point, he may lose the game, but he’ll almost certainly lose his gammon chances.
Safety and Risk: Not a Clear-Cut Choice
This last example raises an important point about clearing points and bearing off checkers. Clearing points isn’t purely a safety play: it can also win gammons when you avoid getting hit and then roll well to score a gammon you would otherwise have missed. At the same time, bearing extra men off can help you win games you would otherwise have lost when you got hit. As a result of these two factors, the safe play and the risky play can have equities that are more similar than you might expect.
The Power of the Small Double
One last theoretical point to make: small doubles are especially powerful rolls when playing for a gammon. Not only do they take off up to four checkers, but they’re bearing off checkers that would otherwise come off later with fours, fives, and sixes. Small doubles guarantee minimum wastage, so bearing off checkers with these numbers should be your default choice in unclear positions.
Back to Our Original Problems
With the theory out of the way, let’s look again at our four original problems.
Problem 105a is clear-cut: bearing off four checkers is hugely correct. Black has a general safe position, he’s squarely in Zone 3, and double-aces gain a full roll compared to clearing the 5-point. Rollouts show that playing safe with 5/4(2) 1/off(2) increases winning chances by about 2% (93.5% to 95.5%), but playing 1/off(4) bumps the gammon chances by 9% (31.4% to 40.4%).
Problem 105b is more intriguing. Black has four different plays – he can take off either one, two, three, or four checkers, using the rest of his roll to make his position incrementally safer. Playing 1/off(4) leaves three points to clear plus three checkers on the 4-point. Playing 1/off(3) and 4/3 leaves him even-ended on the top two points. Playing 1/off(2) and 5/4(2) actually clears a point. Finally, playing 1/off 5/4 5/3 clears a point and gets even-ended.
The right play is still 1/off(4), but now it’s a little closer. If we compare it to 1/off(3) and 4/3, the safer play wins only 1% more games, but taking the fourth checker off gets 4.5% more gammons. Clearing the 5-point but taking two checkers off is worse still – it doesn’t win any more games than just playing 4/3 (an interesting point) but it wins 10% fewer gammons than ripping four men off! Cleaning up the position entirely with 5/3 5/4 1/off is the worst choice of all – it wins 2% more games than taking four men off, at the cost of 14% fewer gammons.
Problem 105c is the first position where we are seriously odd-ended. Now pausing to patch our position with 5/4 1/off(3) does beat 1/off(4), but just barely – 1.6% more wins, 2.4% fewer gammons. (At a match score where gammons were more important for the trailer, Black would be correct to stay odd-ended and rip off four men.) The plays that are safer than this, however, still trail by a lot.
Problem 105d is the first position where taking four men off is actually a serious mistake. Now 1/off(2) 5/3 and 1/off(3) 5/4 are essentially a dead heat, which is itself a little surprising given the awkwardness of the lone spare on the 4-point.
Conclusions
The lesson you want to take away from this problem is that when you’re in the zone, playing safe and clearing points is rarely a foregone conclusion. This is especially true when you have small doubles to play. Note that in this problem we gave White an excellent position, with a perfect board, no gaps or weaknesses, and outer checkers that could get home with aces. As you can imagine, any weakness in White’s position would make ripping checkers off even more correct.
Solutions:
Problem 105a: 1/off(4)
Problem 105b: 1/off(4)
Problem 105c: 1/off(3) 5/4
Problem 105d: 1/off(2) 5/3 or 1/off(3) 5/4
