I'm a re-invent the wheel type of a player. When I started playing backgammon I was immediately hooked but many many things have come up over the years that I simply did not know or it was assumed I should know that I couldn't find explained anywhere. This lesson I'm going to go a lot of things, some of them you may already know especially if you're proficient in math, but I think it's definitely worth going over to make sure you do know everything you need to know. The first thing I want to tackle in today's racing lesson are some reference positions you should know. It's always better to be familiar with as many reference positions as possible and some of them you absolutely need to know. They first few are very simple, you will see they are roll versus roll positions and the bot will spit out the win percentage for you. I will then briefly show you the math and add any additional notes I deem worthy.

2 rolls v. 1 roll - immediate doubles needed to win
is Player 2

score: 0
pip: 2
Money session
Jacoby Beaver
pip: 4
score: 0

is Player 1
XGID=-D----------------------b-:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 16.67% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 83.33% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:-0.667
Double:-1.333
 
Cubeful Equities
No Double:-0.667
Double/Take:-2.667 (-2.000)
Double/Drop:+1.000 (+1.667)
 
Best Cube action: No Double / Beaver
 

eXtreme Gammon Version: 1.21

Obviously you aren't cubing here for money but it is possible that a match score situation would come up where you'd have a proper double, and a big one at that! I am putting this out that not because that's likely but to make sure you know that you get off with 6 rolls, all doublets, or 6/36 rolls. What win percentage does that yield? Hopefully you can do 6/36 which reduces to 1/6 in your head. I studied languages in college and math is far from natural for me but over time I have come to be able to easily do this and many more calculations in my head that I would have deemed *difficult* before my backgammon career. 1/6 = ~16.67% as you can see in the output. If my math ever seems to go astray feel free to check it against a calculator.

One of the first things I remember figuring out is that every roll is worth approximately 2.8% (2.77%) so if you feel more comfortable saying "I have 6 winning rolls" and then doing 6 x 2.8 you will get nearly the same answer.

That was easy enough, let's take another step. How about a pure 2 roll v. 2 roll position?

2 roll v. 2 roll
is Player 2

score: 0
pip: 4
Money session
Jacoby Beaver
pip: 4
score: 0

is Player 1
XGID=-D----------------------d-:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 86.11% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 13.89% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.722
Double:+1.444
 
Cubeful Equities
No Double:+0.722 (-0.278)
Double/Take:+1.444 (+0.444)
Double/Drop:+1.000
 
Best Cube action: Double / Drop
 

eXtreme Gammon Version: 1.21

Again, everyone would know this is a double and a pass but knowing the exact percentages or at least being able to figure out what they would be will come in handy one day over the board, I promise you. Here for your opponent to win two things need to happen.

  1. You need to not throw doubles
  2. He needs to throw doubles

This is known as a compound event and to get the probability of a compound event happening we need to multiply the probabilities of the components together. The odds of you not throwing doubles is 30/36 rolls which reduces to 5/6 and the odds of your opponent throwing doubles is 6/36 or 1/6. So we would take 5/6 x 1/6 which gives us 5/36. This transforms into 13.89% as the bot tells you. If you get to 5/36ths and think "Well I can't do that in my head!?" I will tell you two things. First, yes, you can. Secondly, even if you can't initially, you can fall back on the trick of each roll being worth 2.8% and take 5 times 2.8% to get your winning percentage which will work out to (close enough) 14%.

Let's push on until the math fries our brain or until we have a money take, 3 roll v. 3 roll.

3 roll v. 3 roll
is Player 2

score: 0
pip: 6
Money session
Jacoby Beaver
pip: 6
score: 0

is Player 1
XGID=-F----------------------f-:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 78.78% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 21.22% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.576
Double:+1.151
 
Cubeful Equities
No Double:+0.766 (-0.234)
Double/Take:+1.148 (+0.148)
Double/Drop:+1.000
 
Best Cube action: Double / Drop
 

eXtreme Gammon Version: 1.21

If I remember correctly I owe the fact that I can mathematically figure out a 3 roll position to Walter Trice and his excellent book Backgammon Boot Camp. The odds of rolling doubles over the course of two turns is 11/36, the same probability as hitting a direct shot coincidentally enough. However, for you to win as the person not on roll initially in a 3 roll v. 3 roll position not only do you have to roll a set, but the opponent can't roll a set on either of his turns either. This breaks down to the two turns of your opponent not rolling a set, 5/6 and 5/6, times the odds of you rolling a set, 11/36. Better expressed as (5/6) x (5/6) x (11/36) which equals 275/1296.

Let me stop a second and explain a couple things before you think I'm crazy when I start calculating these big numbers. There are certain numbers in the backgammon world that are known. You ask any good player what 36 x 36 is and there will be no hesitation in answering 1296, no calculations, nothing. The other important part of this equation comes from what the numerator needs to be for you to take in a position like this where you have no recube vig. In a 3 roll v. 3 roll position you never get any use out of owning the cube so you need the full money dead cube take point of 25% if you're going to take. What is 25% of 36 games? 9 games, easy enough. If we needed to know how many games out of 1296 we needed to win we could simply divide that by 4 and our answer would be 324 games. So for us to be able to take in this example we would need to win 324/1296 games. We just showed that we only win 275/1296 games so this is a routine pass. Forging ahead -

4 roll v. 4 roll
is Player 2

score: 0
pip: 8
Money session
Jacoby Beaver
pip: 8
score: 0

is Player 1
XGID=-H----------------------h-:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 74.53% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 25.47% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.491
Double:+0.981
 
Cubeful Equities
No Double:+0.761 (-0.158)
Double/Take:+0.919
Double/Drop:+1.000 (+0.081)
 
Best Cube action: Double / Take
 

eXtreme Gammon Version: 1.21

I am going to spare you and myself the pain of trying to show you the math to this one. Memorize it since it falls under the take for money umbrella and will spawn off into other useful bearoff reference positions. Not only do we win just above the needed dead cube take point of 25% here finally, but we also have the possibility of getting use out of the cube.

4 roll v. 4 roll should also be noted as the bearoff reference position of when to redouble. You never have a proper redouble and take in a pure N v. N roll position except for 4 roll v. 4 roll. Notice I didn't say initial double, redouble. You can see the common sense behind this if you stop and think about it for a second. At 4 rolls versus 4 rolls you're on the verge of losing your market. Any sequence where your opponent doesn't roll doubles (and the rare ones where you both roll doubles) and you've lost your market, albeit not by an overwhelming amount. It only makes sense for you to close in on your market as much as possible before recubing. If you recube at any point earlier and things go wrong you lose a lot more equity wise than you would have gained on the times where you did lose your market.

Here is a chart rounded to the nearest full percent of your cubeless winning chances in any pure roll v. roll situation.

# of Rolls Cubeless win %
2
86%
3
79%
4
75%
5
72%
6
70%
7
68%
8
67%

Now that you have accrued a certain amount of knowledge, let's put you to the test.
is Player 2

score: 0
pip: 15
Money session
Jacoby Beaver
pip: 15
score: 0

is Player 1
XGID=-CCB------------------bcc-:1:1:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

The take is what is in question here. The problem looks harmless enough on the surface doesn't it? Standard 4 roll v. 4 roll position ... but wait. Did you notice that neither double aces nor double twos saves either player a roll here? If you did, bravo. If you also didn't think that the fact it saves neither person a roll cancels out double bravo with some whip cream on top. As we showed above, a pure 4 roll v. 4 roll position is a take with you winning 74.5% of the time. The problem here is that those missing doubles hurt you and not your opponent. He doesn't need to throw doubles to catch up, you do. It was a close decision in a pure 4 roll position and this minor change is enough to push it back into pass territory. Analysis below.

Analyzed in XG Roller+
Player Winning Chances: 76.99% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 23.01% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Redouble:+0.540
Redouble:+1.080
 
Cubeful Equities
No Redouble:+0.847 (-0.153)
Redouble/Take:+1.037 (+0.037)
Redouble/Drop:+1.000
 
Best Cube action: Redouble / Drop
 

eXtreme Gammon Version: 1.21

Allow me to bore you with a few 2 checker v. 2 checker (and usually thus 2 roll v. 2 roll) positions before moving on. I want to make sure you can do the math and let me tell you, you'd be surprised how profitable things can turn out to be. To prove it to you I will first show you a 3 checker v. 3 checker position that I played out as a prop against one of the top players in the world and won $5,000 on.

is Player 2

score: 0
pip: 12
Money session
Jacoby Beaver
pip: 12
score: 0

is Player 1
XGID=---AAA--------------aaa---:0:0:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in XG Roller+
Player Winning Chances: 76.01% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 23.99% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.520
Double:+1.040
 
Cubeful Equities
No Double:+0.624 (-0.362)
Double/Take:+0.987
Double/Drop:+1.000 (+0.013)
 
Best Cube action: Double / Take
 

eXtreme Gammon Version: 1.21

That's it, simple. We played it out as being worth +.96 when it's actually worth closer to +.99. How this worked was for every 100 games we played out I would pay my opponent 96 points to take. Since it is worth nearly .03 more than that, that was my edge. We probably played 500 games or so and I was pretty lucky in the short term but none of that really matters. I had a simple bearoff prop that over time I couldn't lose. There is no play in this position, there are no real cube decisions, you roll the dice and hope you roll well. I would have never quit this prop but my opponent having not only the bad end of the prop but the bad end of the dice on top of that eventually realized maybe he didn't have the best of it and quit. Still, it shows you how useful knowledge of 'boring' positions' can be.

Back to work. Below I've given you a position, how would you figure out the win percentage for either player?
is Player 2

score: 0
pip: 6
Money session
Jacoby Beaver
pip: 6
score: 0

is Player 1
XGID=---B------------------b---:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

I didn't stipulate but hopefully you figured out on your own that the only win percentage there is is cubeless. If the cube is taken the opponent will never be a favorite in this position so it's a dead cube. A dead cube means we need to win 1/4th of the time or 324 games out of 1296, can we do that?

What needs to happen again for us to win is our opponent not to bear off in one roll and then we bear off on our turn. Our opponent does not bear off immediately with 19 rolls meaning he wins immediately with 17 rolls. We will obviously win with the same number of rolls so 19/36 x 17/36 will give us the magic take or drop answer. Come on big bucks, no whammy.....stop! Just short, 19x17 is 323/1296, 1 game short of what's needed to take the position. As you can see in the analysis below by taking you would only cost yourself .003 in equity. In other words if you're playing for $10 a point it will cost you 3 fat cents if you take.

Analyzed in XG Roller+
Player Winning Chances: 75.08% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 24.92% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.502
Double:+1.003
 
Cubeful Equities
No Double:+0.502 (-0.498)
Double/Take:+1.003 (+0.003)
Double/Drop:+1.000
 
Best Cube action: Double / Drop
 

eXtreme Gammon Version: 1.21

Shifting the above position by a pip for each player what does it change if anything?
is Player 2

score: 0
pip: 7
Money session
Jacoby Beaver
pip: 7
score: 0

is Player 1
XGID=---AA----------------aa---:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

If you go through the math again it will work out the same. The player on roll doesn't get off with 19/36 rolls and then you yourself will get off in 17/36 rolls or 323 games it appears, so where's the beef? The schtick of it that comes in to play is now White isn't guaranteed to be off in 2 rolls, he could throw back to back 21s! This minor alteration is enough to justify an actual take since it will swing about 2 more games in your direction.

Analyzed in XG Roller+
Player Winning Chances: 74.92% (G: 0.00% B: 0.00%)
Opponent Winning Chances: 25.08% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.498
Double:+0.997
 
Cubeful Equities
No Double:+0.502 (-0.495)
Double/Take:+0.997
Double/Drop:+1.000 (+0.003)
 
Best Cube action: Double / Take
 

eXtreme Gammon Version: 1.21

I will now skip ahead to the other end of the racing spectrum, medium to long races with no contact left and relatively low wastage positions. If you don't know what low wastage is fret not, we will get to that soon enough. There have been many racing counts devised to try to help backgammon players figure out when they can cube in a race, when they can recube in a race, when they can take or drop, and even go as far as giving you your exact win percentage which becomes very useful in match play where your doubling points and take points can fluctuate wildly from money. The count that I use for the majority of races is called the Keith Count named after its originator and maintainer of the resourceful online site bkgm.com Tom Keith.

I will be borrowing some of the terrific graphs from Tom's article to include here. I have found this count is one of the top 2 in terms of ease of use and accuracy. First a comparison of the Keith Count to other known counting methods.

Chart of Cube Errors

The following chart shows the total errors by each cube-handling method for every position in the database. The three possible types of errors (doubling errors, redoubling errors, and take/pass errors) are shown separately.

Cube Method Doubling Errors Redoubling Errors Take/Pass Errors
8-9-12 Rule 2063.64 2380.06 7493.83
Lamford/Gasquoine 794.13 840.84 1475.70
Thorp 752.82 808.13 2097.53
Keeler/Gillogly 490.17 593.30 1466.11
Ward 435.98 526.23 1202.67
Keith 348.66 353.63 633.89


As you can see the Keith Count fairs far better than any of the other counts listed here. I hope that's enough to convince you so let's dive right in. Here is how the it works.

Summary of the Keith Count

For each player, start with the basic pip count and:

  • add 2 pips for each checker more than 1 on the one point;
  • add 1 pip for each checker more than 1 on the two point;
  • add 1 pip for each checker more than 3 on the three point;
  • add 1 pip for each empty space on points four, five, and six.

Increase the count of the player on roll by one-seventh (rounding down).

  • A player should double if his count exceeds the opponent's count by no more than 4.
  • A player should redouble if his count exceeds the opponent's count by no more than 3.
  • The opponent should take if the doubler's count exceeds the opponent's count by at least 2.

This is easier in application than in text so let's have a few examples.

is Player 2

score: 0
pip: 76
Money session
Jacoby Beaver
pip: 70
score: 0

is Player 1
XGID=-AABCCCA-A---a--a--cdbbaa-:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

So as White on roll here should we double? If we do double should our opponent take? Using the Keith Count and going through the steps we get:

What if the question was to redouble or not, then what? You don't need to work through everything again, you take your numbers, 80 and 76, and compare them to the line that states "A player should redouble if his count exceeds the opponent's count by no more than 3.". 80-76 is still more than 3 in my book, so we hold off on the recube. A rollout confirms this is a bare no redouble but a clear center cube. The center cube rollout is below.

Analyzed in Rollout
No Double
  Player Winning Chances: 73.04% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 26.96% (G: 0.00% B: 0.00%)
Double/Take
  Player Winning Chances: 73.02% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 26.98% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Double:+0.461
Double:+0.921
 
Cubeful Equities
No Double:+0.715 (-0.037)
Double/Take:+0.752
Double/Drop:+1.000 (+0.248)
 
Best Cube action: Double / Take
 
Rollout details
648 Games rolled with Variance Reduction.
Dice Seed: 24983242
Moves and cube decisions: 3 ply
 
Confidence No Double: ± 0.004 (+0.711...+0.719)
Confidence Double: ± 0.005 (+0.747...+0.757)
 
Double Decision confidence: 100.0%
Take Decision confidence: 100.0%
 
Duration: 1 minute 02 seconds

eXtreme Gammon Version: 1.21

Let's see how the Keith Count works with some adjustments.
is Player 2

score: 0
pip: 79
Money session
Jacoby Beaver
pip: 70
score: 0

is Player 1
XGID=-C--CDC-AA-----aa-adcc--b-:1:1:1:00:0:0:3:0:10
on roll, cube action?


Analyzed in Rollout
No Double
  Player Winning Chances: 77.04% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 22.96% (G: 0.00% B: 0.00%)
Double/Take
  Player Winning Chances: 77.02% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 22.98% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Redouble:+0.541
Redouble:+1.081
 
Cubeful Equities
No Redouble:+0.847 (-0.099)
Redouble/Take:+0.946
Redouble/Drop:+1.000 (+0.054)
 
Best Cube action: Redouble / Take
 
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 24983242
Moves and cube decisions: 3 ply
 
Confidence No Redouble: ± 0.003 (+0.844...+0.849)
Confidence Redouble: ± 0.004 (+0.942...+0.949)
 
Double Decision confidence: 100.0%
Take Decision confidence: 100.0%
 
Duration: 1 minute 46 seconds

eXtreme Gammon Version: 1.21

Starting pip counts of 70 for the player considering the recube and 79 for his opponent. The player on roll has two extra checkers on the one point so you add 2 pips for each extra checker to the original count, or 70+2+2=74. After that there is no other checker adjustments so we divide by seven rounding down, 74/7 = 10 and add that to our adjusted count, 74+10=84. For our opponent he has 79 to start with and also has an extra checker on his one point for which we need to adjust his count by two pips, 79+2=81. We now compare our counts, 84 to 81 and go to the rules of Keith Count cubing. It's a recube we're considering here so to redouble our count needs to exceed our opponent's count by no more than 3. 84 to 81, ship it! Since our count exceeds our opponent's by at least 2 he also has a take.

One more Keith Count position with more adjustments then we'll move on. You can create racing position after racing position in your own time to practice. If it seems hard now trust me once you get down what you're supposed to be doing it's like a reflex.

is Player 2

score: 0
pip: 59
Money session
Jacoby Beaver
pip: 52
score: 0

is Player 1
XGID=-BDC-CC------------ddabac-:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

We are considered a center cube so we take our leader's original pip count of 52 and add 2 pips for the extra checker on the ace point, 1 pip each for each of the 3 extra checkers on the two point (1x3=3), and don't forget to add another pip for the gapped four point. That gives us 52+2+3+1 for a total of 58. We now divide by 7 rounding down and add. 58/7=8. 58+8=66 for our leader's adjusted count. The opponent has 59 plus 4 for the two extra checkers on the ace point and no other deductions. 59+4=63. We are comparing 66 to 63 and we know that's both a (re)double and a take

The truth is that every racing count has its strengths, its weaknesses, and will be better suited for this position or that position. You can never know too much. Most of the time I use Keith straight away but you can hardly be blamed for having access to too much knowledge and knowing the value of a pip can come in handy. Again a chart I borrowed from Tom's article -

Keith Count ~80 to ~100 pips can double when leading by 5
is Player 2

score: 0
pip: 102
Unlimited Game
Jacoby Beaver
pip: 94
score: 0

is Player 1
XGID=---ABCCBABA---aabbaccaa---:0:0:1:00:0:0:3:0:10
on roll, cube action?

Keith Count ~100 pips to 120pips can double when leading by 6
is Player 2

score: 0
pip: 108
Unlimited Game
Jacoby Beaver
pip: 100
score: 0

is Player 1
XGID=---ABBCBABAA--babbacbaa---:0:0:1:00:0:0:3:0:10
on roll, cube action?

Keith Count 120 pips + can double when leading by 7
is Player 2

score: 0
pip: 130
Unlimited Game
Jacoby Beaver
pip: 120
score: 0

is Player 1
XGID=----BBCAAAAAdCa-bbabba----:0:0:1:00:0:0:3:0:10
on roll, cube action?

Value of a Pip

When the race is even (in terms of pip count), the player whose turn it is can expect to win about 60% of the time. This advantage is because being "on- roll" is equal to about 4 pips, and each extra pip is worth about 2.5% CPW in a close race.

Cubeless Probability
of Winning		 Each Pip
is Worth
10% to 20%about 1.5%
20% to 30%about 2%
30% to 70%about 2.5%
70% to 80%about 2%
80% to 90%about 1.5%

When might something like this come in handy? Well, if you know your match scores you know that at some scores your window opens much sooner than it does for money and your opponent should pass much sooner than for money also. A good example is 4 away 3 away where you are trailing and are holding a 2 cube. In this situation your opponent's take point is a whopping 40%! That means according to the value of a pip chart above if you're on roll with an equal pip count you'd probably better send that cube over in a hurry. An example:
is Player 2

score: 2
pip: 76
5 point match
pip: 76
score: 1

is Player 1
XGID=--AADDC-AA------aa-cddaa--:1:1:1:00:1:2:0:5:10
on roll, cube action?


Analyzed in Rollout
No Double
  Player Winning Chances: 59.70% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 40.30% (G: 0.00% B: 0.00%)
Double/Take
  Player Winning Chances: 59.70% (G: 0.00% B: 0.00%)
  Opponent Winning Chances: 40.30% (G: 0.00% B: 0.00%)
 
Cubeless Equities
No Redouble:+0.194
Redouble:+0.990
 
Cubeful Equities
No Redouble:+0.791 (-0.199)
Redouble/Take:+0.990
Redouble/Drop:+1.000 (+0.010)
 
Best Cube action: Redouble / Take
 
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 24983242
Moves and cube decisions: 3 ply
 
Confidence No Redouble: ± 0.001 (+0.791...+0.792)
Confidence Redouble: ± 0.000 (+0.990...+0.991)
 
Double Decision confidence: 100.0%
Take Decision confidence: 100.0%
 
Duration: 16.6 seconds

eXtreme Gammon Version: 1.21, MET: Rockwell-Kazaross

As you can see, with equal pips counts in a race to 76 and being on roll being worth about 4 pips you are the 60/40 favorite the chart predicted you'd be. This would be a monster recube at the score and a narrow technical take. If you only knew the Keith Count and nothing else you would truly be lost if such a position came up. Every lit bit of information helps.

We have now covered roll v. roll positions and pip v. pip positions, what about when we get those funky positions when one side has a 'roll' position and the other side has a 'pip' position? To try to tackle such positions we need to understand what the effective pip count or epc is and how to apply it. This has also become known as the Trice Count named after Walter Trice who did so much work in this area. Imagine you have all 15 of your checkers stacked up on the ace point, a pure roll situation. Your actual pip count is only 15 but this doesn't tell us much really does it? To see what I mean, take this position:
is Player 2

score: 0
pip: 18
Money session
Jacoby Beaver
pip: 15
score: 0

is Player 1
XGID=-O-----------------aaaa---:0:0:-1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

Your opponent is cubing you here, what is he thinking, he's down 3 pips! Beaver! This is how deceptive the flat pip count can be. You clearly almost never win from this position (rounds to 100%). What the effective pip count does is try to account for the wastage of a position and convert the wastage + pip count into something more useful. Wastage is basically pips that go unused. If you have a stack of 15 checkers on your ace point and you roll 65 for example you're going to take 2 checkers off and decrease your actual pip count by only 2 pips even though you rolled 11 pips. You are said to have 'wasted' 9 pips.

Thanks to bg junkies of the past and computers we know certain things nowadays like the average number of rolls required to bear 15 checkers off of your ace comes to almost 7 rolls. It was also found by Walter that if you multiply the average number of rolls it takes to bear 15 men off the ace point and multiply that number by 49/6 the product is 57. (49/6 is the average roll, also known as 8.167 pips or "just above 8" for practical use) The fact that these numbers turn out to be a whole number, 57, is quite a coincidence and gives us the equation for the formula for the effective pipcount of any pure roll position.

Rule 1:  For an N-roll position, the EPC is 7N + 1.

In other words in our example of 15 checkers to bear off on the ace point otherwise known as a pure 8 roll position the effective pip count would be 7n+1 where n equals the number of rolls. Substituting the values we get 7(8)+1 or 57 pips as our epc. If you to average the number of pips it took to bear off having all your checkers stacked on the ace point it would be equal to 57

If you extend the formula to look at all pure roll positions here are the epcs:

Here is a chart rounded to the nearest full percent of your cubeless winning chances in any pure roll v. roll situation.

# of Rolls Effective Pip Count
1
8
2
15
3
22
4
29
5
36
6
43
7
50
8
57

When bearing in you want to keep your checkers on the high points (4, 5, and 6) to avoid future wastage. The position that wastes the least amount of pips, just over 7, is shown below and what you should aim for when bearing in your checkers.
is Player 2

score: 0
pip: 79
Money session
Jacoby Beaver
pip: 79
score: 0

is Player 1
XGID=----CEG------------gec----:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

While the actual pip count is 79 pips the epc would be the sum of the pip count + the wastage or 79+7=86. In positions like this which are referred to as low wastage positions you can assume the wastage is between 7 and 7.5 pips. Walter also referred to these positions as 'nice' positions and thus rule #2:

      Rule 2:  Nice positions waste 7 pips.

A 'nice' position could be defined as a smooth distribution of checkers on your 4, 5, and 6 points. So to get the epc of such a position you take the regular pip count and add 7, voilą. Let's try to get some use out of this knowledge from a position that Walter used to play as a prop to anyone foolish enough to take him up on it. If memory serves me he called it 'Prop 57' for obvious reasons.
is Player 2

score: 0
pip: 50
Money session
Jacoby Beaver
pip: 30
score: 0

is Player 1
XGID=-EEE---------------cdc----:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

We have one side, White, with basically an 8 roll position having all his checkers smoothly across the lower points. We know that the epc of an 8 roll position is 7(8)+1 or 57. What about Brown? He seems to qualify as having a 'nice' position which per rule #2 has 7 pips of wastage. We add those 7 pips on to his actual pip count of 50 and also get 57. Prop 57, where's the money if both players have the same effective pip count!? Walter fleeced his sheep by using the epc effortless to make his cube decisions. The formula he gave for these types of positions is that the point of last take for the trailer is when he is down a number of effective pips equal to the number of rolls to go minus three.

For example, if the leader has a 3 roll position which is an epc of 22 then the trailer can take only if his epc is 22 or less. The leader can double within' one pip of the point of last take and give an initial double within' two.

An easy practice problem. Give both players epc and the proper cube action for the bottom player (White) on roll.
is Player 2

score: 0
pip: 26
Money session
Jacoby Beaver
pip: 12
score: 0

is Player 1
XGID=-DD----------------bba----:0:0:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

Relatively easy I hope. White's position is a 4 roll position so his epc is 7(4)+1 or 29. Brown has an actual pipcount of 26 and with only five checkers left his wastage is reduced to about 6 making his epc 26+6 or 32. Brown's point of last take at 4 rolls would be down one. He is down three so it's a clear drop.