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Hypothetical Scenario in a Pure Race

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08-19-2013 , 12:21 PM

Frank the Tank

old hand

Join Date: Oct 2008 Posts: 1,458

Friend of mine and I got into a discussion after I said "there can't be a situation where bearing off 4 checkers in a pure race is a mistake." He thinks there could be but neither of us have thought of a scenario yet.

Quote

08-19-2013 , 05:20 PM

Robertie

old hand

Join Date: May 2004 Posts: 1,954

Quote:

Originally Posted by Frank the Tank

Your friend is right. I'll post a couple of positions in a day or two, but let's see if someone else wants to take a stab at the problem first.

Quote

08-19-2013 , 05:58 PM

kruidenbuiltje

journeyman

Join Date: Mar 2010 Posts: 270

Hi,

i am thinking of a sort of situation where taking 4 off leaves one checker on the 4-point,
while taking 3 off leaves checkers on the 1- and 2-point.

But this doesn't work, since the spare pips must be equal in the problem.

Quote

08-19-2013 , 06:02 PM

kruidenbuiltje

journeyman

Join Date: Mar 2010 Posts: 270

But what about the situation where taking 4 off leaves 6-1
and taking 3 off leaves 5-2,

also not possible since the number of checkers needs to have a difference of 1 checker.

Quote

08-19-2013 , 06:10 PM

kruidenbuiltje

journeyman

Join Date: Mar 2010 Posts: 270

Maybe something with leaving 3 or 4 checkers on the board, where doubles work efficiently?

for example leaving 2-2-4 or 2-2-1-3?

Quote

08-19-2013 , 09:14 PM

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

Quote:

Originally Posted by kruidenbuiltje

But this doesn't work, since the spare pips must be equal in the problem.

Quote:

Originally Posted by kruidenbuiltje

Also not possible since the number of checkers needs to have a difference of 1 checker.

I do not understand what you mean by these comments. It seems as though you are imposing a requirement that both players have the same number of pips, or something along those lines. My reading is that there is only one restriction, and that is that there be no contact.

Quote

08-19-2013 , 09:24 PM

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

A related question pertains to effective pip count (EPC). Are there any positions in which bearing off four checkers results in a higher EPC than an alternative move?

In that case, I would be willing to bet there are at least 35 such positions, but certainly no more than that.

Quote

08-19-2013 , 11:12 PM

yogiman

adept

Join Date: Jun 2012 Posts: 784

I propose that everyone gets out of the laboratory, except for kruidenbuiltje, who is advised to go first into the laboratory.

Last edited by yogiman; 08-19-2013 at 11:20 PM.

Quote

08-19-2013 , 11:29 PM

yogiman

adept

Join Date: Jun 2012 Posts: 784

Having said that, this can only be a mistake to an egotistic prig.

Quote

08-20-2013 , 08:14 AM

#10

kruidenbuiltje

journeyman

Join Date: Mar 2010 Posts: 270

Hi Taper_Mike,

what i meant was the following:

say you have 2 variations, 1) you take 4 checkers off, 2) you take 3 checkers off and move one checker.

In both variations the pips you move are equal, so the pips you leave on the board (spare pips) must be equal in both variations.

My second remarks means that in the first variation you remove 1 more checker than in the second variation.
So the number of checkers left on the board (after the move) has a difference of 1.

Quote

08-20-2013 , 08:39 AM

#11

kruidenbuiltje

journeyman

Join Date: Mar 2010 Posts: 270

Let's see if we can logically construct (rest)positions after the variations
that are equal in pips but have a difference of 1 in checkers.

1+1 vs. 2, 1+2 vs. 3,

what about 4x3 vs. 3+4+5, D3 works with the first but not with the second
or 4x3 vs. 3x3 +2+1

But how do we get to the original position?

Quote

08-20-2013 , 11:14 PM

#12

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

Quote:

Originally Posted by kruidenbuiltje

Makes perfect sense.

Mike

Last edited by Taper_Mike; 08-20-2013 at 11:25 PM.

Quote

08-20-2013 , 11:35 PM

#13

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

Here are a few ideas that might help solve this.

Ripping off a checker is a powerful move. Ripping 4 can save a full turn. Therefore, it makes sense that bearing off fewer than 4 is more likely to be viable when you are odd. That way, you can save a full turn even when you bear off only 3 checkers. A search for solutions where there are 15, 13, 11, etc., checkers remaining is more likely to be fruitful than the alternatives.
What circumstance would motivate you not to bear off? The obvious answer is so that you can fill a gap, or prevent one from appearing.
When will a gap matter? Probably not when there are only one or two checkers behind the gap. Filling a gap is more likely to be productive when there is a stack of checkers behind it.

To make this last point, let me use the example of 11. It has been proven that it is never wrong to bear off a checker with a 1, so this example is not going to give away any of the solutions.

Suppose you roll 11 when you have 7 checkers on the 1pt, and 8 more on the 3pt. There are so many checkers on the 3pt, that you might consider trying to protect against the roll of a 2. If you are odd anyway, perhaps ripping three checkers off, and using the fourth ace to play 3/2 will lead to fewer misses later. In fact, it won't when the roll is 11, but you get the idea.

Given the ideas above, consider how many checkers might still be in play. Is is a lot, or just few?

Mike

Last edited by Taper_Mike; 08-20-2013 at 11:50 PM.

Quote

08-22-2013 , 04:34 AM

#14

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

There is a series of problems near the end of Advanced Backgammon, Volume 2, that are similar to the original problem of this thread. See Problems 396-399.

Bill Robertie makes the observation that if there are going to be opportunities to fill a gap later, there is usually no need to forego bearing off a checker. On the other hand, if you have reached the point where you may not have another chance to fill a gap, then it sometimes pays to do so in lieu of bearing off.

In practice, this means that the highest point, perhaps the 6pt or 5pt, is down to 1, and less often, 2, checkers. Suppose for instance, there are 4 checkers on the 3pt, a stack on the 4pt or 5pt, and only 1 checker on the 6pt. Other checkers fill out a full complement of 15. On a roll of 33, that’s the sort of position where ripping 3 can beat pulling 4.

Mike

Quote

08-30-2013 , 10:23 AM

#15

Robertie

old hand

Join Date: May 2004 Posts: 1,954

In the bearoff, when is it right to fill a gap instead of bearing off a checker?

The short answer is “almost never”. A few such positions do exist. The first correct example in print is on page 292 of Magriel’s classic, Backgammon. In the late 1980s Hugh Sconyers and Walter Trice independently constructed lists of all such positions, using recursively-generated databases of bearoff positions. Walter supplied me with the set of positions a little later, and I used them for the basis of Problems 396-399 in Advanced Backgammon, vol.2. Some but not all of these positions are solutions to the question posed here, “When is it right to bear off three checkers with a double but not four?”

First, take a look at Magriel’s position:

Black to play a 2.

Notice two characteristics:

(1) Playing 2/off leaves Black with 11 checkers. Playing 4/2 leaves him with 12 checkers. In either case he needs six rolls (without a double) to bear off, so playing 2/off won’t save a roll.

(2) Black has a lot of checkers above the gap, so the chance of missing several times with a deuce is non-trivial.

In this position playing 4/2 bears off, on average, a tiny bit faster than 2/off.

Putting three more checkers on the 2-point in Magriel’s position gives us a solution to the poster’s original question:

Black to play 2-2. Best play: 2/off(3) 4/2.

All the cases in Trice’s database involved the play of a 2, 3, or 4. There are no positions where it’s incorrect to bear off with a 1, 5, or 6.

Here’s a position where you don’t bear four checkers off with a 4-4. It’s based on Position 396B in Advanced Backgammon.

Black to play 4-4. Best play: 4/off(3) 6/2.

Quote

09-01-2013 , 03:41 AM

#16

Taper_Mike

adept

Join Date: Sep 2009 Posts: 1,030

Quote:

Originally Posted by Taper_Mike

Suppose for instance, there are 4 checkers on the 3pt, a stack on the 4pt or 5pt, and only 1 checker on the 6pt. Other checkers fill out a full complement of 15. On a roll of 33, that’s the sort of position where ripping 3 can beat pulling 4.

Black to Play 3-3, Ripping 3 beats pulling 4

A couple of years ago I wrote a program to create a one-sided bear-off database. It is not much to brag about—it is completely duplicative of the work done by Walter Trice and Hugh Sconyers almost 25 years ago. It is also very slow. My program takes about ten times as long as GnuBg or XG to generate a bear-off database.

It did serve one purpose, however, that of getting my feet wet. I was forced to invent a way of representing a board, and provide machinery for rolling dice, moving checkers, and so forth. One of the more challenging aspects was simply enumerating all the possible moves for a given dice roll and position. With blocked points, entry from the bar, rules regarding forced moves, and the restriction that the bear off cannot begin until all checkers are in the inner board, that turned out to be more difficult than I imagined. Another hurdle was learning how to systematically identify all the possible positions that can occur in a non-contact race/bear-off position. Working off and on, it took me about four months to finish the project.

With those tools available, as soon as I read the the post by Frank the Tank that begins this thread, I knew that I could provide an answer for him. It took only couple of hours to get a quick-and-dirty result. Since then, I've put in another 20 hours adding some polish, and generalizing the kinds of questions it can answer.

My “algorithm” uses brute force. Counting the position where all checkers have been borne off, there are 54,264 distinct checker arrangements that are possible during a bear off. For each one, there are 21 distinct rolls. Thus, there are a total of 1,139,544 bear-off positions. I already had the routines to identify them all.

For each one, here is what I did:

Set up the position on a virtual board.
For the given roll and checker arrangement, determine all possible moves.
Try out each one by moving the checkers on the board.
Look up the resulting position in the bear-off database to retrieve its EPC (effective pip count).
See whether the move with lowest EPC failed to bear off as many checkers as one of the alternatives.

In the case where the roll is a doublet, I needed to check two things.

Was it possible to bear off four checkers?
Did the move with the lowest EPC do so?

Out of the over-1-million positions I searched, I found 35 that qualify. The following table gives a breakdown of the more general result.

Code:

Category                             No. of Positions 
----------------------------------   -----------------
Rip None, on a Roll of Doublets, 
When You Could Rip 1                      1,751  

Rip None, on a Roll of Doublets, 
When You Could Rip 2                         24  

Rip 1, on a Roll of Doublets, 
When You Could Rip 2                      6,149  

Rip 2, on a Roll of Doublets, 
When You Could Rip 3                      2,078  

Rip 3, on a Roll of Doublets, 
When You Could Rip 4                         35  

Rip None, on a Roll of Non-doublets, 
When You Could Rip 1                      3,825  

Rip 1, on a Roll of Non-doublets, 
When You Could Rip 2                      1,911  

Total                                    15,773

Here are all 35 that satisfy Frank’s original question. Number 27 is diagrammed above. The ones cited in Bill’s post are numbers 5 and 11. The one where ripping off 4 is the largest error is number 20. That mistake will cost you about a third of a pip.

Code:

Position 1. XGID=-CDEA----------------aedc-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
  O | O  O  O  O       |   |                  |  Pips  : 30
  O | O  O  O          |   |                  |  EPC   : 50.7809
    | O  O  O          |   |                  |
    |    O  O          |   |                  |
    |       O          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       X          |   |                  |
    |    X  X          |   |                  |
    | X  X  X          |   |                  |
  X | X  X  X          |   |                  |  EPC   : 50.7809
  X | X  X  X  X       |   |                  |  Pips  : 30
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   38.1162   --           --
  2.   2/off(4)          38.2219   0.105702     0.277315%
-------------------------------------------------------------




Position 2. XGID=-GDCA----------------acdg-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 28
    | O  O  O          |   |                  |  EPC   : 57.0799
    | O  O  O          |   |                  |
    | O  O             |   |                  |
    | 7                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | 7                |   |                  |
    | X  X             |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 57.0799
    | X  X  X  X       |   |                  |  Pips  : 28
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   43.5966   --           --
  2.   2/off(4)          43.6218   0.0251712    0.0577365%
-------------------------------------------------------------




Position 3. XGID=-FDDA----------------addf-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 30
    | O  O  O          |   |                  |  EPC   : 57.1895
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    | 6                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | 6                |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 57.1895
    | X  X  X  X       |   |                  |  Pips  : 30
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   44.0051   --           --
  2.   2/off(4)          44.1604   0.155292     0.352896%
-------------------------------------------------------------




Position 4. XGID=-FBDB-A------------a-bdbf-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 36
    | O  O  O  O       |   |                  |  EPC   : 58.3631
    | O     O          |   |                  |
    | O     O          |   |                  |
    | 6                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | 6                |   |                  |
    | X     X          |   |                  |
    | X     X          |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.3631
    | X  X  X  X     X |   |                  |  Pips  : 36
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   44.346    --           --
  2.   3/off(4)          44.3846   0.0385377    0.0869023%
-------------------------------------------------------------




Position 5. XGID=-EDEA----------------aede-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 32
    | O  O  O          |   |                  |  EPC   : 57.3692
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    | O     O          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X     X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 57.3692
    | X  X  X  X       |   |                  |  Pips  : 32
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   44.5545   --           --
  2.   2/off(4)          44.8406   0.286081     0.642092%
-------------------------------------------------------------




Position 6. XGID=-EDE--A------------a--ede-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O        O |   |                  |  Pips  : 34
    | O  O  O          |   |                  |  EPC   : 57.7362
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    | O     O          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X     X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 57.7362
    | X  X  X        X |   |                  |  Pips  : 34
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/4, 2/off(3)   46.1716   --           --
  2.   2/off(4)        46.2051   0.0334885    0.0725305%
-----------------------------------------------------------




Position 7. XGID=-EDDB----------------bdde-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 33
    | O  O  O  O       |   |                  |  EPC   : 57.3623
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    | O                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X                |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 57.3623
    | X  X  X  X       |   |                  |  Pips  : 33
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   44.5436   --           --
  2.   2/off(4)          44.5871   0.0434839    0.0976211%
-------------------------------------------------------------




Position 8. XGID=-ED-EA--------------ae-de-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O     O  O    |   |                  |  Pips  : 38
    | O  O     O       |   |                  |  EPC   : 60.8951
    | O  O     O       |   |                  |
    | O  O     O       |   |                  |
    | O        O       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X        X       |   |                  |
    | X  X     X       |   |                  |
    | X  X     X       |   |                  |
    | X  X     X       |   |                  |  EPC   : 60.8951
    | X  X     X  X    |   |                  |  Pips  : 38
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   5/3, 2/off(3)   48.393    --           --
  2.   2/off(4)        48.4073   0.0142786    0.0295055%
-----------------------------------------------------------




Position 9. XGID=-ECDB-A------------a-bdce-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 37
    | O  O  O  O       |   |                  |  EPC   : 58.175
    | O  O  O          |   |                  |
    | O     O          |   |                  |
    | O                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X                |   |                  |
    | X     X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.175
    | X  X  X  X     X |   |                  |  Pips  : 37
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   44.3416   --           --
  2.   3/off(4)          44.3722   0.0305624    0.068925%
-------------------------------------------------------------




Position 10. XGID=-EBDC-A------------a-cdbe-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 39
    | O  O  O  O       |   |                  |  EPC   : 58.9159
    | O     O  O       |   |                  |
    | O     O          |   |                  |
    | O                |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X                |   |                  |
    | X     X          |   |                  |
    | X     X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.9159
    | X  X  X  X     X |   |                  |  Pips  : 39
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   45.1625   --           --
  2.   3/off(4)          45.3088   0.1463       0.323942%
-------------------------------------------------------------




Position 11. XGID=-E-ED-A------------a-de-e-:0:0:1:44:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O     O |   |                  |  Pips  : 42
    | O     O  O       |   |                  |  EPC   : 62.0076
    | O     O  O       |   |                  |
    | O     O  O       |   |                  |
    | O     O          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X     X          |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |  EPC   : 62.0076
    | X     X  X     X |   |                  |  Pips  : 42
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 44.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/2, 4/off(3)   46.1716   --           --
  2.   4/off(4)        46.2051   0.0334885    0.0725305%
-----------------------------------------------------------




Position 12. XGID=-E-DEA--------------aed-e-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O  O    |   |                  |  Pips  : 42
    | O     O  O       |   |                  |  EPC   : 60.8347
    | O     O  O       |   |                  |
    | O     O  O       |   |                  |
    | O        O       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    | X        X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |  EPC   : 60.8347
    | X     X  X  X    |   |                  |  Pips  : 42
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   5/2, 3/off(3)   48.393    --           --
  2.   3/off(4)        48.4073   0.0142786    0.0295055%
-----------------------------------------------------------




Position 13. XGID=-DDFA----------------afdd-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 34
    | O  O  O          |   |                  |  EPC   : 57.7007
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    |       6          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       6          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 57.7007
    | X  X  X  X       |   |                  |  Pips  : 34
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   45.3285   --           --
  2.   2/off(4)          45.6076   0.279129     0.615792%
-------------------------------------------------------------




Position 14. XGID=-DDEB----------------bedd-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 35
    | O  O  O  O       |   |                  |  EPC   : 57.6838
    | O  O  O          |   |                  |
    | O  O  O          |   |                  |
    |       O          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 57.6838
    | X  X  X  X       |   |                  |  Pips  : 35
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   45.2698   --           --
  2.   2/off(4)          45.3279   0.0581102    0.128364%
-------------------------------------------------------------




Position 15. XGID=-DDC-CA------------ac-cdd-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O     O  O |   |                  |  Pips  : 42
    | O  O  O     O    |   |                  |  EPC   : 61.3633
    | O  O  O     O    |   |                  |
    | O  O             |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    | X  X             |   |                  |
    | X  X  X     X    |   |                  |
    | X  X  X     X    |   |                  |  EPC   : 61.3633
    | X  X  X     X  X |   |                  |  Pips  : 42
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/4, 2/off(3)   49.752    --           --
  2.   2/off(4)        49.7679   0.0158458    0.0318495%
-----------------------------------------------------------




Position 16. XGID=-DCDC-A------------a-cdcd-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 40
    | O  O  O  O       |   |                  |  EPC   : 58.8091
    | O  O  O  O       |   |                  |
    | O     O          |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    | X     X          |   |                  |
    | X  X  X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.8091
    | X  X  X  X     X |   |                  |  Pips  : 40
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   45.2337   --           --
  2.   3/off(4)          45.3666   0.132847     0.293691%
-------------------------------------------------------------




Position 17. XGID=-DBDD-A------------a-ddbd-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 42
    | O  O  O  O       |   |                  |  EPC   : 59.7146
    | O     O  O       |   |                  |
    | O     O  O       |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 59.7146
    | X  X  X  X     X |   |                  |  Pips  : 42
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   46.2601   --           --
  2.   3/off(4)          46.503    0.242931     0.525141%
-------------------------------------------------------------




Position 18. XGID=-DADE-A------------a-edad-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 44
    | O     O  O       |   |                  |  EPC   : 61.1904
    | O     O  O       |   |                  |
    | O     O  O       |   |                  |
    |          O       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |  EPC   : 61.1904
    | X  X  X  X     X |   |                  |  Pips  : 44
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   48.1754   --           --
  2.   3/off(4)          48.1839   0.00843757   0.0175143%
-------------------------------------------------------------




Position 19. XGID=-D-CDCA------------acdc-d-:0:0:1:44:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O  O  O |   |                  |  Pips  : 50
    | O     O  O  O    |   |                  |  EPC   : 63.9028
    | O     O  O  O    |   |                  |
    | O        O       |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    | X        X       |   |                  |
    | X     X  X  X    |   |                  |
    | X     X  X  X    |   |                  |  EPC   : 63.9028
    | X     X  X  X  X |   |                  |  Pips  : 50
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 44.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/2, 4/off(3)   49.752    --           --
  2.   4/off(4)        49.7679   0.0158458    0.0318495%
-----------------------------------------------------------




Position 20. XGID=-CDGA----------------agdc-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 36
    | O  O  O          |   |                  |  EPC   : 58.292
    | O  O  O          |   |                  |
    |    O  O          |   |                  |
    |       7          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       7          |   |                  |
    |    X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 58.292
    | X  X  X  X       |   |                  |  Pips  : 36
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   46.2726   --           --
  2.   2/off(4)          46.5831   0.310501     0.671026%
-------------------------------------------------------------




Position 21. XGID=-CDFB----------------bfdc-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 37
    | O  O  O  O       |   |                  |  EPC   : 58.2705
    | O  O  O          |   |                  |
    |    O  O          |   |                  |
    |       6          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       6          |   |                  |
    |    X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.2705
    | X  X  X  X       |   |                  |  Pips  : 37
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   46.2109   --           --
  2.   2/off(4)          46.3174   0.106511     0.230488%
-------------------------------------------------------------




Position 22. XGID=-CDF--B------------b--fdc-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O        O |   |                  |  Pips  : 41
    | O  O  O        O |   |                  |  EPC   : 61.5799
    | O  O  O          |   |                  |
    |    O  O          |   |                  |
    |       6          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       6          |   |                  |
    |    X  X          |   |                  |
    | X  X  X          |   |                  |
    | X  X  X        X |   |                  |  EPC   : 61.5799
    | X  X  X        X |   |                  |  Pips  : 41
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/4, 2/off(3)   50.047    --           --
  2.   2/off(4)        50.0518   0.00472563   0.00944237%
-----------------------------------------------------------




Position 23. XGID=-CDDC-A------------a-cddc-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 41
    | O  O  O  O       |   |                  |  EPC   : 58.9773
    | O  O  O  O       |   |                  |
    |    O  O          |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    |    X  X          |   |                  |
    | X  X  X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 58.9773
    | X  X  X  X     X |   |                  |  Pips  : 41
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   45.4638   --           --
  2.   3/off(4)          45.5561   0.0923395    0.203106%
-------------------------------------------------------------




Position 24. XGID=-CDD-CA------------ac-ddc-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O     O  O |   |                  |  Pips  : 44
    | O  O  O     O    |   |                  |  EPC   : 61.8513
    | O  O  O     O    |   |                  |
    |    O  O          |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    |    X  X          |   |                  |
    | X  X  X     X    |   |                  |
    | X  X  X     X    |   |                  |  EPC   : 61.8513
    | X  X  X     X  X |   |                  |  Pips  : 44
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/4, 2/off(3)   50.5146   --           --
  2.   2/off(4)        50.5499   0.03536      0.0699996%
-----------------------------------------------------------




Position 25. XGID=-CCDD-A------------a-ddcc-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 43
    | O  O  O  O       |   |                  |  EPC   : 59.725
    | O  O  O  O       |   |                  |
    |       O  O       |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    |       X  X       |   |                  |
    | X  X  X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 59.725
    | X  X  X  X     X |   |                  |  Pips  : 43
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   46.3772   --           --
  2.   3/off(4)          46.6117   0.234576     0.505801%
-------------------------------------------------------------




Position 26. XGID=-CBDE-A------------a-edbc-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 45
    | O  O  O  O       |   |                  |  EPC   : 60.831
    | O     O  O       |   |                  |
    |       O  O       |   |                  |
    |          O       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          X       |   |                  |
    |       X  X       |   |                  |
    | X     X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 60.831
    | X  X  X  X     X |   |                  |  Pips  : 45
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   47.5641   --           --
  2.   3/off(4)          47.8623   0.298159     0.626858%
-------------------------------------------------------------




Position 27. XGID=-CADF-A------------a-fdac-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 47
    | O     O  O       |   |                  |  EPC   : 62.4666
    | O     O  O       |   |                  |
    |       O  O       |   |                  |
    |          6       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          6       |   |                  |
    |       X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X       |   |                  |  EPC   : 62.4666
    | X  X  X  X     X |   |                  |  Pips  : 47
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   49.5558   --           --
  2.   3/off(4)          49.6347   0.0789109    0.159236%
-------------------------------------------------------------




Position 28. XGID=-C-FD-B------------b-df-c-:0:0:1:44:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O     O |   |                  |  Pips  : 49
    | O     O  O     O |   |                  |  EPC   : 65.6783
    | O     O  O       |   |                  |
    |       O  O       |   |                  |
    |       6          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       6          |   |                  |
    |       X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X     X |   |                  |  EPC   : 65.6783
    | X     X  X     X |   |                  |  Pips  : 49
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 44.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/2, 4/off(3)   50.047    --           --
  2.   4/off(4)        50.0518   0.00472563   0.00944237%
-----------------------------------------------------------




Position 29. XGID=-C-DF-B------------b-fd-c-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O     O |   |                  |  Pips  : 51
    | O     O  O     O |   |                  |  EPC   : 66.1144
    | O     O  O       |   |                  |
    |       O  O       |   |                  |
    |          6       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          6       |   |                  |
    |       X  X       |   |                  |
    | X     X  X       |   |                  |
    | X     X  X     X |   |                  |  EPC   : 66.1144
    | X     X  X     X |   |                  |  Pips  : 51
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   53.038    --           --
  2.   3/off(4)          53.0479   0.00986852   0.0186065%
-------------------------------------------------------------




Position 30. XGID=-C-DDCA------------acdd-c-:0:0:1:44:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O  O  O |   |                  |  Pips  : 52
    | O     O  O  O    |   |                  |  EPC   : 64.6862
    | O     O  O  O    |   |                  |
    |       O  O       |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |                  |   |                  |
    |       X  X       |   |                  |
    | X     X  X  X    |   |                  |
    | X     X  X  X    |   |                  |  EPC   : 64.6862
    | X     X  X  X  X |   |                  |  Pips  : 52
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 44.

-----------------------------------------------------------
 No.   Move            EPC       Difference   % Difference
-----------------------------------------------------------
  1.   6/2, 4/off(3)   50.5146   --           --
  2.   4/off(4)        50.5499   0.03536      0.0699996%
-----------------------------------------------------------




Position 31. XGID=-BDHA----------------ahdb-:0:0:1:22:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O       |   |                  |  Pips  : 38
    | O  O  O          |   |                  |  EPC   : 59.1823
    |    O  O          |   |                  |
    |    O  O          |   |                  |
    |       8          |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |       8          |   |                  |
    |    X  X          |   |                  |
    |    X  X          |   |                  |
    | X  X  X          |   |                  |  EPC   : 59.1823
    | X  X  X  X       |   |                  |  Pips  : 38
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 22.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   4/off, 2/off(2)   47.699    --           --
  2.   2/off(4)          47.7794   0.0803447    0.168441%
-------------------------------------------------------------




Position 32. XGID=-BCDE-A------------a-edcb-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 46
    | O  O  O  O       |   |                  |  EPC   : 61.0309
    |    O  O  O       |   |                  |
    |       O  O       |   |                  |
    |          O       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          X       |   |                  |
    |       X  X       |   |                  |
    |    X  X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 61.0309
    | X  X  X  X     X |   |                  |  Pips  : 46
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   47.9675   --           --
  2.   3/off(4)          48.0948   0.127291     0.265369%
-------------------------------------------------------------




Position 33. XGID=-BBDF-A------------a-fdbb-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 48
    | O  O  O  O       |   |                  |  EPC   : 62.371
    |       O  O       |   |                  |
    |       O  O       |   |                  |
    |          6       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          6       |   |                  |
    |       X  X       |   |                  |
    |       X  X       |   |                  |
    | X  X  X  X       |   |                  |  EPC   : 62.371
    | X  X  X  X     X |   |                  |  Pips  : 48
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   49.2919   --           --
  2.   3/off(4)          49.4605   0.168611     0.342067%
-------------------------------------------------------------




Position 34. XGID=-BADG-A------------a-gdab-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O  O  O  O     O |   |                  |  Pips  : 50
    | O     O  O       |   |                  |  EPC   : 64.2066
    |       O  O       |   |                  |
    |       O  O       |   |                  |
    |          7       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          7       |   |                  |
    |       X  X       |   |                  |
    |       X  X       |   |                  |
    | X     X  X       |   |                  |  EPC   : 64.2066
    | X  X  X  X     X |   |                  |  Pips  : 50
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   51.3387   --           --
  2.   3/off(4)          51.3575   0.018792     0.036604%
-------------------------------------------------------------




Position 35. XGID=-B-DG-B------------b-gd-b-:0:0:1:33:0:0:3:0:10


     24 23 22 21 20 19      18 17 16 15 14 13    Player O
    -------------------------------------------
    | O     O  O     O |   |                  |  Pips  : 54
    | O     O  O     O |   |                  |  EPC   : 67.7228
    |       O  O       |   |                  |
    |       O  O       |   |                  |
    |          7       |   |                  |
    |                  |BAR|                  |v Cash game, Jacoby (Cube: 1)
    |          7       |   |                  |
    |       X  X       |   |                  |
    |       X  X       |   |                  |
    | X     X  X     X |   |                  |  EPC   : 67.7228
    | X     X  X     X |   |                  |  Pips  : 54
    -------------------------------------------
      1  2  3  4  5  6       7  8  9 10 11 12    Player X


X to play 33.

-------------------------------------------------------------
 No.   Move              EPC       Difference   % Difference
-------------------------------------------------------------
  1.   6/off, 3/off(2)   54.7274   --           --
  2.   3/off(4)          54.7348   0.0074757    0.0136599%
-------------------------------------------------------------

Mike

Last edited by Taper_Mike; 09-01-2013 at 03:55 AM.

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09-01-2013 , 11:53 AM

#17

yogiman

adept

Join Date: Jun 2012 Posts: 784

Quote:

Working off and on, it took me about four months to finish the project.

Mike the tank is possibly frank.

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