INSIDE BEAROFF

 

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SUMMARY

With some experience, one can become almost infallible in the bearoff, apart from the cube handling.

Nevertheless some bewildering checker plays exist, that may occasionnally be overlooked in practical play.

Some of them are shown here.

In chapter 1, we enumerate and illustrate all possible grounds which could motivate the stratagem we call "Booster Brakes" (consisting of wasting pips on purpose).

In chapter 2 we enumerate and illustrate all possible grounds which may make bearing a man off straightforwardly definitely not the best play.

In chapter 3 we deal superficially with the question of stacking the checkers in the bearoff.

In chapter 4 we show various positions in which the best possible roll is definitely not double 6 and we show a situation of zugzwang : a rare bird in the bearoff.

In chapter 5 we show how extending the dimensions of the game can induce unexpected curiosities.

 

For sake of simplicity, all the positions shown here are supposed to be cubeless.

The position of the player on move and his opponent's one are shown separated by a / . The number on the left indicates the number of men on the ace-point. The number on the right indicates the number of men on the 6-point.

 

The contents of this paper may be useful in computer backgammon as it is in the bearoff program we have written (still being developed).

 

 

- 0 - PRELIMINARIES

 

Backgammon is definitely not a game of solitaire.

Even after contact has been broken, a player should consider his opponent's position before making a play. He should evaluate the relative positions in the race as he often has to choose between safe plays (paying insurance against the misfortunes most likely to happen) and bold plays (hoping for favorable events and trying to squeeze out of them the most profitable rewards). Cube also interferes although it is not dealt with here.

This is particularly true in the positions shown in this paper.

As a consequence, the position of the player not on roll is always illustrated, even if often without comment.

This is wonderfully illustrated by the following five positions where roller has to play double aces.

 

0 0 0 4 0 4 / 1 1 1 1 1 0

Best play : (6/5) * 4 - Roller wants to be off in 2 rolls.

Opponent's CPW = 99.4018 %

 

0 0 0 4 0 4 / 2 2 1 1 1 0

Best play : 4/Off - Roller wants to roll 66 once and take off the rest in 2 rolls.

Opponent's CPW = 96.9803%

 

0 0 0 4 0 4 / 3 3 1 1 1 0

Best play : (6/5) * 3 , 4/3 - Roller wants to maximize efficiency of his big doubles (66 and 55, but also 44).

Opponent's CPW = 86.6856 %

 

0 0 0 4 0 4 / 4 4 1 1 1 0

Best play : (6/5) * 2 , (4/3) * 2 - The probably most natural move. Roller wants to make efficient use of all his big numbers, not only doubles.

Opponent's CPW = 64.5662 %

 

0 0 0 4 0 4 / 5 5 1 1 1 0

Best play : 6/5 , 4/3 , 4/2 - Roller is comfortably ahead and just wants to make all his numbers work at best.

Opponent's CPW = 36.2515 %

 

Five different plays for a same bearoff position is really amazing. Note however how many legal moves are never optimal.

 

 

- 1 - BOOSTER BRAKES

 

Booster brakes (wasting pips) are a common stratagem when bearing off against contact, and they should also sometimes be used when bearing off without contact.

 

1 0 1 0 0 1 / 1 1 1 0 0 0

52 should obviously be played : 6/4/Off in order to bear a checker off.

 

2 0 2 0 0 1 / 2 2 1 0 0 0

The right play for 54 is : 6/2 3/Off filling the gap on the 2-point.

 

3 1 0 4 0 1 / 3 3 3 0 0 0

63 should be played : 6/3 4/Off, although wasting two pips.

 

One may also desire to avoid the creation of a gap.

3 1 2 0 0 1 / 2 2 1 0 0 0

The right play for 52 is : 6/4/Off, not  6/1 2/Off.

Although a double is needed, it is necessary to sacrify the double aces, for one cannot afford to lose immediately by rolling a single 2.

 

3 3 1 4 1 1 / 5 4 4 0 0 0

The right play for 63 is : 6/3 5/Off.

Here the correct play bears only one checker off although 2 checkers off are possible.

 

Sometimes it is necessary to select the harmless gap

6 0 0 7 1 0 / 5 4 4 0 0 0

The right play for 52 is : 5/3 4/Off , not 5/Off 4/2

There is no remedy against a gap on the 3-point, while a gap on the 2-point is only temporary.

 

When late in the race, one must sometimes pay special attention to preparing doubles.

1 3 2 1 0 0 / 2 2 1 0 0 0

42 should be played : 4/2 3/Off, not  4/Off 2/Off

Such way, 22 becomes efficient next turn. We suspect that even an advanced player playing too quickly might miss this play.

 

0 3 2 0 0 1 / 2 2 1 0 0 0

The right play for 52 is : 6/4/Off, not  6/1 2/Off

The reason is the same and the play even less obvious (X-22 let himself caught in the trap).

 

0 4 0 4 0 1 / 2 2 1 0 0 0

The right play for 52 is : 6/4/Off, not  6 /1 2/Off.

Now 22 preserves a small chance instead of losing immediately.

 

For the end, booster brakes may be necessary in order to improve the flexibility of the position.

3 0 3 0 1 1 / 3 3 3 0 0 0

The right play for 62 is : 6/4 5/Off, not  6/Off 5/3 or 6/Off 3/1, so that subsequent 32, 42, 52, 62 can be played : 4/2 3/Off using the booster brakes a second time!

 

 

- 2 - DECLINING TO BEAR A CHECKER OFF

 

This is an extremely rarely correct play.

 

Walter TRICE states that such plays must be made only with at least 10 checkers on the board, when being ahead in the race (actually we bring the following correction : being on the favorable side of a break-even point, the break-even point depending on the position of the player on move) and necessarily with the idea of filling a gap.

 

As we shall see, there exist another family of positions.

 

With a 2, one can fill a gap on the 2-point ...

5 1 5 1 0 0 / 5 4 4 0 0 0 (an example familiar to MAGRIEL's readers)

The right 2 is : 4/2

because of the sequences 43/53/63/54/64/65 followed by two 2s.

 

... or on the 3-point...

5 1 0 5 1 0 / 5 4 4 0 0 0

The right play for a 2 is : 5/3  , not  2/Off

because of the subsequent rolls containing 5s or 6s followed by various rolls containing 2s and 3s

 

... or even on the 4-point

5 1 2 0 5 1 / 0 5 5 5 0 0

4 1 6 0 0 2 / 0 2 3 2 3 3

The right play for a 2 is 6/4, not 2/Off in both positions.

 

With a 3, one can fill a gap on the 2-point, if the 3-point is also sensitive ...

5 0 1 5 1 0 / 5 4 4 0 0 0

... and also if it is not ...

6 0 6 0 3 0 / 4 3 0 1 0 7

32 should be played 5/2 5/3

2 0 2 0 2 0 / 0 0 1 3 2 7

31 should be played 1/0ff 5/2 !!!

This is incredible, and naturally the obvious 3/Off is correct in front of almost every opposite position. Here the reason is a subsequent 42 which could possibly played 5/3/Off.

After 1/Off the non bearing off play is made with only 5 men on the board, which contradicts seriously Trice's first statement

(Note : this position has been found by ourselves many years ago).

One can also fill a gap on the 3-point :

4 1 1 6 0 2 / 0 2 2 3 4 4

The right play for 3 is 6/3, not 3/Off

 

With a 4, one can fill a gap on the 2-point :

4 0 6 1 0 2 / 0 2 3 3 2 3

The right play for 4 is 6/2, not 4/Off

 

Most of the previous positions have been found by Walter TRICE and reported in W.ROBERTIE's ADVANCED BACKGAMMON (second edition).

                                   

In fact there exist another family of positions where bearing off a checker is not optimal, and these positions may even have a higher practical interest due to frequency.

In these positions, the player on move is far behind in the race and drastically needs to roll doubles (most usually two or three doubles). These positions contain at least 8 men (10 in most cases), usually (and probably always) an even number of men, and the goal is to avoid  a critical point be deprived of bearoffable men when rolling doubles.

 

This may occur when having a 2 to play...

... from the 4-point...

1 3 2 4 0 0 / 4 3 0 0 0 0

The right play for a 2 is : 4/2 , not  2/Off

This makes 44 + 22 a winning sequence (avoiding missing with 22).

 

... from the 5-point...

1 4 3 3 1 0 / 4 3 0 0 0 0

The right play for a 2 is : 5/3 , not  2/Off

because of the same 44 + 22 (here three consecutive doubles are needed).

 

... or from the 6-point.

1 4 1 0 3 1 / 4 3 0 0 0 0

0 4 4 3 0 1 / 4 3 0 0 0 0

In both positions, the right play for a 2 is : 6/4 , not 2/Off because of 55 + 22.

 

This may also occur when having a 3 to play...

... from the 5-point...

1 3 4 3 1 0 / 4 3 0 0 0 0

The right play for a 3 is : 5/2 , not  3/Off

because of 44 + 22 (the position reached is the same as a few lines above).

 

... or from the 6-point.

0 0 3 1 3 1 / 2 2 1 0 0 0

0 0 3 8 0 1 / 4 3 0 0 0 0

In both positions, the right play for a 3 is : 6/3 , not  3/Off

because of 55 + 33.

 

At last this may occur when having a 4 to play from the 6-point.

1 3 1 1 3 1 / 4 3 0 0 0 0

The right play for a 4 is : 6/2  , not  4/Off (the position reached is again the same as one seen above).

 

We finish this chapter with TRICE's deceptive position

7 3 3 1 0 0 / 0 2 3 3 3 3

The best 2 is 4/2

According to Walter TRICE, one gains when rolling 61 or 51 or 41 followed by 22

This seems to be irrefutably logical and the position deserves a 1st price.

If we change opponent's position so that it becomes 0 14 0 0 0 0, the calculation of opponent's CPW gives :

27.740 % after 2/Off

27.581 % after 4/2

Some neighbouring positions exist where the play of the 2 depends on opponent's position.

In fact these positions belong to the first family, despite TRICE's explanation. Not only has the roller to beware of double 2s, but also of a succession of single 2s when he his far ahead. Conversely, when he his far behind, his best play is 2/Off in order to squeeze the best out of a series of two double aces !

 

 

CHAPTER 3.                        HANDLING STACKS

 

Every backgammon player has learnt to avoid stacks and this rule would seem to apply with even more strength in the bearoff.

However, during the bearoff, many exceptions must be made.

 

            3.1. Playing aces

 

A general common sense rule could be :

"When no double is needed, move a checker one pip only if, before the play, the start point is more crowded than the target point (this could be applied for all aces of double 1s)."

 

A striking counter-example is :

0 3 2 0 0 0 / 0 0 0 0 1 3

The right play for an ace is : 3/2 so that 22 becomes efficient next turn. This is surprising because opponent seems so late in the race, but in fact, the only factor favoring the play 2/1 is that it protects against three consecutive aces (it would be the right play should opponent's position be 0 0 0 0 0 4).

 

Other examples :

0 0 0 1 1 2 / 2 2 2 0 0 0

The right ace is 5/4 !

This is due to the virtual necessity to roll a double.

(Incidentally, if opponent's position is just a bit worse, the right ace is 4/3, but the natural 6/5 is never the best play)

 

0 1 2 2 2 2 / 0 0 9 0 0 0

The right 1 is 6/5, preparing 55 in addition to 66. The gap on the 6-point could prove regrettable if several 6s were rolled, but any other ace would leave an even more costly potential inside gap.

(According to Hugh SCONYERS's databases, if opponent's position is significantly worse, the right play is 3/2)

 

            3.2. Selecting the harmless gap

 

2 0 2 1 1 1 / 3 3 3 0 0 0

The right play for a 2 is : 6/4 , not  4/2 for sake of flexibility. Now two consecutive 2s are not dangerous, while two future 4s could be fatal with the wrong play.

 

            3.3. Preparing small doubles in the last turns

 

2 0 2 1 0 0 / 1 1 1 1 1 0

The right play for a 2 is : 3/1 , not  4 /2

With a man on point 4 and an odd number of checker, one has not to fear the gap on the 2-point. In fact one must take care of double aces preceding or followed by 42/52/62. This kind of positions has been known for years.

 

0 2 0 0 1 0 / 3 2 0 0 0 0

The right play for an ace is : 5/4 , not 2/1

If we knew opponent were not to roll a double in the next two rolls (probability 25/36), unstacking would be better by saving the sequence 21 + 32 (prob=1/324). However, it is better advised to buy 22 (prob=1/36) as an immediate winning roll for the case opponent rolls a double in the next two rolls (prob=11/36).

1/36 * 11/36 > 1/324 * 25/36      QED !

 

            3.4. Importance of the number of checkers

 

3 4 0 1 1 0 / 3 1 2 3 3 2 (Robertie - Barabino, WORLD CLASS BACKGAMMON move by move, FRIEDMAN, match 3 game 12, move 20)

Robertie has a 3 to play.

He plays carelessly 5/2 "clearing from the back" while 4/1 is better. Of course, Robertie's lead is tremendous : his CPW is 98.671 % after his play, 98.692 % after the correct play.

Being left with an odd number of checkers, which allows a tempo to bring to home the straggler on the 5-point, Robertie's goal must be to extract the tiniest chances provided by 11. With 5 checkers on the point 2 (i.e. with an even number of checkers), then 5/2 would be correct, in order to avoid misses !

Let's suppose a closer race, Barabino's position becoming 0 2 3 3 2 0.

Then the difference between the two plays would be slightly more significant : 27.624 % versus 27.910 % for Barabino after the plays.

 

 

CHAPTER 4.                        CURIOSITIES

 

            4.1. 66 is not always the best roll

The main reasons are flexibility and gaps.

 

                        4.1.1. 55 is better than 66

5 0 0 0 5 1 / 3 3 3 0 0 0

After 55, 42 next turn bears a man off. It is not the case after 66.

 

                        4.1.2. 44 is better than  66

1 0 0 5 1 0 / 2 2 1 0 0 0

After 44, 32 next turn bears a man off. It is not the case after 66.

 

                        4.1.3. 33 is better than 66

5 0 9 1 0 0/ 5 5 5 0 0 0

It is obvious for the exerted player that the 2-point can be filled in with a 2 next roll.

 

                        4.1.4. 22 is better than 66

You don't believe us and we suspect you are right : such a position is unimaginable.

 

                        4.1.5. 11 is better than 66

You believe us even less, but this time we are not sure you are right !

At least, we don't  know such positions in the backgammon the way it is played on Earth.

See Chapter 5.

 

            4.2. Zugzwang

This german word, well known by chess players, means : obligation to play.

This means that it would be better to stand on a position than to play a number.

Very common in middle game, is this situation possible in backgammon bearoff ?

 

0 0 2 0 4 1 / 2 2 1 0 0 0

You have to play an ace.

If you move 6/5 or 3/2 , you lose immediately your last chances provided by 33.

If you move 5/4 , you lose the chances provided by 55 followed by  33.

You can easily check that no compensation exists.

As a consequence, in the position 1 1 2 0 4 1 / 2 2 1 0 0 0 , 12 is a better roll than 11 !

 

 

- 5 -            BEAROFF IN THE SEVENTH DIMENSION

 

We can get new curiosities by extending the dimensions of the game : number of checkers, figurs on the dice, dimensions of the inner boards.

 

            5.1. With 7 figurs dice and 7 arrows inner boards

 

The "booster brakes" manoeuvre of chapter 1 can now be executed successfully with a 1.

0 0 2 0 4 1 1/ 2 2 1 0 0 0 0

The correct play for 17 is : 7/6/Off , not  7/Off and what ????

One can easily recognize the zugzwang position shown in 4.2.

 

            5.2. With more than 15 checkers on each side

 

Under these conditions, 11 might be better than 66 !!!

Since we do not have at our disposal a software capable to check such a statement, this represents only a suggestion from our part.

The position we propose is :

89 0 8 4 0 0 / 99 0 0 0 0 0

or some similar position with an even larger number of checkers.

66 leads to : 89 0 8 0 0 0

11 leads to : 85 0 8 4 0 0

We believe that it may be better to be protected against two future 2s than to win 9 pips in the race.

The reason is there are enough checkers on the 1-point to play conveniently all 1s by bearing checkers off therefore buying the time to roll enough 3s, 4s, 5s, 6s to bear the checkers off from the 3-point. So that only 2s need care.

However there is no certainty about this without a checking calculation.

Your are challenged to validate or invalidate this open assumption !

 

 

                                                            GIF SUR YVETTE - January 6th, 1995

                                                            Revised, 1999, 2003

                                                           

                                                            Jean-Luc SERET - Bernard BIGOT