INSIDE BEAROFF
------------
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