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 inpractical 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 straight forwardly 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).
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 event sand 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.
Five different plays for a same bearoff position is really amazing. Note however how many legal moves are never optimal.
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.
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 playfor 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 beplayed 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 bedeprived of bearoffable men when rolling doubles.
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 ofopponent's CPW gives :
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 aseries of two double aces !
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.
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)."
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).
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)
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.
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).
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 a3 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 hisplay, 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 evennumber 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.
You believe useven 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.
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 ?
We can get new curiosities by extending the dimensions of the game: number of checkers, figurs on the dice, dimensions of the inner boards.
Under these conditions, 11 might be better than 66 !!! Since we donot 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 :
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 bearthe 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 !