Initial position
Figure 1. The gaming board with checkers in initial position.
NB: Initial position symmetrically mirrored to the represented one – when home is situated to the left and yard to the right – is also possible.
Backgammon is basically a game for two players. The gaming board consists of 24 triangles, called points. The triangles have alternating colors (usually black and white) and form 4 groups of 6 triangles each (see Figure 1).
These groups are called home, yard, opponent’s yard and opponent’s home.
Points (triangles) are numbered for each player in a separate way, starting with home of the corresponding player. The most distanced point is – 24, which is also the first point for the opponent.
Each player has 15 checkers of one colour (usually, black or white), whose initial position is represented in Figure 1, a couple of dice and a special dice cup, where the dice are shaken (it is possible to use only one pair of dice).
Goal of the game
Goal of the game consists in the following: each player must move their checkers into their home boards and then to bear them off on the board. The player who bears off all of his/her checkers first, wins the game.
Making a move
Figure 2. Move of white checkers (black checkers shall be moved in the opposite direction).
At the beginning of the game, both players throw a single die. The player throwing the higher number moves first (if the same number comes up on both dice, the players re-roll until they are different). The player, getting the higher number of points, moves his/her checkers according to the number of points from both dice.
The checkers can be moved only in one direction (from the points with higher numbers towards those with lower numbers – see Figure 2).
After that, each player throws both dice. The checkers move according to the numbers shown on the dice: one checker can be moved the full amount or each of the two numbers can be moved with separate checkers – one checker moves according to the number of points shown on one die and another – according to the number shown on the second die. For instance, if in the initial position (see Figure 1) white checkers start the game with the move [4*2], the following moves could be made:
- 8 → 4, 6 → 4, i.e. a checker from point 8 and a checker from point 6 move to point 4 (the checker from 8 moves by 4 spaces, whereas the one from 6 moves by 2 spaces).
- 13 → 7, i.e. the checker from point 13 moves by 6 spaces (4+2) to point 7.
- 24 → 18, 8 → 2, similar with the previous point.
There are, however, some prohibitions. If there are at least two opponent’s checkers on the same point, your own checker cannot move to this point (possible only if your score allows that, but over that point). For instance, in the initial position, upon obtaining the combination [3*2], white checkers cannot move 24 → 19 or 6 → 1, whereas 13 → 8 or 13 → 11 and 13 → 10 are allowed to move.
Figure 3.
If an opponent’s checker landed on the point and combination of dice allows the move of your checker (or several checkers) to the point occupied by a single checker, the opponent’s checker can be pulled out of the game, if the leading player wishes to do that. The opponent may introduce it back into the game only upon his lead, starting from the opponent’s home, if the numbers shown on the dice allow him to obtain a certain point in the opponent’s home. For instance, if in the indicated position (see Figure 3) white checkers get [3*1] on dice, they can pull out of the game a black checker by the move 13 → 9, which enters back into the game upon the move of the black checker, when it gets number 2 or number 5 on one of the dice (for instance, upon [2*6], the opponent introduces a pulled out checker onto point 2 and makes another move by 6 spaces by means of another checker (18 or 14), at [2*5] the black checkers introduce a pulled out checker either to point 2 or point 5 and afterwards score the remaining points); if the black checkers score neither 2 nor 5, the move is taken over by the white checkers.
If a double is shown on the dice (same numbers on both dice, for instance, [1*1] or [2*2], …, [6*6]), the score of points is doubled, i.e. upon [4*4] white checkers in the position indicated in Figure 3 may play by 2 checkers 13 → 9, 9 → 5, pulling out of the game the black checker 9 or may move by 16 spaces one or several checkers, for instance 13 → 9 by one checker and 6 → 2 by 3 checkers (NB a weak move in this position!)
Figure 4.
Once the player introduced all of his/her checkers into his/her own home, he/she starts to bear off his/her checkers from the gaming board according to the number of points shown on the dice. For instance, in the position indicated in Figure 4, white checkers (see Figure 4)
- at [6*4] move, pull out a checker from point 6 and another one from point 4 (or pull out a checker from point 6 and make a 6 → 2 move, which is a weaker move!).
- at [1*1] move, four checkers can be pulled out from point 1 or a weaker move can be chosen, i.e. other checkers can move by four spaces: 6 → 5 can move by 3 spaces and pull out a checker from point 1, or 4 → 3 by 4 spaces (3, 2, 1), etc.
- at [5*3] move, white checkers cannot pull out any checker, they can move by a checker 6 → 1 (in a forced manner) and by another one 6 → 3 or 4 → 1 (other moves are not possible in this situation).
The black checkers in the same position:
- at [3*3] move two checkers can be pulled out from point 22 and moved by 2 checkers 21 → 24 or a weaker move can be chosen, i.e. can move by three checkers 21 → 24 and pull out a checker from point 22.
- At [6*3] move, a checker can be pulled out from point 21 (the closest one to 6 points) and a checker from point 22 or one checker can move 21 → 24.
- At [5*5] move pull out at 3 checkers from point 21 and one checker from point 22.
If a player is able to make a move, he/she should do that. If points can be realized either from 1 die or another (but not both simultaneously), a maximal move takes place.
The game is over once one of the players (the winner) bears off all of his/her checkers out of his/her home, whereas another player (the beaten one) still has some checkers on the gaming board.
Dave, Mart and Koks
Usually backgammon is played for money, a fixed stake having been set up for one won point. Each game starts with a stake fixed at 1 point. If during the game a player deems that he/she has a greater chance to win the game, he/she can propose to double the stake by announcing Dave (i.e. Double). The opponent may accept this and continue the game (only him/her being able to double the stake from now and on) or may not accept this (in this case he/she loses the game with an initial stake). The Double Number is basically not limited. The Double shall be announced before the dice are thrown upon own lead.
If a player throws all of his/her checkers, whereas an opponent did not any of them, he/she loses with Mart and therefore loses a double stake. Moreover, if a beaten player has his/her checkers in the winner’s home or has checkers, which should be introduced back into the game from the winner’s home, he loses Koks (or Royal Mart) and thus loses a triple stake.
Probabilities
[1*1] [2*1] [3*1] [4*1] [5*1] [6*1]
[1*2] [2*2] [3*2] [4*2] [5*2] [6*2]
[1*3] [2*3] [3*3] [4*3] [5*3] [6*3]
[1*4] [2*4] [3*4] [4*4] [5*4] [6*4]
[1*5] [2*5] [3*5] [4*5] [5*5] [6*5]
[1*6] [2*6] [3*6] [4*6] [5*6] [6*6]
Table 1.
Upon throwing the dice, only one out of those 36 variants, indicated in the Table 1, is possible. This means that the probability of each variant equals to 1/36, which is to be interpreted as follows: at long series of throws each variant appears (in average) once per 36 throws and the greater the number is – the ratio between the number of realization of the corresponding variant to the total number of throws approaches to 1/36.
Attention! The interpretation that, for instance, variant [1*1] is possible once per 36 throws in a forced manner, is erroneous!!!
| Sum of points | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Probabilities | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |
Conclusions: The most frequent sum of points obtained on both dice equals to 7, followed by 6 and 8, then 4 and 9, 3 and 11, whereas the most rare appearance is that of double 1 (2 points) and double 6 (12 points). Information provided in the Table 1 allows to minimize the risk of pulling out your checkers by the opponent with no cover.
Doubles: All doubles have the same probability: 1/36
Numbers: All numbers have the same probability of appearance: 1/3 (both dice are thrown!)