a circle in the European Russian government of Catherinoslav. It is very extensive, including within it a very large portion of the steppes, which are destitute of wood and scantily supplied with water, but feed great numbers of sheep and oxen. The chief town of the same name, contains about 5000 inhabitants, who are great dealers in and breeders of horses. Long. 38. 3. E. Lat. 48. 32. N.

BACKGAMMON, an ingenious game played with dice, upon a table, by two persons. The table is divided into two parts, upon which there are twenty-four black and white spaces, called points. Each adversary has fifteen men, black and white, to distinguish them, and they are disposed of in the following manner: Supposing the game to be played in the right-hand table, two are placed upon the ace point in the adversary's table, five upon the six point in the opposite table, three upon the cinque point in the hitherto table, and five on the six point in the right-hand table. The grand object in this game is for each player to bring the men round into his right-hand table, by throwing with a pair of dice those throws that contribute towards it, and at the same time to prevent the adversary doing the like. The first best throw upon the dice is esteemed aces, because it stops the six point in the outer table, and secures the cinque in the thrower's table; so that the adversary's two men upon the thrower's ace point cannot get out with either quatre, cinque, or six. This throw is an advantage often given to the antagonist by the superior player. When he carries his men home in order to lose no point, he is to carry the most distant man to his adversary's bar point, that being the first stage he is to place it on: the next is six points farther, viz. in the place where the adversary's five men are first placed out of his tables; and he must go on in this method till all his men are brought home except two, when by losing a point he may often save the gammon, by throwing two fours or two fives. When a hit is only played for, he should endeavour to gain either his own or adversary's cinque point; and if that fails by his being hit by the adversary, and he finds him farther advanced than himself; in that case he must throw more men into the adversary's tables, which is done in this manner: He must put a man upon his cinque or bar point; and if the adversary neglects to hit it, he may then gain a forward game instead of a back game. But if the adversary hits him, he should play for a back game; and then the greater number of men which are taken up makes his game the better, because by these means he will preserve his game at home. He should then endeavour to gain both his adversary's ace and trois points, or his ace and deuce points, and take care to keep three men upon the adversary's ace point, that in case the latter hits him from thence, that point may remain still secure to himself. A back game should not be played for at the beginning of a set, because it would be a great disadvantage, the player running the risk of a gammon to win a single hit. The subsequent calculations will show the odds of entering a single man upon any certain number of points. Upon two dice there are thirty-six chances, and upon these chances the following points, viz.

| Points | |--------| | 2 Aces | 4 | | 2 Deuces | 8 | | 2 Trois | 12 | | 2 Fours | 16 | | 2 Fives | 20 | | 2 Sixes | 24 | | 6 and 5 twice | 22 | | 6 and 4 twice | 20 | | 6 and 3 twice | 18 | | 6 and 2 twice | 16 | | 6 and 1 twice | 14 | | 5 and 4 twice | 18 | | 5 and 3 twice | 16 | | 5 and 2 twice | 14 | | 5 and 1 twice | 12 | | 4 and 3 twice | 14 | | 4 and 2 twice | 12 | | 4 and 1 twice | 10 | | 3 and 2 twice | 8 | | 3 and 1 twice | 6 |

Divide by 36

This proves, that upon an average the player has a right to 8 points each throw.

The chances upon two dice calculated for backgammon are as follow:

| Chances | |---------| | 2 Sixes | 1 | | 2 Fives | 1 | | 2 Fours | 1 | | 2 Trois | 1 | | 2 Deuces | 1 | | 2 Aces | 1 | | 6 and 5 twice | 2 | | 6 and 4 twice | 2 | | 6 and 3 twice | 2 | | 6 and 2 twice | 2 | | 6 and 1 twice | 2 | | 5 and 4 twice | 2 | | 5 and 3 twice | 2 | | 5 and 2 twice | 2 | | 5 and 1 twice | 2 | | 4 and 3 twice | 2 | | 4 and 2 twice | 2 | | 4 and 1 twice | 2 | | 3 and 2 twice | 2 | | 3 and 1 twice | 2 | | 2 and 1 twice | 2 |

Total, 11

When deducted from 36

There remains 25

From which it appears that it is 25 to 11 against hitting an ace upon a certain or flat die. And this method holds good with respect to any other flat die. For example, what are the odds of entering a man upon 1, 2, 3, 4, or 5 points?

Answer.

To enter it upon for against for agt. 1 point is 11 to 25 or about 4 to 9 2 ... 20 ... 16 ... 5 ... 4 3 ... 27 ... 9 ... 3 ... 1 4 ... 32 ... 4 ... 8 ... 1 5 ... 35 ... 1 ... 35 ... 1

The following table shows the odds of hitting with any chance, in the reach of a single die.

To hit upon for against for agt. 1 is 11 to 25 or about 4 to 9 2 ... 12 ... 24 ... 1 ... 2 3 ... 14 ... 22 ... 2 ... 3 4 ... 15 ... 21 ... 5 ... 7 5 ... 15 ... 21 ... 5 ... 7 6 ... 17 ... 19 ... 8½ ... 9½

The odds of hitting with double dice are as follow:

To hit upon for against for agt. 7 is 6 to 30 or about 1 to 5 8 ... 6 ... 30 ... 1 ... 5 9 ... 5 ... 31 ... 1 ... 6 10 ... 3 ... 33 ... 1 ... 11 11 ... 2 ... 34 ... 1 ... 17 12 ... 1 ... 36 ... 1 ... 35 How to find out the odds of being hit upon a six, by the table of thirty-six chances.

| Sixes | 1 | |-------|---| | Trois | 1 | | Deuces | 1 | | 6 and 5 twice | 2 | | 6 and 4 twice | 2 | | 6 and 3 twice | 2 | | 6 and 2 twice | 2 | | 6 and 1 twice | 2 | | 5 and 1 twice | 2 | | 4 and 2 twice | 2 |

Which deducted from 36

There remains 19

By which it appears to be 19 to 17 against being hit upon a six.

The odds on the hits.

- 2 love is about 5 to 2 - 2 to 1 is 2 to 1 - 1 love is 3 to 2

If a player has taken up two of the adversary's men, and happens to have two, three, or more points made in his own tables, he should spread his men, that he may either take a new point in his tables, or be ready to hit the man which the adversary may happen to enter. If he finds, upon the adversary's entering, that the game is upon a par, or that the advantage is on his own side, he should take the adversary's man up whenever he can, it being 25 to 11 that he is not hit; except when he is playing for a single hit only, because then, if playing the throw otherwise gives him a better chance for it, he ought to do it.

It being five to one against his being hit with double dice, he should never be deterred from taking up any one man of the adversary's.

If he has taken up one of the adversary's men, and happens to have five points in his own tables, yet is forced to leave a blot out of his tables, he should endeavour to leave it upon doublets in preference to any other chance, because in that case the odds are 35 to 1 that he is not hit; whereas it is only 17 to 1 that he is hit upon any other chance.

When the adversary is very forward, a player should never move a man from his own quatre, trois, or deuce points, thinking to bear that man from the point where he put it, as nothing but high doublets can give him any chance for the hit. Instead of playing an ace or a deuce from any of those points, he should play them from his own six or highest points, so that, throwing two fives or two fours, his six and cinque points being eased, would be a considerable advantage to him; whereas if they had been loaded he would have been obliged to play otherwise.

It is the interest of the adversary to take up the player as soon as he enters. The blot should be left upon the adversary's lowest point; that is to say, upon his deuce point rather than upon his trois point, or upon his trois point rather than his quatre point, or upon his quatre point preferably to his cinque point, for a reason before mentioned. All the men the adversary plays upon his trois or his deuce point are deemed lost, being greatly out of play; so that those men not having it in their power to make his cinque point, and his game being crowded in one place and open in another, the adversary must be greatly annoyed by the player.

If the player has two of the adversary's men in his tables, he has a better chance for a hit than if he had more, provided his game is more forward than that of his antagonist; for if he had three or more of the adversary's men in his tables, he would stand a worse chance to be hit.

When a player is running to save the gammon, if he should have two men upon his ace point and several men abroad, although he should lose a point or two in putting his men into his tables, it is his interest to leave a man upon the adversary's ace point, because it will prevent his adversary from bearing his men to the greatest advantage, and at the same time the player will have a chance of the adversary's making a blot, which he may chance to hit. However, if a player finds upon a throw that he has a probability of saving his gammon, he should never wait for a blot, as the odds are greatly against his hitting it, but at once embrace that opportunity to escape.

Suppose the adversary has so many men abroad as to require three throws to put them into his tables, and at the same time that the player's tables are made up, and that he has taken up one of the adversary's men; in this case it is about an equal wager that the adversary is gammoned; for in all probability the player has borne two men before he opens his tables, and when he bears the third man he will be obliged to open his six or cinque point. It is then probable that the adversary is obliged to throw twice before he enters his men in the player's tables, twice more before he puts that man into his own tables, and three throws more to put the men which are abroad into his own tables; in all seven throws. Now the player having twelve men to bear, he may be forced to make an ace or a deuce twice before he can bear all his men, and consequently will require seven throws in bearing them; so that, upon the whole, it is about equal whether the adversary is gammoned or not.

Suppose a player has three men upon his adversary's ace point and five points in his own tables, and that the adversary has all his men in his tables, three upon each of his five highest points; has the player a probability of gammoning his adversary or not?

For bearing three men from his 6th point is 18 From his 5th point 15 From his 4th point 12 From his 3d point 9 From his 2d point 6

In all 60

Bringing his three men from the adversary's ace point to his six point in his own tables, being 18 points each, and making together 54

There must remain 6

From which it is plain that the player has much the best of the probability of the gammon; exclusive of one or more blots which the adversary is liable to make in bearing his men, and supposing at the same time the throws to be upon an equality. But suppose two blots are left, either of which cannot be hit but by double dice; one must be hit by throwing eight, and the other by throwing nine; so that the adversary has only one die to hit either of them. What, then, are the odds of hitting either of them?

The chances of two dice being in all 36 The chances to hit 8 or 6 and 2 twice 2 5 and 3 twice 2 Deuces 1 Fours 1 The chances to hit 9 or 6 and 3 twice 2 5 and 4 twice 2 Trois 1

For hitting in all 11 Chances for not hitting remain 25

So that the odds are 25 to 11 against hitting either of these blots. This method may be taken to find out the odds of hitting three, four, or five blots upon double dice, or blots made upon double and single dice at the same time. After knowing how many chances there are to hit any of these blots, they must be added all together, and then subtracted from the number 36, which are the chances of the two dice; and the question is solved.

The laws of backgammon are as follow: 1. If a man is taken from any point, it must be played; if two men are taken from it, they also must be played. 2. A man is not supposed to be played till it is placed upon a point and quitted. 3. If a player has only fourteen men in play, there is no penalty inflicted, because by his playing with a lesser number than he is entitled to, he plays to a disadvantage for want of the deficient man to make up his tables. 4. If he bears any number of men before he has entered a man taken up, and which of course he was obliged to enter, such men so borne must be entered again in the adversary's tables as well as the man taken up. 5. If he has mistaken his throw and played it, and his adversary has thrown, it is not in the choice of either of the players to alter it, unless they both agree so to do.