Book Transcription

Chap. I.

Because it is necessary for a Learner to know, how many Points he ought to throw upon the two Dice, one Throw with another; we shall take the following Method to demonstrate it.

Example. I would know how many Chances there are upon two Dice?

The answer is thirty-six.

I would also know how many Points there are upon the thirty-six Chances?

The Answer to which take as follows:

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		10
3 and 1 twice		8
2 and 1 twice		6
Divided by 36  )		294 288	8
		6

294 divided by 36, solves the Question; by which it appears, that one Throw with another you may expect to throw 8 upon 2 Dice.

I would know how many Chances there are upon 2 Dice?

The Answer is 36, which are as follows:

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
		36

Because a Learner may be at a Loss to find out by this Table of 36 Chances, what are the Odds of being hit, upon a certain, or flat Die, let him take the following Method.

Example. To know the Odds of being hit upon an Ace?

Look in the Table, where you will find thus * marked,

Chances.
*	2 Aces	1
*	6 and 1 twice	2
*	5 and 1 twice	2
*	4 and 1 twice	2
*	3 and 1 twice	2
*	2 and 1 twice	2
	Total	11
	Which deducted from	36
	The Remainder is	25

By this Method it appears, that it is 25 to 11 against hitting an Ace, upon a certain, or flat Die.

The like Method may be taken with any other flat Die, as you have seen with the Ace.

I would know what are the Odds of entering a Man upon 1, 2, 3, 4, or 5 Points?

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

I would know what are the Odds of hitting, with any Chance, in the Reach of a single Die?

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

I would know what are the Odds of hitting with double Dice? which are as follows:

Answer.							Reduced.
			for.		agt.			for.		agt.
To hit		upon 7 is upon 8 upon 9 upon 10 upon 11 upon 12 (or 2 6’s)	6 6 5 3 2 1	to	30 30 31 33 34 36		or about	1 1 1 1 1 1	to	5 5 6 11 17 35

To explain further to a Learner how to make use of the Table of 36 Chances, when at a loss to find the Odds of being hit upon any certain, or flat Die, this second Example is here added to shew how to find by that Table the Odds of being hit upon a 6.

2 Sixes	1
2 Trois	1
2 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
	17
Which deducted from	36
The Remainder is	19

By the foregoing Example it is evident, that it is 19 to 17 against being hit upon a 6.

The Odds of	2 Love	is about	5	to	2,
and of	2 to 1	is	2		1,
and of	1 Love	is	3		2.

Chap. II.

1. If you play 3 up at Back-Gammon, your principal View, in the first place, is, either to secure your own, or your Adversary’s Cinq. Point; when that is effected, you may play a pushing Game, and endeavour to gammon your Adversary.

2. The next best Point (after you have gain’d your Cinq. Point) is to make your Barr Point, thereby preventing your Adversary’s running with 2 Sixes.

3. After you have proceeded thus far, you are, in the next place, to prefer the making your Quatre Point in your own Tables, rather than the Quatre Point out of them.

4. Having gained these Points, you have a fair Chance to gammon your Adversary, if he is very forward: For, suppose his Tables are broke at home, it will be then your Interest to open your Barr Point, and to oblige him to come out of your Tables with a 6; and having your Men spread, you not only may catch that Man which your Adversary brings out of your Tables, but you will also have a Probability of taking up the Man left in your Tables, (upon Supposition that he had two Men there.) And suppose he should have a Blot at home, it will then be your Interest not to make up your Tables; because, if he should enter upon a Blot, which you are to make for the Purpose, you will have a Probability of getting a third Man; which, if accomplished, will give you, at least, 4 to 1 of the Gammon; whereas, if you have only two of his Men up, the Odds are in his Favour that you do not gammon him.

5. If you play for a Hit only, 1 or 2 Men taken up of your Adversary’s, makes it surer than a greater Number, provided that your Tables are made up.

6. Directions how to carry your Men home.

   When you carry your Men home, in order to lose no Point, you are to carry the most distant Man to your Adversary’s Barr Point, that being the first Stage you are to place it on; the next Stage is 6 Points farther, viz. in the Place where your Adversary’s 5 Men are first placed out of his Tables; the next Stage is upon the six Point in your Tables. This Method is to be pursued till all your Men are brought home, except 2, when, by losing a Point, you may often save your Gammon, by putting it in the Power of 2 Fives, or 2 Fours to save it.

7. If you play to win a Hit only, you are to endeavour to gain either your own or your Adversary’s Cinque Point; and if that fails, by your being hit by your Adversary, and you find that he is forwarder than you, in that Case, you must throw more Men into his Tables. The manner of doing it is thus: Put a Man upon your Cinque or Barr Point, and if your Adversary neglects to hit it, you may then gain a forward Game instead of a back Game; but if he hits you, you must play for a back Game, and then the greater Number of Men which are taken up, makes your Game the better, because you will, by that Means, preserve your Game at home; and you must then always endeavour to gain both your Adversary’s Ace and Trois Points, or his Ace and Deuce Points, and take Care to keep 3 Men upon his Ace Point, that, if you chance to hit him from thence, that Point may remain still secure to you.

8. At the Beginning of a Set do not play for a back Game; because by so doing, you would play to a great Disadvantage, running the Risk of a Gammon to win a single Hit.

Chap. III.

Directions for playing at setting out the 36 Chances of Dice, when you are to play for a Gammon, or for a single Hit.

1. Two Aces, to be play’d on your Cinq. Point, and Barr Point, for a Gammon, or for a Hit.

2. Two Sixes, to be play’d on your Adversary’s Barr Point, and on your own Barr Point, for a Gammon, or for a Hit.

3. *  Two Trois; two to be played on your Cinq. Point, and the other two on your Trois Point in your own Tables, for a Gammon only.

4. †  Two Deuces, to be played on your Quatre Point in your own Tables, and two to be brought over from the five Men placed in your Adversary’s Tables, for a Gammon only.

5. ‡  Two Fours, to be brought over from the five Men placed in your Adversary’s Tables, and to be put upon the Cinq. Point in your own Tables, for a Gammon only.

6. Two Fives, to be brought over from the five Men placed in your Adversary’s Tables, and to be put on the Trois Point in your own Tables, for a Gammon, or for a Hit.

7. Size Ace, you are to take your Barr Point, for a Gammon, or for a Hit.

8. Size Deuce, a Man to be brought from the five Men placed in your Adversary’s Tables, and to be placed on the Cinq. Point in your own Tables, for a Gammon, or for a Hit.

9. Six and three, a Man to be brought from your Adversary’s Ace Point, as far as he will go, for a Gammon, or for a Hit.

10. Six and four, a Man to be brought from your Adversary’s Ace Point, as far as he will go, for a Gammon, or for a Hit.

11. Six and Five, a Man to be carried from your Adversary’s Ace Point as far as he can go, for a Gammon, or for a Hit.

12. Cinq. and Quatre, a Man to be carry’d from your Adversary’s Ace Point, as far as he can go, for a Gammon, or for a Hit.

13. Cinq. Trois, to make the Trois Point in your Table, for a Gammon, or for a Hit.

14. Cinq. Deuce, to play two Men from the 5 placed in your Adversary’s Tables, for a Gammon, or for a Hit.

15. *  Cinq. Ace, to bring one Man from the five placed in your Adversary’s Tables for the Cinq. and to play one Man down on the Cinq. Point in your own Tables for the Ace, for a Gammon only.

16. Quatre Trois, two Men to be brought from the five placed in your Adversary’s Tables, for a Gammon, or for a Hit.

17. Quatre Deuce, to make the Quatre Point in your own Tables, for a Gammon, or for a Hit.

18. †  Quatre Ace, to play a Man from the 5 placed in your Adversary’s Tables for the Quatre, and for the Ace, to play a Man down upon the Cinq. Point in your own Tables, for a Gammon only.

19. Trois Deuce, two Men to be brought from the five placed in your Adversary’s Tables, for a Gammon only.

20. Trois Ace, to make the Cinq. Point in your own Tables, for a Gammon, or for a Hit.

21. *  Deuce Ace, to play one Man from the five Men placed in your Adversary’s Tables for the Deuce; and for the Ace, to play a Man down upon the Cinq. Point in your own Tables, for a Gammon only.

Chap. IV.

Directions how to play the Chances that are mark’d thus (*) when you are only to play for a Hit.

1. *  Two Trois, two of them are to be played on your Cinq. Point in your own Tables, and with the other Two you are to take the Quatre Point in your Adversary’s Tables.

2. †  Two Deuces, two of them are to be played on your Quatre Point in your own Tables, and with the other two you are to take the Trois Point in your Adversary’s Tables.

The two foregoing Cases are to be played in this manner, for this Reason, viz. That thereby you avoid being shut up in your Adversary’s Tables, and have the Chance of throwing high Doublets to win the Hit.

3. ‡  Two Fours, two of them are to take your Adversary’s Cinq. Point in his Tables; and for the other two, two Men are to be brought from the five placed in your Adversary’s Tables.

4. 1. *  Cinq. Ace, play the Cinq. from the five Men placed in your Adversary’s Tables, and play the Ace from your Adversary’s Ace Point.

5. 2. *  Quatre Ace, play the Quatre from the five Men placed in your Adversary’s Tables, and play the Ace from the Men on your Adversary’s Ace Point.

6. 3. *  Deuce Ace, play the Deuce from the five Men placed in your Adversary’s Tables, and play the Ace from your Adversary’s Ace Point.

N. B. The three last Chances are play’d in this manner, for the following Reason: By laying an Ace down in your Adversary’s Tables, you have a Probability of throwing Deuce Ace, Trois Deuce, Quatre Trois, or Size Cinq. in two or three Throws; in any of which Cases you are to take a Point, which gives you vastly the better of the Hit.

You may observe, by the Directions given in this Chapter, that you are to play 9 Chances out of the 36 in a different manner for a single Hit, to what you would do when playing for a Gammon.

In this, and the foregoing Chapter, plain Directions are given, how, at setting out, to play all the Chances on the Dice, either for a Gammon, or a single Hit; but it is impossible to give full Instructions for the whole Progress of the Game; all that can be done towards helping a Learner, is to communicate some Observations, Hints, and Cautions, which if attended to, may be of Use to him.

Chap. V.

Some Observations, Hints, and Cautions, which are to be attended to.

1. By the Directions given to play for a Gammon, you are voluntarily to make some Blots, the Odds being in your Favour, that they are not hit; but, should it so happen, that any Blot is hit, as in this Case you will have three Men in your Adversary’s Tables, you must then endeavour to secure your Adversary’s Cinque, Quatre, or Trois Point, to prevent a Gammon, and must be very cautious how you suffer your Adversary to take up a fourth Man.

2. Take care not to crowd your Game at any time, if possible. What is meant by crowding a Game, is the putting many Men either upon your Trois or Deuce Point in your own Tables; which is, in effect, losing of those Men, not having them in Play.

   Besides, by crowding your Game, to attempt to save a Gammon, you are often gammoned; because when your Adversary finds your Game open, by being crowded in your own Tables, he may then play his Game as he thinks fit.

3. By Recourse had to the Calculations, you may know what are the Odds of your entering a single Man upon any certain Number of Points, and by that means you may play your Game accordingly.

4. If you are obliged to leave a Blot, by Recourse had to the Calculations for hitting it, you will find the Chances for and against you; and consequently you will be enabled to judge how to play your Game to the greatest Advantage.

5. You will also find by the Calculations, the Odds for and against you, upon being hit by double Dice, and consequently you will have it in your Power to chuse such a Method of Play as is most to your Advantage.

6. If it is necessary to make a Run in order to win a Hit, and you would know to a Point which is forwardest, your Adversary or you, take the following Method:

   Begin with reckoning how many Points you must have to bring home to your Size Point in your own Tables, the Man that is at the greatest Distance from it, and do the like by every other Man that is abroad; when the Numbers of those Absentees are summed up, add to them the following Numbers for those already in your own Tables, (supposing the Men that were abroad as on your Six Point for bearing) namely, six for every Man on the Size Point, five for every Man on the Cinque Point, four for every Man on the Quatre Point, three for every Man on the Trois point, two for every Man on the Deuce Point, and one for every Man on the Ace Point. Do the like to your Adversary’s Game, and then you will know which of you is forwardest, and likeliest to win the Hit.

Chap. VI.

Observations and Directions for a Learner that has made some Progress at Back-Gammon; particular Directions for bearing their Men.

1. If your Adversary is greatly before you, never play a Man from your Quatre, Trois, or Deuce Points, in order to bear that Man from the Point were you put it, because that nothing but high Doublets can give you any Chance for the Hit; therefore, instead of playing an Ace or a Deuce from any of the aforesaid Points, always play them from your Size or highest Point; by which means you will find, that throwing two fives, or two fours, will upon having eased your Size and Cinque Points, be of great Advantage to you: Whereas, had your Size Point remained loaded, you must, perhaps, be obliged to play at length those Fives, and Fours.

2. Whenever you have taken up two of your Adversary’s Men, and that you happen to have two, three, or more Points made in your own Tables, never fail spreading your Men, in order either to take a new Point in your Tables, or to be ready to hit the Man your Adversary may happen to enter. As soon as he enters one of his Men, you are to compare his Game with your’s; and if you find your Game equal to his, or better, never fail taking his Man up, if you can, because it is 25 to 11 against his hitting you; which Chance, being so much in your Favour, you ought always to run that Risk, when you have already two of his Men up.

   There is this Exception to this Rule, that if you play for a single Hit only, and that your playing that Throw otherwise, gives you a better Chance for the Hit, you ought not to take up that Man.

3. Never be deterred from taking up any one Man of your Adversary’s by the Apprehension of his hitting you with double Dice, because the fairest Probability your Adversary has of hitting you, is five to one against him.

4. If you should happen to have five points in your Tables, and to have taken up one of your Adversary’s Men, and are obliged to leave a Blot out of your Tables, take care, if it is in your Power, rather to leave it upon Doublets, than any other Chance, because Doublets are 35 to 1 against his hitting you, and any other Chances if but 17 to 1 against him.

5. Two of your Adversary’s Men in your Tables are better for a Hit, than any greater Number, provided you Game is forwardest, because his having three or more Men in your Tables, gives him more Chances to hit you, than if he had only two Men in them.

6. If you are to leave a Blot, upon your entering of a Man in your Adversary’s Tables, or otherwise, and have it in your Choice to leave it upon what Point you please, always chuse that which is the most disadvantageous to him. To illustrate this by an Example, let us suppose it is his Interest to hit you or take you up a soon as you enter, in that Case you are to leave the Blot upon his lowest Point; that is to say, upon his Deuce Point, rather than upon his Trois Point, or upon his Trois Point preferably to his Quatre Point, or upon his Quatre Point preferable to his Cinque Point, because, (as has been mentioned before) all the Men your Adversary plays upon his Trois, or his Deuce Points, are deemed as lost, being in a great Measure out of Play, those Men not having it in their Power to make his Cinque Point, and consequently his Game will be crowded there and open elsewhere, whereby you will be able also much to annoy him.

7. To prevent your Adversary from bearing his Men to the greatest Advantage, when you are running to save your Gammon; as, for instance, suppose you should have two Men upon his Ace Point, and several other Men abroad, tho’ you should lose one Point or two in putting your Men into your Tables, yet it is your Interest to leave a Man upon your Adversary’s Ace Point, which will have this Consequence, that it will prevent his bearing his Men to his greatest Advantage, and will also give you the Chance of his making a Blot, which you may chance to hit. But if, upon a Calculation, you find that you have a Throw, or a Probability of saving your Gammon, never wait for a Blot, because the Odds are greatly against hitting it.

Chap. VII.

Cases, put by way of Examples, to show how to calculate the Odds of saving or winning a Gammon.

1. Suppose your Tables are made up, and that you have taken up one of your Adversary’s Men; and suppose your Adversary has so many Men abroad as require three Throws to put them in his Tables;

   Query, Whether you have the better of a Gammon, or not?

   Answer, It is about an equal Wager that you Gammon him.

   Because, in all Probability, you will have bore two Men before you open your Tables, and when you bear the third Man, you will be obliged to open your Size or Cinque Point; in that Case it is probable, that your Adversary must take two Throws before he enters his Man in your Tables, and two Throws more before he puts that Man into his own Tables, and three Throws more to put into his own Tables the Men which he has abroad, which in all make seven Throws: And as you have twelve Men to bear, which probably will take seven Throws in bearing, because you may twice be obliged to make an Ace, or a Deuce, before you can bear all your Men.

   N. B. No mention is made of Doublets of either Side, that Event being equal to each Party.

   The foregoing Case being duly attended to, shews it is in your Power to calculate very nearly the Odds of saving or winning a Gammon upon most Occasions.

2. Suppose I have three Men upon my Adversary’s Ace Point, and five Points in my Tables, and that my Adversary has all his Men in his Tables, three upon each of his five highest Points;

   Query, Whether the Probability is for his gammoning me, or not?

   Answer.		Points.
   For his bearing 3 Men	from his 6 Point, is		18
   	from his 5 Point,		15
   	from his 4 Point,		12
   	from his 3 Point,		9
   	from his 2 Point,		6
   	Total		60
   To bring my 3 Men from my Adversary’s Ace Point, to my Size Point in my Tables, being for each 18 Points, make in all			54
   	The Remainder is		6

   And as besides the six Points in your Favour, there is a further Consideration to be added for you, which is, that your Adversary may make one or two Blots in bearing, as is frequently the Case: You see by this Calculation, that you have greatly the better of the Probability of saving your Gammon.

   N. B. This Case is supposed upon an Equality of throwing.

3. Suppose I leave two Blots, either of which cannot be hit but by double Dice, to hit the one that Cast must be eight, and to hit the other it must be nine; by which means my Adversary has only one Dye to hit either of them.

   Query, What are the Odds of his hitting either of these Blots?

   Answer, The Chances on two Dice are in all, 36.

   The Chances to hit 8 are,		6 and 2 twice, 5 and 3 twice, 2 Deuces, 2 Fours,	2 2 1 1
   The Chances to hit 9 are,		6 and 3 twice, 5 and 4 twice, 2 Trois,	2 2 1
   Total Chances for hitting,			11
   Remain Chances for not hitting,			25

   So that it is 25 to 11 that he will not hit either of those Blots.

4. To give another Example, let us suppose that I leave two other Blots than the former, which cannot be hit but by double Dice, the one must be hit by eight, and the other by seven.

   Query, What are the Odds of my Adversary’s hitting either of these Blots?

   Answer. The Chances on two Dice are in all, 36.

   The Chances to hit 8 are,		6 and 2 twice, 5 and 3 twice, two Fours, two Deuces,	2 2 1 1
   The Chances to hit 7 are,		6 and 1 twice, 5 and 2 twice, 4 and 3 twice,	2 2 2
   Total Chances for hitting,			12
   Remain Chances for not hitting,			24

   Therefore it is two to one, that I am not hit.

   The like Method is to be taken with three, four, or five Blots upon double Dice; or with Blots made upon double and single Dice at the same time; you are then only to find out (by the Table of 36 Chances) how many there are to hit any of those Blots, and add all together in one Sum, which subtract from the Number 36, which is the whole of the Chances upon two Dice: So doing resolves any Question required.

5. The following Cases are to shew a way of calculating, which may be called a mechanical way of solving of Questions of the like Nature.

   I would know what are the Odds of throwing 7 twice, before 10 once.

   Answer. It is 5 to 4 that 10 is thrown once before 7 is thrown twice, which is demonstrated, as follows:

   Suppose the Stake depending is nine Pounds, my first Throw entitles me to have one third Part of that Money, because 7 has six Chances for it, and 10 has but three Chances, and therefore it is two to one.

   		l.	s.	d.
   For the first Throw		3	0	0
   Having taken 3 l. out of the 9 l. for the first Throw, the Remainder is 6 l. out of which a third Part is to be taken for the second Throw,		2	0	0
   The Total is,		5	0	0
   Remains,		4	0	0
   The whole Stake is,		9	0	0

6. I would know what are the Odds of entering a Man upon any certain Point in two Throws?

   Answer. Suppose 36 Shillings is the whole Stake depending, I would know what is my Share of that Stake, having laid 18 Shillings that I enter in two Throws? By the Calculations in the Table of 36 Chances, it is found that I have 11 Chances out of the 36 for entering the first Throw, for which therefore I am entitled to 11 out of the 36 Shillings.

   		s.	d.
   The Stake is,		36	0
   For the first Throw,		11	0
   Remains,		25	0
   The Remainder, being 25 Shillings, is to be divided into 36 equal Parts, of which I am entitled to eleven of those Parts, which makes 7 s. 7 d. ½ for the second Throw,		7	7½
   Adding this to the other 11 Shillings, makes my Share of the Stake to be,		18	7½
   Then my Adversary’s Share will be,		17	4½
   Total of the Stake,		36	0

   Therefore it is very nearly 15 to 14 in favour of entering a Man upon any certain Point in two Throws.

Chap. VIII.

Critical Cases for a Back-game.

1. Let us suppose A plays the Fore-Game, and that all his Men are placed in the usual Manner.

   For B’s Game, let us suppose, that 14 of his Men are placed upon his Adversary’s Ace Point, and one Man upon his Adversary’s Deuce Point, and that B is to throw.

   Query, Which Game is likeliest to win the Hit?

   Answer, A’s is the best by Gold to Silver, or 21 for, to 20 against; because, if B misses an Ace to take his Adversary’s Deuce Point, which is 25 to 11 against him, A is, in that Case, to take up B’s Men in his Tables, either singly, or to make Points; and if B secures either A’s Deuce or Trois Point, in that Case, A. is to lay as many Men down as possible, in order to be hit, that thereby he may get a Back-Game.

   When you are pretty well versed in the Game of Back-Gammon, by practising this Back-Game, you will become a greater Proficient in the Game than by any other Method, because it clearly demonstrates the whole Power of the Back-Game.

2. Back-Game.

   Let us Suppose A to have five Men placed upon his Size Point, five Men upon his Quatre Point, and five Men upon his Deuce Point, all in his own Tables.

   And let us suppose B to have three Men placed upon A’s Ace Point, three Men upon A’s Trois Point, and three Men upon A’s Cinque Point; let B also have three Men upon his Size Point in his own Tables, in the usual Manner.

   Query, Who has the better of the Hit?

   Answer, It is an equal Game; but to play it critically, the Difficulty lies upon B who is in the first place to endeavour to gain his Cinque and Quatre Points in his own Tables, and when that is effected, he is to play two Men from A’s Cinque Point, in order to oblige his Adversary to blot, by throwing an Ace, which if B hits, he will have the fairest Probability of winning the Hit.

3. Back-Game.

   Suppose A has three Men upon B’s Ace Point, and three Men upon B’s Deuce Point, also three Men upon his Size Point in his own Tables, and three Men upon his usual Point out of his Tables, and three Men where his five Men are usually placed in his Adversary’s Tables.

   And let us suppose B has his Men placed in the same Manner, both in his own and his Adversary’s Tables, with this Difference only, viz. instead of having three Men put upon A’s Deuce Point, let him have three Men upon A’s Trois Point.

   Query, Who has the best of the Hit?

   Answer, A, because the Ace and Trois Point are not so good for a Hit, as the Ace and Deuce Points in your Adversary’s Tables, for this Reason, because when you are bearing your Men, you have the Deuce Point in your own Tables to play your Men upon, which often prevents your making a Blot, which must happen otherwise to your Adversary; and take care to lay down Men to be hit as often as you can, in order to keep your Game backward, and for the same Reason avoid hitting any Blots which your Adversary makes.

4. As a Case of Curiosity and Instruction.

   Let us supposed A has his 15 Men upon B’s Ace Point, B is supposed to have his Barr Point, also his Size, Cinque, Quatre, and Trois Points in his own Tables.

   Query, How many Throws is A likely to take to bring his 15 Men into his own Tables, and to bear them?

   Answer, You may undertake to do it in seventy-five Throws.

   It is Odds in A’s Favour that he throws an Ace in two Throws; it is also Odds in A’s Favour that he throws a Six in two Throws; when these Events happen, A has a Probability of not wanting above two or three Throws to play, till he has got all his fifteen Men into his own Tables: Therefore, by a former Rule laid down to bring your Men home, and also for bearing your Men, you may be able to find out the Probability of the Number of Throws required. Note, B stands still, and does not play.

5. A Case of Curiosity and Instruction.

   Where A and B shall play as fast as usual and yet A shall make the Hit last, probably, for many Hours.

   We will suppose B to have bore 13 Men, and that A has taken up the two remaining Men.

   And let us suppose that A has his 15 Men in B’s Tables, viz. three Men upon his Size Point, three upon his Cinque Point, three upon his Quatre Point, three upon his Trois Point, two upon his Deuce Point, and one upon his Ace Point.

   The Method, which A is to take, is this: Let him bring his 15 Men home, by always securing six close Points, till B has entered his two Men, and brought them upon any certain Point; as soon as B has gained that Point, A must open an Ace, Deuce, or Trois, or all three; which effected, B hits one of them, and A taking Care to have two or three Men in B’s Tables, is ready to hit that Man; and also, he being assured of taking up the other Man, has it in his Power to prolong the Hit to almost any Length, provided he takes Care not to open such Points, as two Fours, two Fives, or two Sixes, but always to open the Ace, Deuce, or Trois Points for B to hit him.

6. I would know what are the Odds upon two Dice, for throwing two Sixes, two Fives, or two Fours, in three Throws? which by mechanical Calculation, may be found thus:
   

   		s.	d.
   Answer, Supposing 36 Shillings to be the Stake depending, the Thrower will be entitled to have for his first Throw,		3	0
   which deducted out of 36, remains 33; which divided again into 36 Parts, make so many eleven Pence, out of which the Thrower is to have 3 for his second Throw,		2	9
   The Remainder, 30 Shillings and 3 Pence, is again to be divided into 36 Parts; dividing the 30 s. so, make so many Ten-pence, and the 3 Pence divided into so many Parts makes so many Thirds of Farthings, of which the Thrower is to have 3 Parts for his Share for his third Throw		2	6¼
   Total for the Thrower,		8	3¼

   So that it is 27 s. 8 d. ¾ to 8 s. 3 d. ¼ against the Thrower; which reduced into the smallest Number, is very nearly as 10 to 3, that 2 Sixes, 2 Fives, or 2 Fours, are not thrown in three Throws.

7. Back-Game.

   Suppose A to have 2 Men upon his Size Point in his own Tables, 3 Men upon his usual Point out of his Tables, 2 Men upon the Point where his 5 Men are usually placed in his Adversary’s Tables, 5 Men upon his Adversary’s Ace Point, and 3 Men upon his Adversary’s Quatre Point.

   And let us suppose B to have 2 Men upon his Size Point in his own Tables, 3 Men upon his usual Point, out of his Tables, 2 Men upon the Point where his 5 Men are usually placed in his Adversary’s Tables, 5 Men upon his Adversary’s Ace Point, and 3 Men upon his Adversary’s Trois Point.

   Query, Who has the Fairest Chance to win the Hit?

   Answer, A has, because he is to play either an Ace, or a Deuce from his Adversary’s Ace Point, in order to make both those Points as Occasion offers, and having the Quatre Point in his Adversary’s Tables, he may more easily bring those Men away, if he finds it necessary, and he will also have a resting Place by the Conveniency of that Point, which at all times in the Game will give him an Opportunity of running for the Hit, or staying, if he thinks proper. Whereas B cannot so readily come from the Trois Point in his Adversary’s Tables.

Chap. IX.

1. Let us suppose A and B place their Men in the following manner for a Hit.

   Suppose A to have 3 Men upon his Size Point in his own Tables, three Men upon the usual Point out of his Tables, and 9 Men upon his Adversary’s Ace, Deuce, and Trois Points, 3 Men to be placed upon each Point; and suppose B’s Men to be placed in his own, and in his Adversary’s Tables, in the same Order and Manner.

   The Result is, that the best Player ought to win the Hit; and the Dice are to be thrown for, the Situation being perfectly equal in A’s and B’s Game.

   If A throws first, let him endeavour to gain his Adversary’s Cinque Point; when that is effected, let him lay as many Blots as possible, to tempt B to hit him; for every time that B hits them will be in A’s Favour, because it puts him backward; and let A take up none of B’s Men for the same Reason.

   A is always to endeavour to take Care to have three Men upon each of his Adversary’s Ace and Deuce Points; because when B makes a Blot, those Points will remain secure, and by Recourse had to a former Case (Numb. V. in the former Chapter) when A has bore 5, 6, or more Men, yet A may secure 6 close Points out of his Tables, in order to prevent B from getting his Man home, and by Recourse had to the Calculations, he may easily find out (in Case he makes up his Tables) who has the better of the Hit; and if he finds that B is forwardest, he is then to endeavour to lay such Blots to be taken up by his Adversary, as may give him a Chance for taking up another Man, in Case B should happen to have a Blot home.

   Those who play the foregoing Game well, may be ranked in the first Form.

2. A Case of Curiosity.

   A and B play at Back-Gammon, A has bore 13 Men and has 2 Men to bear upon his Deuce Point, B has 13 Men in his own Tables, with two Men to enter. B is to throw — And to name the Throws both for himself and A but not to hit a Blot of either Side.

   Query, What Throws is B to name for both Parties, in order to save his Gammon?

   Answer, B calls for himself 2 Aces, which enters his 2 Men upon A’s Ace Point. B also calls 2 Aces for A and consequently A cannot either bear a Man nor Play one; then B calls for 2 Sixes, for himself, and carries one Man home upon his Size Point in his own Tables, and the other he places upon his Adversary’s Barr Point: B also calls Size-Ace for A so that A has one Man left to bear, and then B calls for himself either 2 Sixes, 2 Fives, or two Fours, any of which bear a Man, in Case he has Men in his own Tables upon those Points, and to save his Gammon.

3. The following Question is to be attended to, as being Critical and Instructive.

   Suppose that both yours and your Adversary’s Tables are made up.

   Also that you have 1 Man to carry home, but that he has 2 Men on your Barr Point to carry home, which lye in wait to catch your Man, and that if you pass him you are to win the Hit. Suppose also that you have it in your Choice to run the Risk of being hit, by 7, or by 8, both of which are Chances upon double Dice.

   Query, Which of these Chances is it best for you to venture?

   Answer, That of 7, for the following Reasons: First, because that the Chances being hit by 7, or by 8, are equal.

   Secondly, If he does not hit 7, you will then have in your Favour 23 Chances to 13, that by your next Throw you either hit him or pass beyond him.

   Thirdly, In Case your second Throw should happen to be under 7, and that consequently you cannot hit him, yet you may play that Cast at home, and consequently leave the Blot upon double Dice.

   Whereas, if on the contrary, you had made Choice of leaving the Blot upon 8, you would have made a bad Choice, for the Reasons following:

   First, Because the Chances for being hit by 7, or by 8, are only equal.

   Secondly, Because if you should escape the being hit by 8, yet you would then have but 17 Chances in your Favour, against 19 for either hitting him, or passing beyond him by your next Throw.

   Thirdly, In case your second Throw should happen to be Size-Ace, which is short of him, you would then be obliged to play the Man that is out of your Tables, not being able to play the six at home, and consequently to leave a Blot to be hit by a single (or flat) Dye; which Event, upon Supposition that you play for 18 Shillings a Game, it would entitle him to 11 Shillings of the whole Stake depending.

The Laws of Back-Gammon

1st, If you take a Man from any Point, that Man must be played; the same must be done if 2 Men are taken from it.

2d, You are not understood to have played any Man, till you have placed him upon a Point and quitted him.

3d, If you play with 14 Men only, there is no Penalty attending it, because by playing with a lesser Number than you are entitled to, you play to a Disadvantage, by not having the additional Man to take up your Tables.

4th, If you bear any Number of Men, before you entered a Man taken up, and which consequently you were obliged to enter, such Men, so born, must be entered again in your Adversary’s Tables, as well as the Man taken up.

5th, If you have mistaken your Throw, and played it, and if your Adversary has thrown, it is not in your or his Choice to alter it, unless both Parties agree to it.

FINIS.

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