**The racing game**
is the basic game plan of backgammon. You roll a couple of high combinations, slip past his defending checkers and make a run for home. Mathematics and pip counting determine your cube decisions. The dice determine how fast you run, but the cube action in the ﬁnal bear oﬀ can be fast a furious.

**The blitz or attacking game**
is when your opponent gets caught without an anchor in your home board and you begin to make point after point on his head. Keeping him on the bar and closing him out is the goal but the sophisticated player always has a plan B to win the game in the event the blitz fails.

**Holding games**
are very common variation on the racing game. You are racing but one or both of you holds a point in the enemy outﬁeld from the bar point to the mid point and your opponent must get his checkers past you before he can bear oﬀ. The three key features of this game are hitting rolls, home board points closed and timing. Timing is the opposite of racing lead - you are the trailer in the race. To you, timing translates to the ability to move elsewhere before the dice force you oﬀ your holding point. The side forced oﬀ their holding point (exposing blots in the process) is at the disadvantage in a holding game.

**High anchor, low anchor, deuce and ace-point games**
. These are variations of the holding game above, except instead of the bar or an outﬁeld point, you hold the enemy 4/5/6 point (high anchor), 3 point (low anchor), 2 (deuce point) or 1 (ace point). The deeper you go into the board (from the 6 to the 1 point) the more important hitting a shot to win becomes. The higher you are in the board, the more important the race is. You must contain an enemy checker if you hit one.

**Back games.**
You have two or more points in the enemy home board. To win, you hit a shot, which is very likely, and then contain the hit man in your home board or close it out. This is sort of a blitz in reverse. The key concept to winning this game is timing in buckets. If you are running a back game and all your other checkers are stacked on your 1 and 2 points, it is hopeless. If you have a 4 point prime and plenty of checkers to roll it forward, you have a chance. They are very diﬃcult to play, but when they work, they can be very satisfying.

**The priming game**
is very interesting. A full prime is 6 closed points in a row, any enemy checkers trapped in front of the full prime are trapped until the prime is broken. With 4 or 5 points in a row, you have a 4 point prime or a 5 point prime. If you have 5 of 6 points in a row (a full prime with one gap for example) you have a broken prime. There are three key concepts in priming games—how many men are trapped behind the prime, how long the prime is (6 > 5 > 4) and our old friend timing which determines who has to break their prime ﬁrst.

Two other aspects of backgammon are the opening rolls and replies. Here you try for an early advantage in making home board points or anchors; and the so-called middle game where you are past the opening but are jockeying for position to settle into a more formal game plan. While there are no formal articles in this series on these aspects of the game, we will touch on them in the discussions of the game plans.

The tactical aspect of the game is the checker play. A main consideration is the decision to play aggressively, or to exercise caution.

If, for example, all your home board points are closed and you have an enemy man on the bar, you can pretty much do whatever you wish - leave blots, hit loose, break anchors, all without fear - so long as you do not have to break your home board due to an awkward dice roll.

Rarely are things so clear cut in an actual game. Generally, it is relative strength compared to your opponent that you use as your guiding principle. For example, if you have three home board points closed, and your opponent has only two of them, a hit will hurt your opponent more than it will you because you have 32 of 36 rolls that allow you to enter and he only has 27. Accordingly, you can accept more risk of having a man hit than he. You can play more aggressively to achieve your goals whereas he must be more cautious.

Likewise, if you have an anchor in his board, you cannot be shut out, even if the other 5 points are owned by your opponent. If you both have anchors, the one higher in the board is stronger, because it is harder to prime and easier to escape from - the best anchor is "Golden Point" which is your opponent's 5 point.

Here are some additional criteria for bold vs. safe play and how to interpret them:

- Home board points. Relative strength in home board points allows for more aggressive play.
- Anchors. One anchor vs. none allows for more aggressive play, this extends to:
- Multiple anchors vs. single anchor.
- A higher anchor vs. a lower anchor.

- Blots in your home board discourage aggressive play (if you hit, they may hit you as they re-enter). Conversely enemy blots in his home board invite an exchange of hits.
- More back men allow for more aggressive play, while counterintuitive, this is based on sound logic:
- More than one man back can combine to form an anchor.
- A single man back can only escape or be attacked.
- A hit with only one man back may critically reduce your racing lead.

Other factors to be evaluated include outer board blocking points (especially points that block the escape of an enemy back man with 5's or 6's), blots exposed (the side with more exposed blots will wish to clean them up rather than expose more in many cases), and racing lead (lead discourages bold play).

When you must expose a weakness, there are several ways to minimize the chances your opponent may capitalize on it. Examples include:

- Take away part of his roll by hitting him and forcing him to use a number to re-enter (or even stay on the bar). This prevents him from using his full roll to hit you, extend his prime or otherwise advance his game plan.
- Duplicate his good numbers. This means that if he has a number that does something he really wants to do (like re-enter from the bar with a 2), he cannot do something else with that number such as hit one of your blots, cover one of his or extend his prime/make a point.
- Keep your blots more than 6 points from an enemy checker that can hit them. Once you are past direct hitting range - 1 to 6 points - the number of rolls that actually hit you goes down dramatically. Plus he must use both rolls to hit you, rather than being able to do something else with the other half of his roll.

Many beginning players have a feeling they lose games because their opponent gets "lucky" rolls. If you deﬁne a lucky roll as one that can be used to do something good for our game, that is probably true. Better players are lucky because they plan better than beginners do, and as a result can use more of the 36 possible rolls than players that don't.

The key to having good rolls is something called ﬂexibility. Previously, we discussed the concept of duplication where we want to make our opponent have to choose between several ways to play the same number. We on the other hand want to have more numbers we can play.

This concept is best demonstrated with a couple of examples:

Red has 2-1 as opening move. White can hit the blot on 5 with any 4, 3-1, 2-2, and 1-1; 6-4 hits both

Note that with my play of 2-1, I can make my 5 point (which not only is good for me by making a key home board point, but disrupts my opponent's game plan by taking his golden point from him) with 3 diﬀerent numbers (6, 3, and 1) plus several combinations 2-2, 4-4 (from the midpoint at 13), If I moved the 2 elsewhere, I would only have 2-2, 4-4, 3's, and 1's to cover my blot on 5. By making the proper move, I increased my covering rolls from 22 to 29 at minimal risk. I would have 7 more "lucky rolls".

Bearoﬀ: Both Red and White have a pip count of 21

Note that we are down to the wire; both sides have a pip count of 21 and 6 checkers left on the board. Red has 6 numbers that will bear oﬀ a man, while White only has two. Red is very ﬂexible and diversiﬁed, whereas White has duplicated his good numbers. Red will bear oﬀ at least two men this roll, with any roll. White will only bear oﬀ two or more men this roll with only 6-6, 5-5, 2-2, 1-1 or 5-2; on the other 30 rolls he will bear oﬀ 1 or 0 men - a similar picture will repeat next roll as well. In a race this close, even one failure to bear oﬀ at least two men with each roll is critical and may be game losing. If I were Red and oﬀered you the cube, are you conﬁdent enough in your winning chances as White to play for twice the stakes, or do you resign?

Finally, a demonstration of thinking through a checker play problem.

How should Red play 4-4?

Let's examine this position to see how we should play 4-4 as Red. First we look at the race by doing a pip count:

Red has 25 pips for the man on the bar, 40 for the men on 13 and 14, 25 for the 3 on 8 and 9, and 38 in his home board (128) and we just rolled 16 pips so we expect to be 112 by the end of the turn.

White has 24 for the one on the 1 point, 66 for the 4 on the 7 and 10 points, 35 for the 4 on the 18, 16 and 15 points and 30 in his home board (recall his pip count is from his aspect, our 7 point as listed is his 18). This totals 155.

So our winning game plans play to our racing lead - blitzing and priming.

His winning plans involve holding games and back games which use his superior timing to best advantage.

To defeat his plan, we need to get past his blocking points and prevent him from forming an anchor in our home board, or continue to hit him and close our board. So ﬁrst on our 4-4, are there any moves we have to make? Yes, we must use one 4 to enter. Now, by moving a man from 8 to 4 we do two good things: we make another home board point and we make a 4-prime that White's man on the 1 point must get past to escape.

Now we have two 4's left. Some of our options are:

1. run with the back man on the 21 to 17 then on to 13

2. run to the 17 as above then come down from the 13 and make the 9 point.

3. shift one or both of the men on 8 and 9 to the 4 and/or 5 point and move one of the back men if needed.

If we run one man 21/17 and move the other man 9/5 (option #3).

The main feature of this position is that we are very well placed to make the 2 point or even make the 1 point on White's head. We are very weak however due to all the blots and white is very well placed to hit them. Because of this, we may well never make the 1 or 2 point before White escapes. There is a better move to be made with the ﬁnal two fours: simply use the ﬁnal two 4's to shift points from the 5 to the 1, putting White on the bar.

Let's look at the advantages of this position

- White is on the bar against a 4-point board.
- White needs a 2 or a 5 to enter. He needs these same numbers to hit, cover his blots and make points in his home board - his good numbers are duplicated. Even a good roll like 5-3 and 5-4 duplicates his hitting numbers.
- Red has 26 rolls to hit white's blots on 18 and 15 to put a second (or third) man on the bar
- Red is well placed to make his 5 point or cover his blots on 8 and 9 if White cannot come in
- If White does come in, Red is well placed to hit loose. If White is fortunate enough to hit if he re-enters, Red should come right back in on 4 open points in White's home board.

Looking at both basic plays - the running option #3 and the switching option - we need to learn to look at the positions and our options realistically from both sides to evaluate our cube decisions and checker play.

Key concept is the ability to understand how many rolls out of 36 will allow you to do something you want to do. Suppose you have a checker on the bar and your opponent has all his home points blocked except for his 5 point. To get back in this turn, you have to roll a 5. How many chances out of 36 possible rolls include a 5? Well, obviously if you roll a 5 on one dice, you can roll any of 6 numbers on the other dice and still have a 5 - that is 6 diﬀerent combinations. By the same token, if you have a 5 on the other dice, you can roll any of 6 numbers on the ﬁrst. That is 12 diﬀerent possibilities. The only ﬂaw with this argument is that you have to make sure that you don't count the same roll more than once. Note that a roll of 5-5 can only be made once out of 36, like any double. Since 5-5 is being counted twice in the discussion above, we need to change our 12 total rolls to 11.

To continue, what if instead of just the 5 point being open in the example above, both the 5 and the 2 point were open? If we applied the same logic, remembering to not double count doubles and paying attention to rolls like 5-2 and 2-5 we ﬁnd that our ﬁrst impression of 24 rolls (recall 1 seemed to have 12 possible rolls, so 2 must seem to have 24) needs to be adjusted to 20.

We see how to calculate it, now we simply memorize the numbers.

- any one number is 11 rolls of 36,
- any two are 20,
- any three are 27,
- any four are 32,
- and any 5 are 35.

Consider this example: say there are three checkers on the board. You have one on your 24 point, and your opponent has one checker on each 23 and 18 point (1 and 6 checkers away from your man). It is your roll, how many shots hit each checker, how many hit both?

The simple answer is easy, 11 rolls contain a 1, 11 rolls contain a 6 and 20 rolls contain either a 1 or a 6. But that is not the whole story. While 11 rolls do contain a 1 and hit the man on the 23 point, not only do 11 rolls contain a 6 to hit the man on the 18 point but so do 5-1, 4-2, 3-3 and 2-2 for a total of 17 possible rolls. To continue, we must add a couple of these rolls to our 20 rolls that hit either the man on the 23 or the man on the 18 - speciﬁcally 4-2, 3-3 and 2-2 because the 5-1 is already counted in the 11 rolls that contain a 1. This gives us a total of 24 rolls that hit the man on the 23 or the man on the 18. Only a 5-1 hits both, and that is only 2 rolls out of 36.

What about when we have to roll a number greater than 6? Take the above example with our man on the 24 point, remove the enemy checker on the 23 and move the checker on the 18 back to the 17 point. That checker is now 7 pips away from our man on the 24.

We can hit it with a 6-1, 5-2 and 4-3 which is 6 rolls. No doubles will hit it.

If we moved it back one more to the 16 point (8 away from our man), we could hit with 6-2, 5-3, 4-4 and 2-2, also 6 rolls. On the 15 point (9 away) we have 6-3, 5-4, and 3-3, 5 rolls of 36.

Why is this information important for us to know?

Because we take risks in backgammon in order to succeed. We need to be able to calculate or at least estimate those risks in order to ﬁnd the best play and the best balance of safety vs aggression in our tactical play.

- If I wish to slot a key point, how likely is my opponent to hit that blot?
- How many rolls out of 36 do I have to complete my prime or hit a shot?
- If I can't do something good this roll, how can I set up my checkers to give me the greatest chance to do something next roll?
- If I have to expose a weakness, how can I make it painful for my opponent to exploit it (if he hits me, can I re-enter and hit him back for example).

A more important aspect, however, may be when we are playing with the cube. A basic premise of the cube is that you can double when you have at least a 50% chance to win (or 18 winning rolls out of 36 if you are in a last roll situation) and your opponent can take when he has at least a 25% to win (or 9 winning rolls of 36).

Let's see an example. You are on the 1 and 4 point, your opponent has 2 on his one point. You are on the roll, if you bring both in you win, if you don't you lose. This is a last roll situation. How many rolls win the game for you? Can you double (do you have at least 18) and can your opponent take your double (do you have at least 9 rolls that don't take both men oﬀ)?

Should Red double? Can White take?

This can be directly calculated over the board in one of two ways. You can calculate your winning rolls or you can calculate your losing rolls and subtract that number from 36.

First the winning rolls - any 6, any 5 and any 4 win, plus 2-2, and 3-3 - 29 rolls.

How about the losing rolls? 1-1 is the only double, 2-1, 3-1, 3-2 or 7 rolls.

You double, your opponent passes. If you fail to double, you give your opponent 7 out of 36 chances (19%) to win a game you could have ended with the cube on your turn (81%).

Another way to calculate your winning chances is to multiply probabilities.

Consider a two roll situation where you and your opponent both have 4 men on your one points and it is your roll. To win you need to have one of two things happen, you have to roll a double or if you roll a non double than your opponent must roll a non double.

Your chance of rolling any double is 1/6. Add to this the probability of rolling a non double and your opponent rolling a non double which is 5/6 × 5/6 or 25/36. Your total winning chances are 31/36.

Your opponent wins only if you roll a non double and he rolls a double. The probability of this is 5/6 × 1/6 or 5/36.

So you should Double and your opponent has to pass.

What about more complicated positions?

Should Red double? Can White take?

Red is on the roll and considers a Double. Does Red have a Double and does White have a take? This is not a two-roll position as all doubles do not take oﬀ 4 men and all non-doubles do not take oﬀ 2 men.

Just by glancing at the position we see Red's winning chances are greater than 50% - he wins outright on doubles greater than 2-2 (4 rolls, 11%) and for the other 32 rolls, White must roll a double other than 1-1 to win immediatly, so Red picks up another 89% × 31/36 = 77% to stay alive after the first rolls exchange. So in the ﬁrst glance Red clearly has a Double.

But there is a complication. If Red roll and take off 2 checkers than White can roll a non-double and take oﬀ 2 checkers too. After that Red can roll and fail to win, then White can roll again and win.

If Red will not high double as his first roll than White can win the game in the following circumstances:

- todo
- todo
- todo
- todo
- White rolls any double except 1:1
- White rolls 1:1
- White rolls any non double

- Red rolled 1:1 and stay with two men on his point 2 (3%)

- Red rolled 2:2 and stay with two men on his point 1 (3%)

- Red rolled 1:3+ and stay with two men on points 1 and 3 (22%)

The doubling cube uses a threat. With the cube a signiﬁcant threat to win is all you really need; you don't need to necessarily to do it, you just need to threaten to from a position of force. How much strength do you need? Let's discuss the concept of **equity**.

Say you and a regular opponent are playing your game. You know from past experience with this player he is exactly at your level, neither stronger nor weaker. At the start of the game, if you are playing for a dollar, you can set a value on your game. The calculation is simple, in two games you will win one and lose one (you are equally matched and both deal with the same dice right?). If you win, you win a buck, if you lose, you win nothing. Since the end result of two games is that you win one dollar, one dollar divided by 2 is 50 cents a game. 50 cents is the value of your game. If I came by and wanted to buy your game from you, I would have to give you 50 cents.

Obviously, as the game goes on and one side gets advantage over the other, the price to buy each side adjusts accordingly. So you play the game and your opponent doubles you. You assess the position and decide, based on a number of factors, that you still have an equal chance to win. In other words, your chances of winning are 50:50. Now you have a choice to make. If you drop, you lose 1 point. If you take, you will either lose another point or you will gain 2 points. Let's look at the value or equity of your two choices.

Choice one is to drop, this will cost you −1 point in one game, which is −1/1 game or −1 points per game. Choice two is to take. Since your chance to win is 50:50, if you play two games you will either win 2 points or lose 2 points. So (2 − 2)/2 games is 0 points per game. Clearly since your choice is between losing a point and breaking even, breaking even is the better choice, so you take.

What if instead you evaluate the game when the double is made by your opponent and decide your opponent is twice as likely to win as you are?

If you play three games from the current position, you can expect to win one and you expect him to win the other two. Again, you have the same two choices. Choice one is to drop, this will cost you −1 point in one game, which is −1/1 game or −1 points per game. This is just the same as the last example. Choice two is to take, with the cube now on 2, you will win one game for 2 points and lose two which will cost you 4 points. So now we have (2 − 4)/3 games or −2/3 point per game. Since your choice is now between losing 1 point (passing) and losing 2/3 point (taking), you are still better to take.

Another way of looking at equity is the total points you stand to gain with one choice vs the total you gain with the other. In this case

- you risk 1 point (the diﬀerence between passing and losing 1 point vs taking and losing 2 points)
- against the possibility of gaining 3 points (the diﬀerence between passing and losing 1 point and taking and winning 2 points)

We already talked about the 75/25 rule of doubling . Now we can see why you can still take with your opponent having a 3/1 advantage over you. Again, if you pass, you lose 1 point. If you take, you win once for 2 points and lose 3 times for 2 points. This is (2 − 6)/4 games or −1 point per game, exactly the same as passing. So if you estimate your winning chances are 25% - you can take or pass; 26% - you take; 24% - you pass.

One term to know is that if you double with a 75% advantage you are giving a maximally eﬃcient cube. The 75% advantage is a target point to get the most from the cube.

The next major consideration is: when do we oﬀer a double?

Technically it is correct to oﬀer a double when you have an advantage over your opponent. So if your winning chances are 50% or more, you can double. This brings up a new term. If you can double at 50% and your opponent can take with 25%, then we say that 50% to 75% is the **doubling window**.

The same considerations aﬀect the other end of the window. A good example is the case in match play where you oﬀer a double, but your opponent can redouble you for enough points to win the match. This would be the case, for instance, when you needed 2 points to win a match and your opponent needed 3. By doubling, you give him the chance to turn the cube to 4 (What does he lose? If you win, you win the match, if he wins he might as well win the match too). Clearly in this case, you want more than a 50/50 chance for the amount of risk you are taking.

How do we note or evaluate cube decisions? Cube action is referred to by a code. If a double is technically correct, and a take is technically correct, a cube decision is referred to as **double/take** (sometimes further deﬁned as **easy take**. **Beavers** are not always allowed, but if they are, you may, if given an improper double, turn the cube again and retain ownership of the cube. This is designed to punish a foolish double.

So we know when we can oﬀer a double - with an advantage. When should we?

Ideally, you want to stack the deck in your favor. The closer you are to the maximum advantage your opponent can accept the better. In other words, you would like a 74-75% advantage.

If you have an advantage but you have a number of potential rolls that will increase it to such a point that your opponent will not be able to accept the cube, you may want to consider turning the cube on the chance you will roll and crush him.

Another factor you can use to guide your cube action is whether or not your opponent will have a take next roll even if your roll is the best roll and he rolls his worse. If the answer is yes, you lose nothing by waiting another shake to turn the cube, as you can turn it next roll when perhaps you have more advantage. This is handy if there is a sudden turn in the game - you reduce your risk if you hold oﬀ on the cube until things are clearer and more decisive.

The ﬁnal bit of advice on cube action is known as Woolsey’s Rule, after Kit Woolsey - one of the best players in the world.

**Woolsey’s Rule: "If you are not sure your opponent has a take or a pass, turn the cube."**

The explanation of this rule is simple. If you are not sure your opponent has a take or a pass, your opponent probably doesn’t either. Put the stress of that decision on him. If the position is a pass, he might incorrectly take the cube. If the position is a take, he might pass it. Either way, your equity goes up.

Another aspect of the cube we need to consider is cube ownership.

Both players can use the cube in the middle, but if a double is made, the side doubled owns the cube. If one side owns the cube and his opponent rolls a very good roll (a joker) or otherwise turns the game around, he can't turn the cube on you and double you out.

Here is the breakdown of the position: Red rolls and wins 19/36 outright. White then rolls on the 17/36 remaining and wins 26/36 of those (34%). Red then rolls on the last 5/36 and wins them. (OK, 2-1 twice in a row would not be good, but what are the chances?)

Here we will look at the equity in a third form, we will compare two series of 36 games, see how many points we win overall with one choice and compare it to how many we win overall with the other choice.

var 1: Red does not double

Red wins 19 games on this roll 1pt each +19 points White doubles and wins 12 games 2pt each -24 points Red redoubles and white passes 5 games 2pt each +10 points ———— +5 points

var 2: Red doubles

Red wins 19 games on this roll +38 points White redoubles and wins 12 games 4pt each -48 points Red redoubles and white passes 5 games 4pt each +20 points ———— +10 points

var 1: Red does not redouble

Red wins 19 games on this roll 2pt each +38 points White wins 12 games 2pt each -24 points Red redoubles and white passes 5 games 2pt each +10 points ———— +24 points

var 2: Red redoubles

Red wins 19 games on this roll 4 pt each +76 points White redoubles and wins 12 games 8pt each -96 points Red redoubles and white passes 5 games 8pt each +40 points ———— +20 points

The ability to control the cube (cube ownership) prevents your opponent from capitalizing on a turn of the game in his favor. This is why you should not be as anxious to redouble as you are to give an initial double: you are giving up control of the cube and it may come back to bite you.

A **pure race**
is when you ﬁnd yourself in a situation where you no longer have contact with enemy checkers. You cannot be hit or blocked, and you are running home for the bearoﬀ.

a long race vs a short race. If all your men are in your home board, it is a **short race**
. If some are in the outﬁeld, it is a **long race**.

Back games, prime battles and other interesting and complicated games occur, but most games do end up in a short race, so we need to understand them and use the cube eﬃciently. There are several key features to short races. The most important is the smoothness of your bearoﬀ compared to your opponents — gaps are very important, if you have none and your opponent has 3 empty points, you will have the advantage because he will not bearoﬀ if he rolls a gap number. Another factor is pip wastage — are there many checkers stacked on the one and two points that will be taken oﬀ with higher number rolls? Finally, do all doubles take oﬀ 4 men and all non-doubles take oﬀ 2? Further modiﬁcations need to be made if one side has more checkers oﬀ than the other. Based on these factors, the pip count is modiﬁed and cube decisions are made. Here is Keith count procedure:

- do a quick pip count for your checkers
- divide it by 7 and add it to your total (forget the fractions)
- look at the 4/5/6 points and add 1 point for each empty point
- look at the 3 point add 1 point for every checker there more than 3
- look at the 1 and 2 points - you should have no more than one checker on each. If you have any more, add 2 points for each extra on the 1 point, and 1 point for each extra on the 2
- do the same for your opponent, but skip step 2 (this step adjusts for the fact you are on the roll)

OK, cube decision time. Assuming that both players have the same number of checkers on the board, if you have a Keith count not more than 4 more than your opponent, you can double. If your count is not more than 3 higher, you can redouble. If your count is at least 2, your opponent can take. Note that if you are on the receiving end of a cube, you need to do the above calculations from the viewpoint of the doubler (you add 1/7 of his pip count to his count and don't add it to yours).

One special situation in the short race is the n-roll position. This special type of race has two conditions:

1. All doubles bear oﬀ 4 checkers.

2. All non-doubles bear oﬀ 2 checkers.

Thus in a perfect bearoﬀ position with 15 checkers there are 7 rolls remaining, this is 6 non- doubles and 1 double (1/6 of all rolls are doubles). Before deciding you are in an n-roll position, you must carefully play devil's advocate to be sure all your doubles and all your non- doubles work, with the same going for your opponent's position. Even one double or non-double that does not meet the conditions invalidates the position.

1-roll position is a 100% win.

2-roll position is a double/pass (84:16)

3-roll position is also a double/pass (77:23)

4-roll position is unique in that it is an initial double/take but it is not a redouble (remember the value of cube ownership)

5/6/7-roll positions are all double/takes, but it is best to simply watch them and act after a couple of rolls bring you to one of the above positions

You can exert some degree of control as you are bringing your checkers to the home board to set up a bearoﬀ position that can work to your advantage. The key to this is to set up as smooth a bear oﬀ as you can. Another feature to pay attention to is the minimum number of rolls to bear oﬀ your last man. As an example, a position with 7 men remaining and one with 8 men both will require 4 rolls (excluding doubles) to bear oﬀ. There is no advantage to taking a risk to bear oﬀ an additional man if you have 8 checkers to go. If there is a question of gammoning your opponent, you might consider taking an extra risk with 7 men to take an additional man oﬀ to get to 6.

The long race has all the same considerations as the short race, with a few twists thrown in.

Since it does have many of the features of a short race, adjustment of the raw pip count with the Keith Count is certainly appropriate.

Another factor that must be considered is the number of crossovers needed to bring your (and your opponent's) checkers home to start the bearoﬀ.

In a blitz, you are trying to keep your opponent on the bar and close him out. Generally, an early double allows you to hit a split enemy checker and make a point or two on your home board. The checker stays out on the bar and the blitz is on. While many are of the opinion that a blitz is a purely dice driven event and an exercise in moving checkers rather than an in depth battle of the wits like a priming game, they sure are fun when you are running one. Fun until you run out of ammunition, your opponent anchors and throws a joker then doubles you out. Or, your opponent anchors and you have not made any preparations to win the game without the blitz.

Social players let the dice dictate their game. Serious backgammon players have a game plan and make their own lucky rolls with ﬂexibility and have a backup plan in case the game changes course. An extension of this concept is applying it to your opponent: if he survives your blitzing attack, how does he plan to win the game and what can you do to dash his hopes?

Roll your dice, but before you move a checker run over these three game plans in your head.

- How do I move to maximize my close out and what can he do to interfere?
- What is my plan to win if he gets out of this?
- And how does he hope to win if he survives and how can I disrupt that plan?

The basic goal of blitzes (from the attacker's viewpoint) is to bring builders to bear on open points, close them, and keep your opponent on the bar until you gammon him. You must prevent him from forming an anchor (especially a high anchor) so hit loose if you must. If you are hit when he re-enters, so be it, just come back in and charge into the fray because in many cases your opponent has not had time to close points on his home board. If not, you may cover the next roll and your position is even stronger.

A key concept is that the number of rolls that make a point is directly related to the number of builders within direct range of the point. This is the concept of **ﬂexibility**
. Flexibility over all means that you have checkers available to do something you want to do with as many diﬀerent dice rolls as possible.

**Duplication**
is the opposite, you don't want to have to re-enter, cover a blot, make a point and escape a prime with the same 5 for example. The graphic shows this concept well.

The number of checker and the pip count are the same, but one side has excellent ﬂexibility and the other has duplicated his numbers. In the ﬁnal bear oﬀ, which side would you rather be? The same concept extends to bringing builders to bear on a point you need to make.

The formula for non-double rolls that make an empty point is n × (n − 1) where n is the number of available spares within 6 pips of the point in question. To complete the roll count, doubles must be evaluated.

example 1

In the opening position, we want to make the 5 point. To do so, we look to our spares within 6 pips of the 5 point (we have one on the 6 and one on the 8). The number of rolls is n × (n − 1) which in this case with 2 spares is 2 × 1 = 2. These two rolls are 3-1 and 1-3. To that we add the doubles 4-4, 3-3 and 1-1 for a total of 5 rolls.

example 2

Now we have a spare on 6, 8 and 9. We have 3 × 2 = 6 non-double rolls 3-1, 4-1, 4-3. Plus our 3 double rolls we now have 9 rolls. Adding only one spare gives us an additional 4 rolls.

example 3

Now with 4 spares (6, 8, 9, and 10) we have 12 nondoubles 3-1, 4-1, 5-1, 4-3, 5-3 and 5-4 plus our doubles for 15 rolls.

Each additional spare nearly doubles our point making rolls.

Another key point is that stacking checkers on a point does not increase the usable spares. Had we made the 9 point in example 2, we would still have 3 usable spares; our only advantages would be the loss of indirect hitting numbers for White and a blocking point for us.

The number of rolls that cover a slotted point go up quickly based on the number of usable spares within direct range. An example would be an opening 2-1, playing this 13/11 6/5 gives you 3 covering numbers (6, 3, 1) for 27 rolls plus 4-2, and 4-4/2-2 for 31 covering numbers against any 4, 2-2, 1-1 and 3-1.

From a strategic standpoint we now have priorities:

- Make home board points (every point made helps but given a choice start high)
- Hit loose if you must to prevent your opponent from getting an anchor
- Put a spare on as many direct range points as you can (especially if they help your backup plan)
- If hit and unable to accomplish 1, 2 or 3, run a man around to re-enter the fray
- If your blitz runs out of steam, continue with your backup plan and interfere with his game plan

If all goes well, we continue to a closeout and a gammon. But what happens if we are stopped? Say our opponent makes a high anchor on our 5 point—the blitz is now over and he is playing a holding game and we are bearing in. Now is when our prior planning comes into play—how are we still going to win and what is his new game plan?

Until now, most of our evaluations have dealt with races. Clearly there are other considerations in contact positions. **PRAT**. This stands for **Position, Race** and **Threats**. Let's go over the elements, recalling that we are talking relative advantage to your opponent. An overwhelming advantage is just that, it carries far more weight than a relative one when it comes to cube decisions. This is not all inclusive, but gets you thinking correctly.

Position

- Inner board points
- Outer board points
- Primes in place
- Back men escaped
- Anchors

Race

- Do a pip count, half crossover count or eyeball it
- Timing and how does it aﬀect the current game plans

Threats

- Blots to be hit
- Points to be made
- Primes to extend
- Anchors to be forced oﬀ
- Blots to be exposed in the bear in

There are three general rules of thumb when we compare these three considerations to our position and our opponents.

1. If we have relative advantage in Race, it is a double/take. If we lead in all three it is a double/pass.

2. An overwhelming advantage in any of them strongly skews the results to double/pass.

3. With an overwhelming advantage, consider playing on to gammon.

Consider this position:

Position

• 3 Red vs 1 White inner board points, 5 point made by Red.

• Red has split back men guarding Red's inner board and spares on 6, 8, 9, and 13.

• White is static, pretty much stacked in the opening position and on the bar but has some builders aimed at the bar and 5 point.

Advantage: Red.

Race

• Red has 24 half crossovers and 150 pips (rolled a 3-3 and 4-1 for 17).

• White has 30 half crossovers and 165 pips (rolled 3-2 and hit on the 22).

Advantage: Red.

Threats

• Red has three builders in direct range of his open 4 point.

• There is an enemy blot on the 15 that is under indirect attack and the enemy blot on 1 can be hit to start that point.

• White must re-enter and may anchor on the 1 point or hit our builder with a 2-6 or come in on the 2 or 4 (and possibly get covered by the blot on 1).

• Double shot on a reentering blot on 4 that can't escape or be covered.

• White can re-enter then hit our back men on the 23 or 24.

Advantage: toss up leaning toward red. Gammon chances for red still good.

So what does this tell us so far? White hopes to re-enter and anchor. Our priority is to make the four point and/or hit the blot and slot the point. Nailing the blot on 15 with 6-3 or 5-3 would get a back man out and put the blot on the roof. If White anchors on the 4, then we go to Plan B — convert the blots on 8 and 9 to landing points from the spares on the midpoint, then get the back men out to safety and convert to more of a racing game, being mindful of how we will get home past his midpoint and 4-point holding game.

Our game plan is to continue the blitz. Our hopes to close him out and get a gammon are reasonable so we hang onto the cube for now. Red does clearly have a double and White has a drop, but rather than take one point we play on for two.

Overall, we did not have much to work with other than an initial double during this blitz. There were not many builders in place and we had two men deep and under threat. But we had a solid winning back up plan — a prime in front of his back men.

A ﬁnal word on cube action in the blitz. If you can gammon your opponent, charge on for the gammon rather than let him away with one point. If you are running out of steam, your cube decisions should be to double your opponent out or you should double him in based on your game plan B and his game plan B.

Finally, remember a bit of human psychology. If your opponent survives your attempt to crush him with a blitz, he may begin to feel a bit invincible and overestimate his chances. Play on this with the cube; you may get some extra points from a bad take. An advanced psychological tactic might be to deliberately leave just enough hope to encourage a back take.