Backgammon Variants

As you recall, wastage is the difference between (1) the number of pips your checkers must move to get off and (2) the number of pips you expect to have to roll to get your checkers off. Last time, I asked what positions in hypergammon have the highest and lowest wastage.

When there are lots of checkers in play, it is easier to get a feel for what positions have high and low wastage. But it is not so easy when there are just a few checkers.

Wastage with One Checker

A checker on the ace point is the highest wastage you can have with one checker. Your pip count is very low (just 1) compared to the number of pips you expect to roll (8.167).

Highest wastage with one checker: 8.1667 − 1 = 7.1667

A checker on p2 or p3 has less wastage than this because the pip count is higher but the rolls to get off is the same. Even a checker on p4 is likely to get off in one roll, so the thin possibility of needing 2 rolls is more than offset by the extra pip of distance.

The farther you move that checker out, the lower the wastage gets, until you reach p7. And that is the position that has the lowest wastage in hypergammon (except for all checkers off, which has no wastage at all). It is also the position that has the lowest wastage in backgammon.

Lowest wastage with one checker: 11.1409 − 7 = 4.1409

Wastage with Two Checkers

It is hard to get a feel for wastage in two-checker positions. There are two competing forces: On the one hand, you’d like to have your checkers farther out where the pip count is greater and you have more flexibility on how to play your checkers. On the other hand you’d like to maximize your chances of getting off in a single roll because a second roll adds another 8.167 pips to the EPC.

I would never have guessed the highest wastage position for two checkers — you combine the lowest wastage position for one checker and the highest wastage position for one checker.

Highest wastage with two checkers: 15.5267 − 8 = 7.5267

Likewise, I would never have guessed how to minimize wastage with two checkers. Just take the highest wastage 2-checker position and slide the checker on p7 down to p3.

Lowest wastage with two checkers: 8.6204 − 4 = 4.6204

Wastage with Three Checkers

If your first instinct for creating a lot of wastage with 3 checkers is to put them all on your ace point, then your instinct is correct. This is the highest wastage postion in hypergammon.

Highest wastage with three checkers: 14.9722 − 3 = 11.9722

And if your instinct says that to create low wastage you should put your checkers on points 6, 5, and 4, that’s good too. It is almost the lowest wastage with three checkers, but following position beats it slightly:

Lowest wastage with three checkers: 28.2950 − 23 = 5.2950

What Is Contact?

So far we have been dealing with wastage in pure races. Now let’s look at contact positions. But before we do, let’s clarify: What exactly is “contact”?

All backgammon players have a good idea what contact is. A contact position is a position where the sides are still engaged with each other — where there is still a chance of hitting and blocking.

But that is not a precise definition. To refine this definition some, let’s look at a series of positions and try to identify which positions have contact and which do not. This isn’t just an academic question — the term “contact” appears in the rules of tournament backgammon. You can resign a game only in a noncontact position.

So which of the following positions have contact in them? (This is really more of a poll rather than a question with a definitive answer. What I’m asking is, what is your definition of contact.)

In each case it is black’s turn to roll.

a. 12,5,1 v 12,1,1

(This one is easy. Both players’ armies have moved past each other.)

b. 1,1 v bar

c. 3 v bar

d. 2,1 v bar

e. 3,1 v 23

f. 4 v 22

g. 5,1 v 22

h. 3 v 24,1,1

i. bar v 1,1

j. 13 v 13,13

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