Strategy--Bearing Off

Forum Archive : Strategy--Bearing Off


From:   David Montgomery
Date:   25 April 1996
Subject:   Re: 15 positions that minimize waste of pips in bear-off
Google:   4loeck$

William C. Bitting writes:
> hmm, it strikes me that the optimal board - 7 5 3 checkers on the
> 6 5 4 points - may be the most efficient in terms of wasting
> fewer pips to bearoff, BUT does this also mean it is the most efficient
> in terms of the number of rolls needed to bear all the checkers off the
> board? I would think a 2 2 2 3 3 3 board (6pt to 1 pt) would take fewer
> rolls on average to bearoff all the checkers versus the optimal board
> for pip efficiency. How does one balance the most efficient in terms of
> fewest rolls versus the most efficient in terms of fewest wasted pips?

You are correct that 333222 will take fewer rolls to bearoff on average
than 000357 -- and 15 0 0 0 0 0 will take even fewer!  The problem is
that you don't often get to choose between these bearoff formations.
(I've tried it -- bearing my last checker into my board to produce
 000357, I offer to move all my checkers to the ace point, pointing out
 how much extra wastage I'll have.  No one's gone for it yet. :-> )

The concept of wastage is relevant for comparing positions with
equal pipcounts, because most checker play problems allow you to
choose among plays with equal pipcounts, and usually the best play
among these is the one with the least wastage.  The lowest wastage
positions will be the best play, except sometimes when your chances
are very good or very poor (Kleinman's bull-offs and bear-offs) and
in some positions with very few checkers left where the ability to use
the doubling cube effectively can changes things.

By knowing the general patterns in wastage, you can know the type of
bearoff position you are aiming for, and modify your bearin technique
accordingly.  We now know that the best bearoff formation (for a given
number of pips) is one which places most of the checkers on the 4, 5
and 6 points, with preferably more on the 5 and the most on the 6.
This is more important than memorizing the wastage of particular

Wastage can also be extremely useful in making cube decisions.

> I guess the question is when should one try to build a 7 5 3 board
> - when even or slightly behind in the race?
> ..wcb   on FIBS

If its just a pure race, and you still have the ability to aim for
000357 (you don't have any checkers yet on the 1, 2 or 3 points),
and you're not too far behind (probably if your chances are more than
a few percent and greater than your chance of getting gammoned)
then this formation should be your goal.

There might be some positions in which you are a huge favorite
to win the race in which its correct to not aim for 000357.  However,
I'm not sure which way would be an improvement, if any.  Perhaps you
should put more checkers on the 4 point to create a lower volatility
race and reduce your opponent's chances.  Or perhaps you should
stack more heavily on the 6 and possibly 5 point, hoping to roll sets
and win a lucky gammon.  In either case, you won't give up much by
just aiming for 000357.

In fact, with the exception of positions with very few checkers where
cube utility changes things, and positions where the trailer *must*
rely on doubles to catch up, if you make the low wastage play, even when
it is wrong you will give up very little.

David Montgomery
monty on FIBS
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Strategy--Bearing Off

Greedy bearoff exception  (Stick+, Jan 2011) 
Playing for gammon  (Max Levenstein+, Aug 2011) 
Rules of thumb  (Rich Munitz+, Oct 2009) 
Wastage  (David Montgomery, Apr 1996) 
Wastage--Positions that minimize  (Stig Eide, Apr 1996)  [Recommended reading]
Which gaps to fill  (Kit Woolsey, Mar 1996) 
Winning the bearoff race  (Dave Flaks+, Jan 2001) 

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