Backgammon End Games and Ace Point Antics

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By Robert Wachtel

Before backgammon, in my universe, there was chess. I had a talent for the game, and had my successes as a junior. Accustomed to study, I found the transition to backgammon (in my late twenties) easy. Very few players at that time worked at all.

There were, of course, a few backgammon books, including Paul Magriel’s Backgammon. I read them all; but one thing that struck me as odd was that there were none devoted to the endgame (as there are in chess). The endings were essential knowledge to a chess player: not only did you learn from them the simplest elements of the game, as playing scales on the piano teaches a beginner the keyboard; but you could also use your familiarity with them to guide your play. From the middle game, and sometimes even from the opening, you would know what position to aim for.

In 1991, after playing backgammon for ten years, I decided to write a chess-style endgame book. I titled it In the Game Until the End, and decided to concentrate on a few of the simplest contact endings: the remnants of well-timed ace point games with only a few checkers left. The theme in most of these positions is: should the ace point side wait for a last (possibly winning) shot? Or should he finally run from his opponent’s home board, conceding a gammon?

This was still the pre-computer era, so I had to rely on exhaustive hand rollouts and proposition play to fix the equity value of a few key positions, which I called ideal closeouts 1 and 2.

Ideal Closeout1

endgame position

endgame position

Ideal Closeout 2

backgammon endgame position

backgammon endgame position

With these values more or less determined, I proceeded by brute mathematical calculation to solve a few of the basic positions. I found for, example that it was right to run in position A:

backgammon position A

backgammon position A

But right to stay in position B:

backgammon position B

backgammon position B

Probably the most interesting position I investigated was this one:

most interesting backgammon position

most interesting backgammon position

The eternal dilemma: should Black play the second half of his 6-2 by hitting on the ace point or by taking another man off? The solution to this problem involved a good deal of extra experiment and calculation; but at least one part of it was clear: I knew from my work on position A that if Black takes the second checker off he does not need to worry about “double jeopardy” – for if White misses he must not stay to try for a second shot. In this and a few other instances, I was able to partially achieve in a backgammon context the chess-like backwards-solving of the game (what the mathematicians call recursion) that had been one of my goals in writing the book.

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