Magriel’s theorem

Throughout this course, we have learned many rules of thumb to help guide our play. We’ve also repeatedly cautioned against blindly applying them, carefully pointing out that there are exceptions to every rule.

Magriel’s theorem is different. In his “bible of Backgammon,” Paul Magriel, the godfather of modern backgammon, states that:

[In non-contact positions,] it may be proved that no possible position can exist when it is wrong to use a 1 to bear a man off the one point.

Stated more simply, in a race, if you can take off a checker with an ace, just do it. Danny Kleinman formally proved this in the 1970s, and we link to an article on the topic by Raymond Kershaw below. Art Benjamin has also stated that this has also been proven using exhaustive computer searches. As far as I know, this rule is the only global one in backgammon.

While researching this topic, I discovered a related theorem, which I’ll call Magriel’s Scholium.

[In non-contact positions,] it may be proved that no possible position can exist when it is wrong to use a 5 to bear a man off the five point.

This is less handy, but it follows pretty straightforwardly from the first theorem, and is no less universal.

I must admit that, for “global rules,” these are fairly underwhelming in terms of their utility. Hopefully this helps highlight how multidimensional a typical backgammon position is — your play needs to consider your board strength, your opponent’s checker distribution, the race, the score, how badly you need to fight for a gammon, and a multitude of other factors all at once. The game doesn’t lend itself to being distilled into a single catch-all rule that can always be applied. That’s part of what makes it challenging — and rewarding, too.

Further reading:

• It’s A One-Off, an article by Raymond Kershaw