Cube Theory

There have been a number of articles, letters, and comments about the existence of a position which is both a double and a beaver. Bill Robertie (October, 1980) points out the flaw with the most common double-beaver position:

Should Black double? Should White beaver?

Although Black will usually win a gammon when he hits, he won't always win a gammon; in fact, he will occasionally lose the game. In order for this to be a double, Black would have to win a gammon more often than he actually does. This position is not a double and is an easy beaver.

Arthur Ramer (September, 1981) shows us a position which is both a double and beaver — almost. His solution (first shown to me by Kit Woolsey) would be correct only if he moves the five-point blockade out into the outfield, guarding against the distinct possibility of a backgammon! The resulting position, shown below, is both a correct double and a beaver (assuming the Jacoby Rule applies).

Should Black double? Should White beaver?

Notice that if Black rolls a six, he will always win a gammon and will never lose the game. If Black fails to roll a six, White will send the cube back and Black will be forced to pass. There, Black should double and White should beaver!

To add further confusion, Burt Simon has an interesting article (December, 1981) with a variation of the original (incorrect) double-beaver position.

Should Black double? Should White beaver?

Black is on roll, with the cube in the center. Should he double? Should White beaver? Even if Mr. Simon could convince me that he will win over 95% gammons when he hits, I fear he has overlooked the possibility of getting gammoned himself if he misses. Although this is unlikely, it is far from impossible. A one in ten chance of being gammoned is enough to "bust" his solution, even if he always wins a gammon when he hits! Black should not double. White should beaver.

So far, the Woolsey (Ramer) solution is the only one I have seen which unquestionably is both a double and a beaver.

More articles by Kent Goulding

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