Some time in the 1920s a lone genius or more likely a group of avid gamblers in New York or Long Island came up with the idea of doubling. Backgammon soon became an exciting game rather than just a good one. Initially matchsticks were used to keep track of the value of the game. The doubling cube itself was a later invention. Despite a lot of research the origin of doubling remains lost in the mists of time. As there is probably no one alive who played in those heady games of the 1920s it may do so for ever. A recent discussion on the backgammon newsgroup on the Internet shed no light on the situation. Similarly the origin of the beaver (your opponent doubles you, you turn the cube to 4 but keep it on your side of the board) is also a mystery. The first reference I can find in a book is Jacoby and Crawford's The Backgammon Book of 1970 where, "beavers have no real part in backgammon, but they give desperate gamblers a chance to turn the cube over faster than otherwise." Of the beaver's cousins, the racoon (to beaver a beaver), the skunk (to beaver a racoon), etc. I can find no mention.
Backgammon bibliography before 1970 is sketchy. If any reader has any old backgammon books, I would like to hear from them.
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