The first note at the end of the page says: <quote>Theoretically, it would be possible to get it down to 64
bits by using Walter
Trice'sD() expressions,
but I think you'd have to be a mathematical masochist
to try it!</quote>
Walter states that the number of possible positions in backgammon is: 18,528,584,051,601,162,496
However 2^64 is 18,446,744,073,709,551,616 which is a bit lower than the theoretical number of positions.
I therefore wonder if the note in the description is a bit wrong.....
This email could end here, but there is more: if it is possible to get the number of legal (and relevant) position below 2^64 it would make hashing of positions more interesting. As disk space and memory space increases a perfect hash of position to 64-bit key would be the ultimate solution to ultimate question of life, the universe and everything.* Backgammon could be "solved" and implemented. (However finding such perfect hashing function seems a bit out of reach at the moment.)
Let's first try to get the number of legal positions below 2^64. The difference is "only" 8.190836056001331E+16 positions. I could divide this differently into contact and non-contact positions. I could also remove irrelevant positions (ie positions already won), but I don't see how I can remove much more. I could find illegal positions like both players closed out. .... still think we're way above 2^64.
If someone sees a brilliant way of representing a backgammon position (just the board) in 64 bit, I would be really interested for pure academic interest.