I don't think they can be. Or, at least, according to the guy I linked to who did a lot of thinking about it, when well-made were good enough for all practical purposes, and as good as many d6s.
But I don't think they can be actually be theoretically fair. I don't have a rigorous proof, but the argument in previous posts, based on the other guy I linked to, went something like this. Have a matrix of the probabilities of going from one side to another. The values depend on the friction and restitution of the surface. So changing the surface will change the probabilities of different transitions by different amounts, so an asymmetric die fair for one surface can't be fair for another.
That's not rigorous, we haven't proved you can't cancel out the changes, but it doesn't seem likely.
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Date: 2007-07-09 02:52 pm (UTC)But I don't think they can be actually be theoretically fair. I don't have a rigorous proof, but the argument in previous posts, based on the other guy I linked to, went something like this. Have a matrix of the probabilities of going from one side to another. The values depend on the friction and restitution of the surface. So changing the surface will change the probabilities of different transitions by different amounts, so an asymmetric die fair for one surface can't be fair for another.
That's not rigorous, we haven't proved you can't cancel out the changes, but it doesn't seem likely.