[Nate]

My solution, since you're all probably getting tired of grasping at straws by now:

1) Draw a six-pointed "compass" onto a sheet of paper with six imaginary "cardinal directions."

2) Label the directions as 1, 2, 3, 4, 5, and 6 in any configuration you like. Order doesn't matter.

3) Flip a coin. Whichever direction the coin is most closely oriented to is the result of the die.

Since the orientation of the coin (i.e. the direction from which the coin can best be read, basically, so just imagine the coin as a glorified spinner for spin-the-bottle or what-not that you flip to determine direction rather than spin) has an infinite number of possible results from your flip (that is, the spectrum of possible orientation angles in relation to your six-pointed compass represents the output of a continuous, rather than discreet, function), the probability of the coin orienting itself exactly on the line between two directions is, literally, zero. (Of course, any statistician will tell you that zero-probability events happen all the time, such as the event of the coin landing at whatever orientation it DID land at when you first flipped it, but that doesn't change the fact that they have zero probability of occuring.) Most results you'll able to read with just your eyes, but for the results that are contestable, a simple protactor would probably suffice as a judging implement (you should probably draw an arrow on the coin in order to make this process easier). It works better if you flip the coin onto a smooth, hard surface, as it will then be more apt to spin around before settling into its final orientation, thus making the orientation less dependent upon the flipper's skill in flipping the coin (this is sort of analogous to using a dice cup or your hands to shuffle dice around before rolling them, and to ensuring that the dice bounce a few times instead of simply landing and stopping their movement immediately).

[Nate]

Originally Posted by RB+Gravity

Oh, mathematically.

Well, that's kind of the point of the exercise. Thus far, we've identified two (with my most recent post and Jonodude's coin-pull method) ways of doing this non-mathematically and one way of doing it semi-mathematically (flipping a billion coins and letting the law of large numbers eliminate biases toward certain results). We have yet to come up with, and there may not be, a rigorous / robust mathematical method for solving this problem.