Copyright © Karl Dahlke, 2019
Did you ever think about a 3 sided coin? A die almost qualifies, since it has 6 sides. We could assign 1 and 2 to one outcome, 3 and 4 to another outcome, and 5 and 6 to the third outcome. Each is equally likely (ignoring tiny deviations due to pips). But is there a coin, or object, of any shape, that can land in one of 3 ways with equal probability? Put this book down and see if you can find a solution - the next paragraph is a spoiler.
Here is a solution, perhaps not the only solution. A standard coin lands on heads or tails, never on edge, except for the Twilight Zone. Might a coin land on its edge in the real world? If you tossed it a billion times, or a trillion? Maybe not, maybe never, but if you thicken it into a cylinder, it might. Imagine a coin that is very thick, thicker than its diameter. It might be a tall cylinder with the diameter of a quarter and the height of a man. More like a rod than a coin. Toss it off of a tall building and it falls, end over end, with only one possible outcome; it will land on its round "edge", and never on one of its two ends.
To recap, a standard quarter lands on heads or tails, and never on edge, but our extended quarter lands on its round edge and never on heads or tails. Apply the Intermediate Value Theorem from real analysis. Somewhere in between the two extremes, a quarter is just thick enough to land on heads, tails, or edge, with equal probability. It would be interesting to calculate the thickness of such a coin using a supercomputer, or, construct coins of various thicknesses and toss them experimentally.
Assume the height h equals the diameter d. Spin the coin end over end in the air. As the coin descends, and strikes the table, it looks like a spinning square. It lands on edge or end with equal probability. But there are two ends, so it's edge ½, heads ¼, and tails ¼. That's not what we want, so shorten the coin just a bit. As a guess, I'd say h should be about 80% of d.
This can be generalized to higher numbers. Suppose you want a coin with 7 different outcomes, each equally likely. Instead of a circle, let your quarter have the shape of a pentagon. Now it has heads, tails, and 5 edge sides. Give it just the right thickness, so that all 7 sides come to rest with equal probability. Perhaps h = 1.4×d.
This may not generalize forever. Imagine a coin with a million edge segments, insearch of a million and 2 outcomes. The coin would have to be extremely tall, so tall that it lands on heads or tails with a probability of one in a million. At that point the coin is so tall that inertia probably keeps it turning, so that it never stands on end at all. If we reach a height where heads and tails are simply impossible, then tossing the coin is equivalent to spinning a wheel having the same number of slots, i.e. the coin in cross section.