I want to make a cylindrical three sided fair coin, with sides: heads, tails, an

Question

I want to make a cylindrical three sided fair coin, with sides: heads, tails, and edge.

What should the area of the edge be in relation to the area of the head of the coin?

Assume it is all made of a uniform material.

Thoughts: I was thinking that so long as the surface areas of all three sides were equal, that would be enough, but this seems to lead to tipping over and landing on one of the other faces. Another thought was that the height of the edge should be equal to the diameter of the face, but this seems much too thick.

I am looking for a rigorous way of approaching the problem, as opposed to using (bad?) intuition.

Explanation / Answer

Seriously, by far the easiest way is to find it empirically. Take several coins and find the number of coins it is required to make a fair one by piling them together. The problem seems to be hard from the theoretical point of view.

However, a bit of googling in google scholar brought out "Probability and dynamics in the toss of a non-bouncing thick coin". The abstract says:

When a thick cylindrical coin is tossed in the air and lands without bouncing on an inelastic substrate, it ends up on its face or its side. We account for the rigid body dynamics of spin and precession and calculate the probability distribution of heads, tails, and sides for a thick coin as a function of its dimensions and the distribution of its initial conditions. Our theory yields a simple expression for the aspect ratio of homogeneous coins with a prescribed frequency of heads/tails compared to sides, which we validate by tossing experiments using coins of different aspect ratios.

As you see "without bouncing on an inelastic substrate" is not a very realistic assumption, so empirical approach would still be better. Nevertheless, I recommend you to read the article, I didn't, but I think there should be quite a lot of useful information.

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