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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.

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  • $\begingroup$ Why does a normal dice not suffice? $\endgroup$
    – Bernhard
    Commented Aug 9, 2012 at 19:28
  • $\begingroup$ @Bernhard I am not looking to actually have one per se, but I am curious as to what would make one fair. $\endgroup$
    – soandos
    Commented Aug 9, 2012 at 19:53
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    $\begingroup$ Ok, I think it is difficult to estimate this a priori. Maybe by trial and error you can find one? Let's see if anyone comes up with a nice answer. $\endgroup$
    – Bernhard
    Commented Aug 9, 2012 at 20:08
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    $\begingroup$ How about a three sided prism which has an equilateral triangle as a cross section. To eliminate the possibility of landing on the triangular ends, put a pyramid made of equilateral triangles on each end. By symmetry this would have to be fair. I know you want a cylinder and this it is more like a die than a coin but I doubt the exact cylinder proportions for equal probability could be calculated theoretically - for example it could depend on the properties of the surface it falls on. $\endgroup$
    – FrankH
    Commented Aug 10, 2012 at 0:17
  • $\begingroup$ @FrankH, assume the surface is perfectly flat and unyielding. The easier solution than a three sided prism is a three sided top, the side that is on the ground is the one that is counted. $\endgroup$
    – soandos
    Commented Aug 10, 2012 at 0:45

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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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