# Creating a fair 3 sided coin

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.

• Why does a normal dice not suffice? Aug 9, 2012 at 19:28
• @Bernhard I am not looking to actually have one per se, but I am curious as to what would make one fair. Aug 9, 2012 at 19:53
• 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. Aug 9, 2012 at 20:08
• 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. Aug 10, 2012 at 0:17
• @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. Aug 10, 2012 at 0:45