What Are the Ideal Proportions for a Cylinder to Act as a Three-Sided Die?

[Helios]

There is a cylinder of certain proportions, that when dropped, lands either on its side or one of its two ends with equal probability. A coin is too short of a cylinder. A solid pipe is too long of a cylinder. Somewhere in between is a cylinder that acts like a three-sided die. I've imagined this problem but don't know how to solve it.

 

[Vanadium 50]

Draw a picture. (Always good advice)

You have a cylinder that touches the ground at one point on the rim, tipped at some angle. Draw the forces acting on the cylinder and ask for what angles does the cylinder tip over to the side and for what angles does it tip back to land on an end. You now have a relation between the length of the cylinder and the random angle it lands at.

 

[sir-pinski]

This is an interesting problem to think about! It seems like the key to solving it would be finding the right proportions for the cylinder, where it is neither too short nor too long, but just the right size to act like a three-sided die when dropped.

One approach could be to use geometry and calculus to determine the ideal dimensions for the cylinder. We would need to consider factors such as the height, diameter, and weight of the cylinder, as well as the surface area and the distribution of mass within the cylinder. By finding the perfect balance between these elements, we could potentially create a cylinder that behaves like a three-sided die when dropped.

Another approach could be to experiment with different cylinder sizes and shapes, dropping them repeatedly and recording the outcomes. Through trial and error, we could potentially find the right proportions that result in an equal probability of landing on either side or end.

Ultimately, solving this problem would require a combination of mathematical analysis and experimentation. It would also be helpful to consider real-world limitations, such as the materials and manufacturing processes available for creating the cylinder. Overall, this is a fascinating problem to explore and could have practical applications in creating fair and random games.