I ran into this today during a school competition...which is now over but I continue to ponder the question.

What are the dimensions of a coin that has an equal, that is, 1/3 probability of landing heads, tails, or on it's edge?

My thought was to set the dimensions such that each possibility, that is, heads, tails, or edge, placed the coin's center of mass at equal heights above the ground. This would require the diameter to equal the height. I'm seriously doubting this answer but I can't seem to come to any conclusion.

Any thoughts?

A cube.

Everyone knows the answer is either

-The plane won't take off
-The squirrels got there by swimming

Unless you're joking........

It's a coin. It needs to be a cylinder. The answer must contain only two measurements, height, and radius.

I suppose the height is some ratio to radius such that it compensates for the fact that a cylinder will not bounce fairly when striking a surface, because the caps are planar but the sides are not.

Lot's of these lately, did I start something of a trend?

diameter == height sounds right to me.

Well, physically this answer is probably terrible, but the first thought I had was to set it so that the surface areas were equal. So:

pi r^2 = 2pi hr

h = r/2

What shape does a rotation describe?

Circumscribe a two-dimensional coin within a circle. Compare the amount of the circumference contained within the edge vertices against the amount of the circumference contained within the face vertices.


[quote=The Mega-ZZTer]diameter == height sounds right to me.[/quote]Pretty close.

You sure that doesn't make it twice as likely to land on the edge (which counts as one surface) as on either of the faces?

No. I'll try to explain better.

Let's pretend that, instead of a coin, you have a spinner. There are three colors on your spinner:



The sizes are calibrated so the spinner has a 1/3rd chance of landing on any single color.

The spinner is a really good analogy for what's happening here. When you randomize a coin by flipping it, you're really rotating it in one plane. You don't care about the other two, because they're irrelevant to the answer you get. You can therefore visualize the coin in two dimensions. The 'arrow' on the spinner is the vector of the force of gravity, extending out from the center of gravity of the coin.

Let's imagine that this two-dimensional coin is circumscribed within this spinner. What does that say about the size of the coin? If you're willing to accept the above reasoning, you get a fairly simple answer:



So, like I said, MZZT and happydud were pretty close.

I suppose cylinder would be more accurate than cube, but yeah.

No, it'd be closer if they had said radius ~= height.

1.73:1 is hardly what I'd call a good approximation of 1:1.

Yeah, well good luck getting equal probabilities on heads and tails, let alone the edge.

I don't think you can just equalize the areas. You'd need to take center of mass and the shape into account to determine which side the coin will end up on. It will likely change drastically depending on the material it's made of and the material it bounces off of.