Quick Rando Question : Fair Coin

[GuyOnTilt]

Exactly how thick would a coin have to be to have an equal probability of landing on all three sides?

GoT

[Nottom]

I'd guess somewhere around 1/Sqrt(3)~0.577 times the diameter of the coin. Assuming the coin is dropped from a random position so that no rotational velocity is involved which would seriously bias a coin this think.

Brief Explanation (hard without a picture).

../// <-coin
.///
///
..^

OK imagine that is a think coin sitting on its corner. Now it should fall to the flat side if its center of mass is to the right of that little arrow, and fall to its edge if its to the left. Now to make it even the angle that the center of mass is directly above the corner, should be 60 degrees. This leaves 60 for the edge and 60 for each side.

A little bit of trig later leads to the thinkness being .577*diameter in order for that to happen.

[Siegmund]

There is no unique answer, unless you specify exactly how you drop the coin.

If you drop it in random orientation *from a sufficiently low height*, choosing the length so that the center of mass is above each side of the coin for 60 degrees is good enough. (But that's not very effective randomizer, if you simply let go of it and then let it fall straight down without bouncing around any at all: your hand position before you release the cylinder determines the whole outcome)

Assuming no angular velocity when the coin is released doesn't help you, if the coin is dropped from a significant height, because it gains considerable angular velocity as soon as it strikes the ground, any time the center of mass isn't exactly over the point of first contact. Anything more than a few coin-diameters above the table and it will strike hard enough to spin past at least one face before coming to rest.

[SheetWise]

I've always been somewhat skeptical of three sided objects. I checked with the Bureau of Standards, and they confirmed that there was no uniform method to warehouse 3 sided objects. As I suspected, this pot-of-gold dissappeared as the dimensions replaced sides and speculators overwhelmed the market.

Please let me know of any dimensional changes in your future marketing. I have a cube I could apply (for the right price).

[irchans]

I would think that you would want the potential energy of lying on an edge to equal the potential energy of lying on a face. If those energies were equal then,

thickness = diameter.

[MikeL05]

[ QUOTE ]

thickness = diameter.

[/ QUOTE ]

[AaronBrown]

The coin doesn't have to be thick, you just have to flip it in zero gravity. If that's too much trouble, flip it in a liquid with density equal to the coin's density.

[sekrah]

It would have to be a perfect triangle for this to happen, no?

[Scotch78]

[ QUOTE ]
The coin doesn't have to be thick, you just have to flip it in zero gravity. If that's too much trouble, flip it in a liquid with density equal to the coin's density.

[/ QUOTE ]

I disagree. The vast majority of the time, the coin's first contact with the ground will be at an angle (to its sides). It will then bounce until coming to rest on a side. Because of the shape, it's center of gravity will almost always direct it to one of the larger sides.

Scott

Edit: The odds of it's initial contact being on a flat side, and thus not bouncing significantly, are also enormously higher for the larger sides.

[Nottom]

[ QUOTE ]
[ QUOTE ]

thickness = diameter.

[/ QUOTE ]

[/ QUOTE ]

For people who say this. Take some poker chips and stack them until they are as tall as a chip on its side.

Now think about whether you still think thickness=diameter.

[irchans]

I taped together a few pennies until the number of side results was approximately equal to the number of heads or tails. At about 11 pennies,

#sides ~= #heads ~= #tails.

A penny has a thickness of about 1/13 of its diameter.

[AaronBrown]

Except that it doesn't really come to rest. It stops moving, but there's no reason to expect it will be flat against a surface.

[mtrubo]

as far as the math goes, i get the same thing as Nottom.
...it's been a while since i've had to bust out the Tan function /images/graemlins/shocked.gif

[DMBFan23]

Nottom,

I would think the action resulting from the coin hitting the surface would impact the orientation of the coin more if it landed on its edge as opposed to the side...after all, on a side the coin has a whole side's worth of surface area touching the surface...very stable.

whereas on an edge, no matter how thick we make the coin, we still only have a very small amount touching the surface at any time

it seems like if the coin were to land initially on its side, it would be more likely to stay there as opposed to initially landing on an edge and staying on an edge. this is probably a function of the energy in the coin and however you can translate that to some sort of bounce coefficient - so every time the coin bounces you have the same process with slightly less energy.

I would think the coin would have to be slightly thicker than geometry would suggest to compensate for this

[DiceyPlay]

Is it landing on carpet or concrete? Is it made of soft metal or is it extremely brittle?

I'll guess 2/5 of it's diameter.

Naw, make that 3/5 - I don't want the coin to topple over too easily before it comes to rest.