Consider a coin which consists of a strip along one side made of a dense metal, and the remainder made of a light metal. This would move the center of rotation to the side of the coin. Now place the coin on your forefinger with heads up and the heavy band facing away from you. Assuming the rate of rotation was high relative to the falling rate of the coin, then it will tend to first impact the surface with the leading edge of the light two thirds, face down.
The article makes assumes that conservation of momentum guaranteeing a constant rotation rate guarantees fairness. But this relies on the assumption that biasing the coin cannot affect the timing of when it lands.
Of course, this doesn't bias the coin in either direction, it just lets the flipper choose which way to bias a particular flip.
That's not biasing the coin itself though, it's rigging the whole flipping procedure.
If I toss a fair die and catch it in mid air with a robotic arm and high-speed camera so that I always catch it with the two face on top, it's still an unbiased die.
kevin_pet's coin would still be biased if it were caught. The article contends that you can bias the bouncing part of a coin flip, but not the falling part - GP proposes a coin with a biased falling motion.
It is very easy to bias a coin. Ask any stage magician. You can bend them, you can taper the outer edge, you can cover one side in a thin layer of something bounce-deadening (like rubber cement), et cetera.
There's a big difference between "you can't bias a coin using weight" and "you can't bias a coin."
The article says that the flip should be caught and not allowed to bounce or spin on the ground.
It still seems to me possible to bias it with weight, based on angular momentum. I think a cavity internally that went from the center of the coin towards one side, with a small weight that moved up and down freely would cause one part of the spin to go faster (when the weight was in the center) than the other part, and would cause a bias if the spin were slow enough not to keep the weight locked at the outside (maybe a spring would be needed? Turns out I don't have a good intuition for what happens when you mix gravity with fast rotation)
Thanks for those links, really interesting. Summary for anyone who hasn't read them: Diaconis did detailed analysis and ran experiments and found that coins are biased to land the way they start, but it's a small bias in normal use (51%). The article also claims that he has trained himself to reliably flip coins so they land the same every time.
Interesting that it seems to contradict the article linked here.
That's a neat trick. Based on the article and rewatching the video, he's not actually flipping the coin.
> In Section Three we prove that the angle ψ between M and the normal to the coin stays constant. If this angle is less than 45 ◦ , the coin never turns over. It wobbles around and always comes up the way it started. Magicians and gamblers can carry out such controlled flips which appear visually indistinguishable from normal flips.
He refers to it at the end of that video clip as tossing. looking closely that's pretty much what he is doing. it looks like he might be flipping it once during the toss for effect. Such a slow rotation allows him to easily catch it at the right moment.
