Unfairly Tossing a Fair Coin

I had a good laugh while reading this CBC article about teaching students how to cheat at coin tossing. The article does not relate the ways by which this can be done, however the techniques have been well-understood and used or abused for a very long time. If you are unfamiliar with the "tricks" involved, it may be difficult to picture how it could be done; that is, how the tosser can cheat without rousing the suspicion of observers.

In a traditional coin toss the coin is launched from a flat position with the spin imparted by an off-axis force from (usually) the thumb. A coin flipped in this way has two superposed motions: a gravitationally-determined parabolic arc and a rotation about an axis that contains a diameter of the coin. The result is a coin spinning rapidly end-over-end and travelling through the air in an arc. Since the spin about the coin's diameter is many times the rate of the (roughly-speaking) half-revolution about an invisible centre -- usually near the parabola's focus -- and both rates are not precisely controlled, the outcome is sufficiently uncertain that the outcome of the toss -- heads or tails -- can be said to be random. It isn't entirely random: it is possible that a tosser with a sufficiently fast eye and hand could interrupt the coin's flight at just the right moment to reach a desired outcome. However it is good enough to be an uncontroversial way to decide the kickoff in a football game.

That's was a simple toss, but there are other ways to spin a coin. Further, these various motions can be combined to create some complex, seemingly-random spins that are often easier to manipulate. If you want to find out all the ways to spin a coin (and cheat) I recommend you read the description provided by the physicist and statistician E. T. Jaynes. He went so far as to train himself in the various techniques and then test the results. From the raw data of many, many hundred of trial runs, he became very accomplished at achieving almost certain outcomes with tossed coins. The book where his description can be found is called Probability Theory: The Logic of Science, which you may want to avoid purchasing since it is heavy, large, expensive and highly technical. Happily, early drafts of the text are available online. One place is here (browse down to chapter 10) provided you have a Postscript viewer handy. If not, do a web search and you will find versions in other formats, though you may have to hunt a bit in each one since there is no consistency in the work's structure and content across its many versions. You'll also need a good imagination since Jaynes provides no diagrams, just precise descriptions.

If you don't want to do all that work, let's review a few of the spin types that a fair coin can exhibit. The most trivial is where the spin axis goes through the coin's centre and is orthogonal to the coin's surfaces; in other words, the spin axis is at a right angle to that in a traditional coin toss. No matter how much the coin spins, the result of the "toss" is trivially identical to the face that is showing at the start.

Next, we can launch a coin spinning on its orthogonal axis along the same parabolic arc as in the traditional coin toss. However, if no additional spin component is employed, this form of travel will not affect the outcome: it will still show the same face as if the coin were not launched. You can try this by placing a coin in your palm and accelerating it upward without changing the attitude of your palm. You could also do it with your thumb but it is more difficult to centre the force so that the coin does not tumble. In either case, this is difficult to do while also imparting spin around the orthogonal axis.

A slight variation is to launch the coin so that the same face of the coin points inward (along the coin's orthogonal axis, and approximately toward the parabola's focus). This is much like the tidally-locked orbit of the Moon around the Earth where the same face (but not quite due to libration) always faces the Earth. This is nearly as easy as a flat toss, but this time the outcome of the coin toss will be opposite to the face shown at the start.

Now it gets more complicated. Consider a spin axis that is intermediate between the two preceding cases of the orthogonal and diameter axes. The resulting motion is a wobble that is much like you get by spinning a plate on a table if you launch it an angle to the surface (where the plate touches the table at only one point). You do this with a coin by centering the force imparted by your thumb off to one side of the coin while simultaneously pulling your thumb towards yourself. You may have to brace the coin with another finger so it doesn't slide off your thumb, or you can direct your thumb's motion at an angle that is off the vertical. It can be learned with only a little practice.

The wobble amplitude (the angular difference between the highest and lowest excursions of the rim at any fixed point at the coin's edge) is important to creating the impression of a fair coin toss; to an untrained eye a large wobble is hard to distinguish from a spin about the diameter axis, if the spin rate is high enough. However if you think about it for a moment I think you'll see that the outcome of the toss is exactly the same as the original, trivial case where the spin axis is orthogonal (no wobble, and the top face doesn't change). Combine this with a flat or tidally-locked arc and you can get any outcome you desire. According to Jaynes, depending on the resiliency of the landing surface you can often let the coin land on the floor without greatly reducing the predictability of the outcome. This tactic also has the advantage of allaying the doubts of observers about the toss's fairness. It is of course better when the surface is soft enough that the coin doesn't bounce, such as a natural-turf football field.

There is more to it if you want to get into even more arcane combinations of spin axes that give highly deterministic outcomes while appearing to have random, unpredictable flights. Jaynes pretty much goes through all the possibilities in his book so I'll leave off here and let the curious read what he has to say on the topic.