Good point! I don't know whether the problem can be fixed, but what if we posit that the spinner has to go around at least once; otherwise, the spin doesn't count & we need to try again ? Presumably there is a minimum energy/angular momentum needed to get the spinner to make a complete circle. Similarly, there is a minimum energy/angular momentum needed to get to each of the different marked off sectors. If the "slowing down" function as a function of the initial "kick" is f(k), k is the kick, then the final value is something like f(k) mod 1 (assuming we measure each full turn as being =1). We now need to allocate the interval [0..1) into N pieces s.t. the chance of landing in each piece is the same. I suppose for the sake of at least one solution, we could require exactly M full turns but less than M+1 full turns, where M is a constant decided upon ahead of time. However, we could just as well have started with a smaller energy & angular momentum in the first place. I guess that the "randomness" here comes from the inability to calibrate the initial kick finely enough to distinguish between M and M+1 turns, much less some intermediate sector. At 10:46 AM 12/17/2015, James Propp wrote:
I suspect I've misunderstood the question, but: if the initial impulse is small, won't the spinner end up pointing at whatever sector it's pointing at before you start spinning it (leaving aside a small proportion of cases where it starts close to a boundary between sectors)?
Jim Propp
On Thu, Dec 17, 2015 at 11:04 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I'm sure that everyone has seen and/or used one of those little spinners that come with many board games. There's usually a piece of cardboard with a circle printed on it and a metal spinning arrow. In order to produce a "random" number, you pluck the spinner arrow in such a way that it spins before stopping on a color or a number.
Now since the spinner is given a certain amount of angular momentum due to an impulse from your finger, it has to dissipate that angular momentum (and energy) in order to stop and indicate the color or number.
There are various models of the physics of friction that one might utilize to estimate where the arrow will stop. But it should be obvious that no matter which model we use, these spinner can't possibly produce unbiased random numbers -- at least so long as each of the colored and/or numbered sectors are identical in size.
So... Let's choose a preferred direction for the spin -- e.g., *clockwise*. Is there a physical model of friction that allows a sector size allocation so that the numbers chosen *will* be less biased -- at least to a first-order approximation?
In particular, is there a model of friction that allows a fixed allocation of sector sizes *independent* of the absolute size of the initial impulse ?