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The frequentists' definition of a "fair coin" is a coin that, tossed a million times, converges toward an equal number of heads and tails.

But what is meant by converges? We might imagine a plot of heads minus tails, in which an erratic line gradually smoothes into right-infinity, hugging the x-axis ever tighter.

But how many tosses is truly enough? How do the frequentists know that after 25 happy lightyears, the plot won't suddenly divorce its beloved and shoot upwards to y-infinity?

Fools.

Let us design a coin to confound these quasi-intellectuals and run their casinos out of business. Create a coin that at first exhibits convergence, but after many, many flips, gradually biases completely in favor of one side.

  • No external forces, e.g. magnets.
  • The coin must work anywhere with Earth-like gravity.
  • The coin must work 1,000 years from now, i.e. no batteries or disintegrating materials.
  • The coin may be as intricate as desired, but throwable by a human, and must land only on one of two sides (provided the ground is solid enough e.g. a thin edge won't pierce the ground).

I have ideas that may produce such a coin, but so far nothing elegant enough to consider an answer. Might be off-topic, but I thought it might be interesting enough to inspire Puzzling.

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  • $\begingroup$ what do you mean by "create"? do you mean in physical term (lighter side, put some heavy elements on some side etc) or some puzzle term? $\endgroup$ – Oray Jun 30 '17 at 7:00
  • $\begingroup$ @Oray - Good question! Imagine you have a team of the most brilliant scientists, engineers, and access to all materials currently known and available on Earth. Since I am looking for the most elegant answers though, reliance on crazy materials or reactions would naturally be favored less than simpler mechanical answers. $\endgroup$ – Andrew Cheong Jun 30 '17 at 7:02
  • $\begingroup$ and how much unfair coin is needed for this? %51 vs %49 or more? because statistically you can determine if the coin is unfair after some trials if the percentage gap is too much... $\endgroup$ – Oray Jun 30 '17 at 7:03
  • $\begingroup$ @Oray - Let's use the same definition used for convergence, but for divergence. It's been 10 years since I spoke "math," but hopefully you will understand: For all $p$ greater than a "turning point" $T$, there exists a $q > p$ such that for all $r > q$, $F(r) > F(q)$. $\endgroup$ – Andrew Cheong Jun 30 '17 at 7:07
  • $\begingroup$ @Oray - Hm, I think that "statistical determination" assumes some property of the shape of convergence, e.g. normal. That is the notion I am trying to dispel with this puzzle (: $\endgroup$ – Andrew Cheong Jun 30 '17 at 7:09

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Frequentists won't be damned. This is a trick question.

In looking at making a coin, you can't bias it by weighting. There is an entire class on such myths and methods of bias at Berkley (at least there was some years ago). I attached a link to a paper that describes actions such as taking a plastic ship and adding heavy putty to one side to see if you could get it to bias. You can't. It has to do with the fact that when the coin is spinning, it shows each side exactly 1/2 of the time. Your only option to make it land differently is to toss the coin unfairly (such as very slowly, or toss it like a Frisbee). A coin tossed fairly is assumed in the coin toss above. To quote a Berkley:

Deterministic physical laws govern what happens in the flip of a coin and the throw of a die, but we consider these events as random. It’s hard to separate the random from the deterministic even in something as simple as the coin flip. What makes a coin toss fair?

The uncertainty of the coin’s initial state is the key. A coin tossing is basically deterministic. The coin obeys Newton’s laws of motion, with its final state depending on its angular velocity (rate of spin) and time traveled (which in turn depends on the upward velocity with which it is flipped).

For tosses where the coin spins rapidly and goes high in the air, the set of initial velocity values that lead to either heads or tails are of equal size. That is, half of the initial conditions lead to heads and half to tails (see, for example, Keller, 1986, and Peterson, 1990, 1997). So, uncertainty in the initial state (for example, a smooth probability distribution on a range of values for the initial state) leads to the coin landing heads half the time.

https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf

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  • $\begingroup$ @AndrewCheong If you read the rest of the paper, it shows how weighting and changing the coin cannot affect the 50/50 outcome. My answer is based on the three tags for the question: Physics, Mechanical Puzzles and Coin Tossing. $\endgroup$ – Keeta - reinstate Monica Jun 30 '17 at 15:28
  • $\begingroup$ (I wanted to rewrite my comment more coherently.) Actually @Keeta, you and I are on the same side (heh) here! You are indeed damning the frequentists in the same manner I mean to damn them. Basically, the Bayesian definition of "fair" is that, as long as you don't know anything about a coin, you can't possibly figure out its bias, and that's where the source of (perceived) randomness is. The frequentist definition, meanwhile, is based on flipping the coin a million times. (See my previous question on math.se for more debate.) $\endgroup$ – Andrew Cheong Jun 30 '17 at 15:29
  • $\begingroup$ Now, it's not like there really are two groups who disagree in reality. I just spun it that way for fun. But you and I certainly say nay to frequentists. They know nothing! Damn them all. $\endgroup$ – Andrew Cheong Jun 30 '17 at 15:29
  • $\begingroup$ I see! I must not have read deeply enough. Let me do so later in the day and get back to you (: I can't seem to load that PDF at work. $\endgroup$ – Andrew Cheong Jun 30 '17 at 15:30
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I have an idea.

You take a normal coin. It is evenly weighted on each side (if you cut it exactly in half along the side the two halves would weigh the same).

Now

You know scratch cards? Those things were you buy 10 and win £1 or if you are really lucky £2? Well the silvery scratch stuff on that is called UV scratch off ink.

So

You take a large portion of this ink and weigh it. You then slice a piece off one side of the coin that is the same weight. You fill in the space with the ink (it dries under UV light to become solid)

ChrisH has provided an excellent idea as an alternative: solder

However

As @BreakingMyself points out - the coin will be larger on one side because the ink will be have a different mass to the metal of the coin. This can be easily fixed by applying a normal ink with the same mass to the other side. It will make it a fair coin.

If we were using solder instead then the counter could be steel. It would be possible to make a high density tin solder with the same density as steel. See here

So

At the start the coin is weighted evenly so will show convergence. But as you keep flicking the coin your nail will slowly scratch away the ink (or solder) and the coin will become biased as it will be more heavily weighted on one side.

Diagram of idea (apologies for bad freehand writing):

enter image description here

It doesn't have to be scratch off ink or solder - it can be any material that can be eventually scratched off by a nail. However it does require a counter material with the same density to use on the opposite side

Checklist:

  • No external forces, e.g. magnets.

Nope - the only force is weight

  • The coin must work anywhere with Earth-like gravity.

The coin would work anywhere that has gravity. Even in almost zero gravity - though it would take longer to flip. This is simply because the coin starts off with the same mass on each side and eventually has more mass on one side.

  • The coin must work 1,000 years from now, i.e. no batteries or disintegrating materials.

As long as the coin doesn't disintegrate from constant flipping this will be fine.

  • The coin may be as intricate as desired, but throwable by a human, and must land only on one of two sides (provided the ground is solid enough e.g. a thin edge won't pierce the ground).

This coin will look simple and he slightly thicker than a usual coin but it will still only land on one side. If necessary the coin could be dyed to look like an actual coin.

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  • $\begingroup$ I would assume that the new 'side' will be of a different thickness to the original due to the mass of the materials. Wouldn't this move the centre of the coin when flipped, so it starts bias anyway? I know it will be balanced at the point between halves but the way the coin is flipped (and other forces acting on the coin) should make a difference to the motion it takes because of the lop-sided shape. $\endgroup$ – BreakingMyself Jun 30 '17 at 8:11
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    $\begingroup$ @BreakingMyself hmmm good point. Perhaps with the tech available we would be able to make a material with same weight but is scratchable. I'll do some research. I'll also see if I can come up with an idea to get round this $\endgroup$ – Beastly Gerbil Jun 30 '17 at 10:19
  • $\begingroup$ You could use a very soft alloy such as solder $\endgroup$ – Chris H Jun 30 '17 at 13:20
  • $\begingroup$ @BreakingMyself proposed a solution to the problem. You get an ink that doesn't scratch with a similar mass and fill in the opposite side until it is evenly weighted. I'll see if I can actually find a good example of one $\endgroup$ – Beastly Gerbil Jun 30 '17 at 13:20
  • $\begingroup$ @ChrisH that would be a good idea too. There are probably lots of materials that work. You'd need a counter material with a very similar mass that doesn't scratch too though. I'm trying to find a good match now $\endgroup$ – Beastly Gerbil Jun 30 '17 at 13:21
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You can avoid the need for degrading materials:

Inspired by the pitch drop experiment, create a hollow coin with an internal reservoir of a highly viscous fluid such as pitch. Place a weight in the center of the pitch. If everything is centered, then this will behave entirely as a fair coin until in the distant future the weight starts moving through the pitch, at which point it will become biased.

As a bonus:

If you are extremely patient, you can reset the coin to fairness by holding it at a precise angle while the weight slowly slides back to the middle.

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  • $\begingroup$ Why would this take such a long time though? Surely the weight would start moving immediately? (disclaimer: not a fluid dynamicist) $\endgroup$ – Rand al'Thor Jun 30 '17 at 14:29
  • $\begingroup$ @Randal'Thor It would start moving immediately, but since pitch is practically a solid on short timescales it would take a while to see any effect. $\endgroup$ – cobbal Jun 30 '17 at 14:34
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Without any materials changing or falling apart—

I see a lot of solutions based on decaying materials (whether radioactively or otherwise, e.g. glue). I want to post one idea that doesn't involve any materials morphing or falling apart, in hopes of inspiring other such answers!

Imagine a coin with a kind of screw chamber in it—a screw with very slight gradient: enter image description here But add miniscule ridges* (depicted by triangles here, since it's a cross-section) that would bias the flow of any material, like extremely fine sand / clay, in one direction. Fill the coin with such material in the center, balanced, perhaps in a special chamber that adds an especially great obstacle for sand to leave. Over time, all or most of the sand is able to leave, and with every coin toss, be driven by centripetal force toward one side of the coin. *After building a "prototype," one would realize that the asymmetry of the wedges themselves create an infinitesimal bias. The final build can adjust the position of some of the wedges, ever so slightly, to compensate for this bias and produce a "perfectly" fair coin (at $t=0$).

This coin is also, theoretically, "resettable"!

You must construct a sort of, specially designed centrifugal spinner, that can measure the center of mass / bias of the coin and constantly calibrate by spinning the coin at high speeds clockwise or counterclockwise, using gravity as a helping force.

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Well, no matter how hard I look at it, this coin needs something that changes, at least slightly, after enough tosses. So, it has to be electronics or degradable material. At least in some way. Sure, the material can and should degrade by mechanical deformations by tosses instead of time, but it has to - it can't be the same thingy to infinity.

Now I propose:

Coin that is made out of 4 pieces. 2 mostly flat sides you see, solidly fixed together. Inside that, 2 weights, one attached to each side. Weights are the same, one weight is strongly attached, the other is lightly glued. Throw it ~100 times and glue will hold to abuse well enough and the coin will be fair. Throw it 10k times and the glue will break, and the side with fixed weight will be heavier.

Now let's take this solution if you want to cheat in casino, and work on it a bit more:

It would be possible to design it to have it go from 0.5:0.5 to 0.55:0.45 to 0.45:0.55 etc - to have it changing which side it is biased towards. By having enough weights inside the coin, held by various strengths of glue.

Another solution, this time not matching your "no electronics" requirements, but without mechanical degradation:

You could have a tiny little coil that keeps recharging a battery, both parts designed to last forever - getting energy from the throws. Now what you need is just a tiny bit of logic like "when biased side gets reached N times in a row, bias switches to the other side". Bias would switch by position of internal weight. With fair coin it will likely take many throws to reach this, so it will be fair. But when you have it biased, say 60% vs 40%, it will be switching sides much more rapidly, making it harder to notice it is biased to one side.

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  • $\begingroup$ Agreed. Some state needs to change. I chose the word "disintegrate" to discourage a certain type of state change, but it may not have been specific enough. Glue will certainly work! At first I was skeptical because of its sudden, "discontinuous" jump in pattern once the glue broke, but statistically (plot-wise), the trend should be smooth indeed... (at least in lower orders). $\endgroup$ – Andrew Cheong Jun 30 '17 at 14:10
  • $\begingroup$ For your electronics-using version, it's impossible to be sure which way it's biased. I'd insert patterns like HTH -> T bias, THT -> H bias; easy to remember patterns and you can restart the game profitably fairly soon after a toilet break ;) $\endgroup$ – JollyJoker Jun 30 '17 at 14:21
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Following other thoughts here, a coin cut in half and "altered".

Have one side with an alpha-emitter with a known half-life, and the other side with the equivalent mass of a non-radioactive material. Over time the alpha side gets lighter, and starts skewing the results.

Unfortunately due to the way coin flipping works, all the solutions involving unbalanced sides would only affect things when the coin lands on its side and falls over.

Another (slight) possibility is

Having the two sides of the coin made out of different materials that appear the same, and have the same mass, but behave like a bimetallic strip - so when the "cheater" wants to they can hold the coin tighter to raise the temperature enough to change how it falls. This one isn't as long-term, but is easier to control when it happens ;-)

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So,

Insert into a coin a two layered rectangle, each layer from different materials but adjusted this way the center of gravity remains the center of gravity of the coin. Let's the half-life of one material differ drastically from the half-life of another - but both be lower then the cover of the coin of course. Some million years passes by - and voila.

The nice thing with this you don't even need to toss, but you can and won't see any anomalies for a while.

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  1. Replace some mass of a standard coin with equal mass of a hard radioactive material.
  2. Make sure the material has a distribution such that, if it weren't there, the weight distribution of the "ordinary" material the coin is made of makes the coin tend towards one side more than the other.
  3. Wait tens of thousands of years and watch as the coin gradually tends more and more towards one side.
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  • $\begingroup$ You beat me by 9 seconds :-P $\endgroup$ – Rycochet Jun 30 '17 at 13:41
  • $\begingroup$ Early bird gets the worm :-) $\endgroup$ – Vegard Jun 30 '17 at 13:51
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    $\begingroup$ You can do much more than just making the coin unfair with this. You can make the coin start fair, then favors heads, then go back to fair, then favors tails. Any combination should be achievable with the correct selection and placement of materials. This is how you make the evilest coin of all. $\endgroup$ – A Bailey Jun 30 '17 at 16:04
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How about a coin which has loads of microscopic holes on one side, too small to see. At first, the coin is perfectly balanced, but as time goes by the cavities fill up with dirt (anything heavier than air will do) and so the coin becomes heavier on the side where the cavities used to be.

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  • $\begingroup$ Only if there's dirt around. What if the coin is kept in a vacuum? $\endgroup$ – Rand al'Thor Jun 30 '17 at 14:29
  • $\begingroup$ I was assuming that the tossing involved something touching the coin. Well, honestly I was thinking "hands", but that might be too simple. ;-) But either way, as long as something touches the coin to toss it, that something might be worn and drop some of it's particles into the coin cavities. In total vacuum, with nothing ever touching the coin - yes that would prolly kill this idea! =) $\endgroup$ – Culme Jun 30 '17 at 14:33
  • $\begingroup$ Regarding vacuum, the coin ought to be LANDING on something even if it's being tossed in vacuum. Particles from the landing surface might wind up in the coin's cavities after a few million tosses... If tossed and landed using some magnetic field or similar, that would be very troublesome. $\endgroup$ – Culme Jun 30 '17 at 14:44
  • $\begingroup$ Hmm, true. This is somewhat reminding me of the bird and the diamond mountain :-) $\endgroup$ – Rand al'Thor Jun 30 '17 at 14:47
  • $\begingroup$ =) The coin must be veeeery durable indeed. $\endgroup$ – Culme Jun 30 '17 at 14:52
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Two solutions:

  1. Put velcro-like microscopic "hairs" on one side of the coin (much smaller than the usual ones, to make them invisible). The coin will be obviously biased towards this side (depends on the texture of the surface it lands on), but gradually the "hairs" will wear down, making the coin fair. So, carefully balance it at the beginning to look fair, so with the hairs wearing down it will be more and more unbalanced.

  2. Put miniature plates sliding out from the rim of the coin, spring-fixed so they slide out only when the coin rotates. Make them stick out by 45° - they'll divert the airflow around the coin, thus biasing it towards one side. Again, carefully balance the coin to look fair. With the time passing, the springs will rust in place and the plates won't slide out anymore, making the coin unbalanced.

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