Does a Fair 3 Sided Coin Exist?
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- เผยแพร่เมื่อ 25 ก.ค. 2024
- UPDATE: Download the simulator: • Three Sided Coin Simul...
Turns out that the real world is kind of complicated, who knew?
And for those of you wondering, no, I didn't use the default cube.
0:00 Intro
1:01 Simulation
1:30 Results
2:56 Conclusion
How thick is a three-sided coin? - Stand-up Maths
• How thick is a three-s...
Fair Dice - Numberphile
• Fair Dice (Part 1) - N...
How Many Digits of Pi Do We Really Need? - Real Engineering
• How Many Digits of Pi ...
My original simulation
• How Thick is a 3 Sided...
Music:
"Head Candy" - William Rosati, from the TH-cam Audio Library
"Jazz Comedy" - www.bensound.com/royalty-free...
"Robots a Cometh" - Dan Lebowitz, from the TH-cam Audio Library
"Hip Jazz" - www.bensound.com/royalty-free...
Intro sound effects from: freesound.org/
I have to say, when i watched the stand up maths video, i felt like way in which the coins got thrown would make the results inconsistent. I never imagined that almost every single variable would affect the necessary thickness. I imagine this took quite a while to figure. Thanks for satisfying my curiosity.
A three sided coin already exists if you think about it enough
A 3-sided "coin" might not exist, but a 3-sided dice does exist. It takes the form of an ovoid. You can modify the ovoid so that it has 3 ridged faces separated by 3 ridged edges. Those faces will still be rounded, somewhat. (I'm sure there is a name for this object, but I am not aware of it.) The ridge ovoid can't give you unfair rolls.
The ovoid shape can be used for n-sides. There is an upper limit as you approach pi. The lower limit is 2, and looks like a coin.
iirc there are actually at least 2 fair 3-sided die.
One is a d6 with the faces joined into groups of two, and the adjoining edge being rounded off.
The other looks kind of like a purse, and I think it is the one you are talking about... but I had a hard time visualizing it based on your description.
A pill shape with 3 equal faces flattened into the shaft of the cylinder?
a triangular prism with pointy ends (so that it wont roll onto them) and you throw really high, i'd say thats a fair 3 sided dice but i dont think there is a 3 sided fair coin
Alternatively, and infinitely long triangular prism. Or, a d6 with 2 sides labeled 1 and so on
@@benjamincruz6633the infinitely long 3 sided prism sounds great
Maybe this is why stand up maths never made a follow up video, turned out the results were just too messy
I had this idea in the car once. I assumed there exists a value since you can write the probability as a function of length and use intermediate value theorem. Neat to see it in action
I think the problem here is that different way of throwing the coin would result in different result. On a symmetric dice or coin this won't be a problem, but symmetry can't help us here so I don't think it's gonna work
@@l1ttel_y699yes but you can still average out with all different kinds of throwing
@@default2826 You are missing the point. It would be really easy to "cheat".
I can’t believe I just found this video now, but those simulations were a surprisingly good way of understanding how the factors actually influence the result. I had emailed Matt Parker a few years back and after the video came out, my only real understandings was that it was messy and probably a single value didn’t exist, but not much understanding of why. This I think really helps give a better idea why those things influence the final value.
Fantastic video!
Fair dice seems to be a modern invention. E.g. most roman dice were wildly "unfair", having a long side up to 1.5 times longer than the short side. I assume they could have made dice much fairer than that, but didn't want to.
If I'm not mistaken, this was because dice were pieces of natural bone, rather than crafted objects. Also, in Europe, probability wasn't really understood as a mathematical discipline until around the Rennaissance. The book " The Drunkards Walk" goes into the history of western thought on probablity
@@dr.limeade3035 Or it's because they wanted to add more strategy into their games. If every possibility is equally likely, then it is purely random, which means there is no strategy: you just hope you get lucky. But if some possibilities are more likely than others, then you can strategize around that. You can take a risky gamble and bet on a lower probability, or play it safe and bet on the higher probability. This creates a risk vs reward mechanic.
@@dr.limeade3035in Polish, even the word for "dice" and "bones" is same word
@@SchemingGoldberg There are a couple problems with what you are suggesting is that this dynamic exists in dice games with equal probabilities. Craps gives bigger payouts for more unlikely rolls. Second, a lot of dice games ARE about just luck, especially in a gambling context. Third, probability was not an understood concept in antiquity, and dice games were often viewed in a metaphysical and supernatural context
@@dr.limeade3035 They must have had some understanding of probability. Humans and many other animals even have an intuitive understanding of probability, look up statistical learning. Their understanding of probability was probably as good as that of any average modern human, no pun intended. They might not have any academic understanding of probability, but neither do 99.99% of modern day humans.
Why not use a triangular prism, but instead of the projection being straight, have the edges pinch into each other on each end of the projection, creating a shape similar to a rugby ball?
Been done, does work, not a coin
That's just triangular pencils
wow, as someone who loves thinking about these kind of problems, you covered everything I would have thought of to discuss, and some things i never considered, in less than 4 minutes. Phenomenal job!
Would have been nice if you've had written the value meanings on the respective axes of your graphs. That would have made it a lot easier to understand and less ambiguous
Realy wanted to do that after watching standup maths video, though couldn't find time and eventaully forgot bout this. Thanks a lot for that vid m8 ;)
Awesome video, and thanks for mentioning other related videos to watch from other channels. That's very cool of you.
Omg thank you so much I have been thinking about that stand up maths video since then
One more interesting test would be to repeat all experiments with different numerical precision. (64bit/32bit/16bit/8bit floating point numbers)
The size of the time steps in the simulation could also affect the outcome.
Testing these could prove the simulation trustworthy or not.
First off, thank you very much for your work on this. Given that the coin flip depends on the precise starting conditions, I think it could be worthwhile exploring the necessary coin thickness when integrating over the range of 'plausible' (human thrown coin conditions? ground is mud, gravel, ...), but different starting parameters. Seems to me, that once you either define a range of starting conditions and assume a prob. distribution (e.g. just constant in the range) for your starting parameters you will get an answer for the coin thickness. Stand up maths assumptions were to calculate the initial point of ground impact (if the coin would be covered by a sphere or circle) and then state that the face the impact point is projected on is where the coin would land finally (always given in understood it correctly, which is not always the case!). These assumptions are quite strong and lack to deal with the involved dissipation, velocities bounciness as you described. I would really appreciate a small follow up or alternatively if you could make the simulation results available for others to crunch the numbers. Thanks again
Thanks for making this video 3 minutes. I saw this video title and was expecting a 30-minute video (and I don't like watching those). Thanks for not being like other channels. Subbed.
You saved me a whole 4min by putting the answer in the thumbnail thx
Great video! Have you seen the d3s that have a cross section almost like the inside if a mercedes emblem? I got some like this for playing DCC and think they're a great solution and kind of keep the spirit of a coin
Nice work here!!
omggg this video is extremely amazing!!!know a lot more on three-sided coin topic
Curiosity box once had a set of dice that contained a different design for a 3 sided dice than the one you used in the video. But then it wouldn't be a coin. Maybe it would be interesting to test that one.
I mean, just a seed shape with 3 sides would be fair wouldnt it?
Yup that would work, there's a whole class of dice which do exactly that (they look like 2 cones stuck together), but I'm not sure that would actually be as "fair" as just dividing a 6 sided dice into 3 groups of sides.
@@sirrandalotOh, goot to know i guess :P
What do you mean by "seed shape?
Awesome vid man! How did you do the 3d graph animations?
Thanks! I used blender for all the 3d animation.
I feel like there’s almost certainly a way to do this rigorously. Assuming any angle is equally likely, the chance of it landing on a certain face is the chance of the centre of gravity being above the opposite face.
With this kind of a result to the simulations, it would be really interesting to repeat them for a known fair die shape. Just for a convenient plausibility check.
0:37 when I was in grade school, textbooks always called dice “number cubes”, I assume due to some moral panic about gambling that was going around (do they still do this?)
initial angle, speed, and rotation of the dice should have been simulation parameters because that is where the pseudo-randomness of a dice resides. You would have gotten rid of multiple parameters, and it may as well have influenced your result curves concerning other parameters. The downside is, you should get more uncertain result, or just as uncertain but with more throws. You could try setting all parameters to random and see if the dices converges to a set of parameters. Also, you could implement machine learning and try n times each dice from a set of q dices with randomized parameters. Those with best scores (closer to 1/3 chances) survive to the next generation, made from small randomizations for all parameters of the best dices. Repeat the generation until all dice converges to a set of parameters assuring approximately 1/3rd of a chance for each face (that is just machine learning applied to dice throwing).
I like you. Great explanation.
@@pressurisedcarbon6238 my pleasure :)
By intermediate value theorum, for every "throw" (initial velocity and angular momentum or whatever the physics word is), there must be some thickness of coin that gives an equal probability for all three
It may be true that it makes it too easy to rig your result by rolling in a certain way though
I looked at the title and thought "a coin pretty much allways lands on the faces, a piece of wire pretty much allways lands on its side. Making the coin thicker influences the likelyhood of it landing on its side, so we can assume we can "select" a value, and there is no reason that we couldn't select equal chances, so the answer must be 'yes' "
Turns out, you wanted to find the ratio and not just figure out if it exists...
When accounting for the angular momentum, does it make a difference which axis you spin the coin at? I assume the answer is yes but you don't specify. Also, I would set as convention that the coefficient of friction and restitution to be set at 0.5. If we follow those conventions, how does the interplay of initial angle, initial linear velocity, as well as angular velocity along the two relevant axis determine the likelihood of the coin landing on its side? You could write an academic paper on this! Another thing is that an extremely heavy mass, would increase the likelihood of landing on the faces, why is that not the case?
For maximum fairness give opposite sides on your d6 the same value, so 1 ahould be opposite 1, 2 should be opposite 2, and 3 should be opposite 3.
Using the cube to represent a 3 sided die is probably the best solution, although you probably have to figure out which values should be opposite. Does having the 1s opposite each other while the 2s and 3s are adjacent have any real impact on probabilities? In some ways I would think the answer is no, but at the same time that does mean if you had consistent roles, you might be able to take advantage of that geometry to slightly bias the results.
I think the biggest discovery made here unstated is that you could also repurpose the cube to create a 2 sided die. Now you'd just need to use 1s and 2s on different faces. I'm not sure you'd even have the same sort of biases you might have with the 3 sided cube because there's not an alternative layout. Completely fair!
For the 2-sided die there's only two layouts: either all faces with the same value share one corner (i.e. one face is perpendicular to each axis & forms an "L" shape if unrolled), or all faces with the same value are parallel to the same axis (i.e. two opposite sides plus one side to connect them by, forms an "I" shape if unrolled).
@@Stratelier oh... I guess so. Something like a tennis ball or on each corner. Still, with both layouts there isn't a difference between them, but it might not be fair.
The labeling doesn't matter because of a symmetry argument.
For what proportions are all three sides equal? Such that πr²=2πrh? For r=1 to find h, π=2πh, 1=2h, ½=h: therefore, the surface area is equally divided between the three sides when the height of the cylinder is equal to the diameter.
You could take a standard die and use a sharpie to write 1, 2, 3, to replace 4, 5, 6 and that's the easiest and most fair three side "coin" you could possibly make. Giving you a mathematical 1/3rd rounded down chance on each roll to get 1-3.
You can even round out pairs of adjacent edges to get 3 true faces.
Finally, the voice of reason!
Could you try modeling a three-sided shape by eroding three edges of a cube? The edges between each pair of sides?
Im no math expert, but I think that would totally work with the understanding that a die is fair
I'm a just roll a die and say: 1 & 2 for A, 3 & 4 for B, and 5 & 6 for C
Cheers (from US)
Lol I love the Adan Neely level of anti-clickbait
I imagine Unity's basic rotation engine does not model such phenomena as nutation and precession, the rotating coins seemed very unstable
Loved the cube joke
"Pi is not a nice value" So you have chosen death?
3:28 Or just round of half of the edges of a cube on each edge that connects with the same number
just wait till this guy finds out there are actual 3 sided dice in the world
Arrange three faces as the sides of an equilateral triangle. Fill the remaining surface in an unstable yet symmetric way. Fair three sided coin.
Try doing the calculations again but shoot it way up in the air and at like >5 rotations per second like a real good coin toss and i reckon your graphs would look quite a bit different. These are more like coin drops (which is still interesting, but I think we both know u wanna see a REAL GOOD coin toss)
How does this not have millions of views?
Why does it need to be a cylinder?
Coins come in many shapes.
3 sides around the center mass, tapering to a point at each end.
Get 2 ‘4’ sided pyramids,
put them together
Smooth the edges that form a square
Put numbers, 1, 2, 3
Kinda like this one at 0:49
could also just do an extruded triangle but have both triangle faces be collapsed if you want a 3 sided die
Is it possible to create a sphere divided in 3 equal parts?
V sauce made a set of odd sided dice starting from 1 side
doesn't this have something to do with mass distribution and by consequence the inertia tensor?
Goated thumbnail
Why wouldnt a triangular prism witv rounded side faces (so its like a pointier football) work?
i really want to know why the values get so weird with bounciness
Opens video by greeting Internet, links to a Numberphile video, doesn’t mention Adelaide . 4/10 Brady points.
cant you just create a cone that is weighted at the pointed end and rounded at the blunted end and give it 3 flat sides?
Make it 6 sided and have the faces repeated. Nice and fair, ezpz to have a 6 sided symmetrical shape.
Or a dodecahedron with 4 copies of each number 1-3. ;)
Without thinking too much about it wouldn’t the theorem of intermediate values of continuous functions prove that there exists a fair three sided coin? A coin with zero thickness would never land on the side (or at least it would be a zero measure set of such events) while an infinitely thick coin would always land on its side, so by the intermediate value theorem there aught to exist a thickness where the coin lands a third of the time on its side.
6 sided block with 2 sides made into one curved side
The more ball shape the thick coin is the closer you'll get to an equal opportunity coin
That'ts a really compicated way to to make a 6 sided die that has 2 of each type.
I may be wrong, but it seems obvious that yes. If the coin is super thick (essentially being a rod), there's a near 100% chance it lands on the edges. If the coin is normal, there's a near 100% chance it lands on the side. By continuity there has to be a middle point where the probabilities are equal.
did you just pull out the Defend your Nuts soundtrack 0_0
triangular prism with 3 sides, and small pyramids on each end.
The answer is yes: A d6 where each number is marked on two faces.
Or just a sign each outcome to two different numbers?
just a long equal sided triangle with curved bits on the end would work
Fair 3 sided "coin" may not exist but can't 3 sided die be created by connecting 3 leaf shaped (oval with 2 pointy ends) surfaces together?
for a 3 sided die, just take 2 coins, cut them in half, then fuse 3 of those halves together
3 sided dice crying in despair
You could just use a dice with numbers 1-3 and have each number appear twice 🤷♂️
Just use a d6 and divide by 2, rounding up
What if you made a weird, straight-banana-shaped die where the edges are arcs between the two poles of the thing? Arguably that’s a harder idea to try to make a model of, never mind to machine, but there’s no reason a design like that one wouldn’t be fair, since it’s totally symmetrical.
I went into this thinking "Wouldn't a tetrahedron with the base ignored be a fair 3 sided die?" lol
Haven't heard something that diasappointing for longer.
nice video though
Can’t you just get a cube and round it’s edges so it turns into a cylinder?
What would the third side be?
You have 50/50 chance of getting heads vs. tails. Third side would make it 33/33/33 chance of getting what?
Id make a 4 sided die with one of the faces saying "roll again"
It would be very interesting to see this tested on a learning model.
I definitely understood this video
can't we just have a 4 sided die but round the fourth side so it cant land on it
Just make a dice and replace numbers above 3 with values 3 lower than them. Easy.
Google image 3 sided die. It's basically a triangular prism with smooth edges such that it cant land on the triangle sides and so that it rolls better.
could make the top and bottom pointed and make the side triangular.
Google image 3 sided coin, as that is what this video is about.
Mass should only be a factor when air resistance is considered
in my mind's eye... I was thinking a sphere with 3 flat sections... equal chance of landing.
Can't you just use a d6 with doubled sides? As in, it has two 1 faces, two 2 faces and two 3 faces
It's called a d6/2 but you probably covered that in the video
I will not be using that form of 3 sided dice, I will instead make a triangular prism to roll across a table.
TRIANGULAR PRIZIM WITH ROUNDED CORNERS BOOOOOM
You can make 6 sided coin, every value is repeater twice
Nice! This has been an open question for me since 2010.
I have some questions though:
1. Which collider did you use? Unity has no default cylinder collider.
2. How did you select a random initial conditions (rotation and angular momentum)? And I see you also added some additional random linear momentum...
3. The Bounciness graph was not presented clearly. You should have added labels on the axes, and spent a little more time on explaining your counter-intuitive results. Are you saying there is some chaotic bifurcation in the high bounciness zone?
Physical experiments are required to validate your results, any "Makers" out there willing to pick up the glove?
1. Pretty sure I just used a mesh collider.
2. Those were gotten mostly with random samples of the unit sphere. As for the linear momentum yes. I had no intention on adding another dimension to everything and it was already clear from these results that it would have made things more complicated, and I didn't want this project to drag on forever.
3. Fair point, I noticed the missing graph axes as I was uploading the video and figured it wasn't the most critical thing but I'll give you the detailed explanation here. When the 3d lines first show up it's showing the relationship between the thickness of the coin (horizontal) and the proportion of how much they land on a given side (vertical). This is why as we make the coin thicker, the "edge" side gets landed on much more frequently and the the "face" sides decrease equally. Then as the graph sort of extrudes towards the camera it's showing those same tests of coin thickness except with more and more "bounciness". You're correct that there seems to be quite a lot of chaos happening when the bounciness is high, and specifically when it's above 0.975. There weirdly seem to be two values of the thickness which have an equal probability of being landed on but with a lower probability in between them. I'm not taking those particular results too seriously because it could be the result of any number of things (physical quirk with the particular shape of the colliders, simulation quirk with energy conservation, just not enough detailed sampling, maybe bad methodology), and besides, there's almost nothing that bounces with near 100% efficiency in real life as well.
@@sirrandalot Thanks for the detailed answer.
BTW, regarding random sampling I suggest the following question:
Draw a circle. Now draw a random cord. What is the chance that the cord you drew is shorter than the length of an edge of an equilateral triangle (that has its points on the circle)?
@@YoniMek that's why I used an even distribution on the unit sphere, all random vectors normalized to length of 1
use a sphere and marker it into 3 symmetrical sides
Rhombic dodecahedron would be fair because all of the sides are the same and the shape is strictly convex but it would be useless because there’s already a 12 sided die
Just saw the title and the thumbnail and the first thing that came to mind is:
Why not, on a sphere, divide its surface into 3 equal regions and spin it? Wouldn't that make a fair three-sided coin?
Maybe, but would that still be a coin?
@@sirrandalot Ah, I completely neglected the shape part over the selection part
I tried to figure this out in the seventh grade, as you might expect, I was unsuccessful. What I tried, was to make the surface area of all sides the same, it did not work.
pi is quite a clean looking number in base π.
1
make a long triangle,and the side facing down is the number
As a theorist, I only care about the surface area of the faces being the same
That cube at teh end was a three *value* die, not a three *sided* one. it still has six faces.