Fair Dice (Part 2) - Numberphile

[@Pewpbawlz]

It made me smile at the end when he explained how the theoretical and real life application made him happy.

[@hobbified]

And just in case it's not obvious to anyone out there: you can have a fair 5-*choice* die even if you can't have a fair 5-*sided* die. You just take a 10-sided die and label 2 of the faces "1", 2 of the faces "2", etc. up to 5.

[@EduPascualSaez]

I checked the comments just to see if someone had brought this up already. There are many wargames that use mainly six-sided dice (commonly abbreviated as "d6") and sometimes require a "d3 roll". This can be applied to any odd number (although a "d1" would be utterly useless): take the fair dice with twice that many sides, roll, half the result, and then round *up*. This obviously assumes that the sides are numbered consecutively starting on one, and there is an important exception: 10-sided dice made for role-playing are often labeled 0 through 9 instead of 1 through 10. In that case, always count the "0" side as "10" (so it's a "5" result once halved). Which brings up another interesting point: there are a few tricks to "roll" dice combinations for ranges that would be awkward to achieve with a single dice. An example used in many RPGs is the "d100" or "d%" roll: take two 10-sided dice (0 through 9, or regard the "10" as a "0"): roll first for the "tens" and then for the units, and you'll get a value between 0 and 99, with fair distribution (assuming the dice were fair to begin with). In many games, the "00" result is regarded as "100", so the range is 1 to 100. This can be extrapolated to any value that can be broken down as the product of the numbers of sides of available dice (for example, you can get a fair distribution between 1 and 36 using two separate "d6" rolls), but that gets a bit messy.

[@HallucigeniaIV]

You could also "merge" the top and bottom triangular sides of bipyramid dice into a curved lune and have an n-sided fair die with hosohedral symmetry

[@TehCorbzor]

look up barrel dice

[@ceruchi2084]

Of course! How else would we calculate damage?

[@sawekbogusawski1649]

there is other option if you need 5 side dice you just roll standard 6 side dice and re-roll when you roll 6

[@jagoandlitefoot]

I love how he pronounces "Tobleroni"

[@John_Ridley]

It's an Italian Toblerone.

[@coosoorlog]

it's a swiss germanic portmanteau where the other part is of italian origin. odds are it's not pronounced like a native american english speaker would automatically assume :).

[@gawainthedane3314]

Peperoni

[@GiuliSnow]

The last e must read as the first e. In italian the letters have always the same sound.

[@adamp3223]

Yeah, it's Swiss it's not real

[@Nurr0]

I really like that this hasn't left me hanging for a week or something while the knowledge of the original video slowly fades from memory.

[@moayadbassam]

:P

[@jojojorisjhjosef]

:(

[@nice3333333333]

+مؤيد بسام moayad bassam' Mah brotha!

[@LazerLord10]

One interesting thing, they can make practically fair dice with any number of sides by using that prism trick. They just make it long enough so it won't land on its edge very often. If it does, re-roll.

[@rastrisfrustreslosgomez544]

I think thats the point. A true fair dice would neglect between surfaces, but the toblerone does depend on the local physics. It would be fair on a flat and elastic surface, things like wood really lack on the elastic department. I do believe you can make a fair dice for any one surface, but the area ratio between the different sides varies between surfaces

[@liammontgomery1825]

This is outside of what they're discussing probably, but you could always round the two flat ends so that it's exceedingly unlikely.

[@chaotickreg7024]

@@liammontgomery1825 Make the ends come to a point, maybe add some feature that makes chaotic bounces on those ends.

[@valle2353]

@@chaotickreg7024 i think that would work. a pentagon-prism with pentagon-pyramids glued to the sides in a way it cant stay on the pyramid sides without falling over. in that way the faces of the pyramid wouldnt count as "sides" of that die and all other sides, i believe, satisfy the gometry group theorem of part 1.

[@DaveCurran]

The maths is quite interesting, but more research needs to be done in how to pronounce Toblerone. I for one volunteer eat sufficient Toblerone in order to be certain.

[@mercronniel3122]

All I've heard my entire life is TOB-le-roan.

[@joelproko]

Tob-lur-OHH-ne(y), would be closer. (The "y" is there to influence how the "e" is pronounced, but it's in brackets because it should be silent)

[@fburton8]

+joelproko I would be surprised if you could find even 1 in 1000 people in Britain who pronounced it "correctly". Here, it's tobe-le-roan. Just one of many subtle differences between US and UK pronunciation.

[@KriegsMeister27]

I am currently taking donations to fund my Toblerone -addiction- experiment

[@RDSk0]

Tub-Lee-Row-Nah

[@nmarbletoe8210]

I like throwing televisions while I watch dice.

[@mountfairweather]

Lolll

[@baazarafa]

When he started talking about the enjoyment of applied math, oh so many feels.

[@johnbehan1526]

To me, the very essence of statistics is finding pure math expressed spontaneously in natural populations. Anyway, fairness isn't a geometry question, it's a probability and mechanics one.

[@Triantalex]

??.

[@Oldiesyoungies]

Trying to explain something in 10 minutes that you've thought about for a lifetime :)

[@ATXpert]

remove 10 years xd

[@Triantalex]

false.

[@Oldiesyoungies]

where have you been all my life ♥@@Triantalex

[@venkatchait007]

awesome job with the animations, they have all been on point recently!

[@numberphile]

thanks - they were were done by Pete McPartlan

[@il2xbox]

On point, you mean on vertex? :D

[@zxana]

ba dum tis

[@matthewrouge]

Very nice work, yes!

[@hps362]

These jokes are very well coordinated. coordinated coordinate (6,9) lol

[@kodymongold]

Man, I never knew I would be this happy to spend 20 minutes learning about dice. Haha. Love your vids! 👍🏼

[@puupipo]

Best excuse ever to buy Toblerone.

[@rosserobertolli]

Too bad the inner weight distribution is not even, so you will always be more likely to end on the bottom side opposed to the upper sides... Guess you can eat the whole Toblerone now

[@MrCheeze]

*Tobleroni

[@natea5225]

MrCheeze aww that was gonna be my joke :(

[@kailomonkey]

and eat the chocolate to the right size

[@jojololo9157]

Ive never seen or even heard of a tolberoni or whatever its called... is it sold in usa? I dont even know what its made out of, its candy im guessing.

[@lostevesy]

Toblerony ?

[@VictorAnsem]

It's Swiss chocolate in the shape of a triangular prism, pretty good actually and you spell it "Toblerone"

[@lostevesy]

Oh i know that, i was just commenting on his pronunciation ~

[@lawrencecalablaster568]

+Ansemthewise94 Love a good Toblerone :)

[@lostevesy]

Mama mia gotta get me one of those tobleronys.

[@UpstairsPancake]

I think he's just so fond of the chocolate that he's given it an affectionate nickname like you might give your partner. "Oh I love you my chocolatey toblerony".

[@bsebire]

"Tobleroneie"?!?! I've never heard anyone pronounce it with the "ie" bit on the end.

[@adityakhanna113]

'xactly

[@Scy]

it's a swiss chocolate (so pronounced italian)

[@ColossalZonko]

it should indeed be pronounced without the "ie"

[@livedandletdie]

It's called Toble, like the spanish Doble but with a t instead of a d, Rone is moan but an R instead of M, and of course the e at the end is pronounced literally e, not the english idiocy of pronouncing e i.. I don't know why it is that way but that's probably why English people can't speak anything but English, your vowels are not vowels most are diphthongs.

[@mattlm64]

Why is it idiocy? What about French and dropping consonants from the ends of words? Most languages do not pronounce letters in words in a literal and precise manner.

[@levi12howell]

I found the format of talking about the math proofs and talking about the applied math in the same video really interesting and informative. Ps this guy seems good at explaining complicated ideas

[@Viruzzz]

I loved the first video I watched with this guy, and he's still by far my favorite personality on the channel.

[@TheAlison1456]

2:15 "you don't have to be perfect at it in order to get some kinda advantage" this goes out to everyone who dislikes imperfect solutions.

[@dhy5342]

There are other fair cubic dice that have numbers other then 1 through 6 on each. The Sicherman dice with faces of 1-2-2-3-3-4 on one die and 1-3-4-5-6-8 on the other. These give the same probabilities of rolling numbers 2 through 12 as standard dice.

[@suit1337]

Speaking of fair 5-sided dice: those are usually manufactured as n-sided prisms with rounded caps and therefore never can end up standing, also some manufacturers make ellipsoids with dents where the die can land - so each face is equaly likely and therefore ideal

[@DanielGolding1337]

A five sided football shape makes for a fair 5 sided dice. Just curve the Toblerone edges so they meet at the end points.

[@zynthio]

This guy is so interesting to listen to that he makes even advanced concepts simple to follow. I wish that he was my math teacher in highschool, I probably would have enjoyed the subject and not have had a horrible time

[@MrNateSPF]

What if you drill 9 holes in every side and fill some of the holes with paint the same color as the die, and other holes a contrasting color. So for example 1 would have the contrasting color in the very middle hole and the other holes colored the same as the die.

[@AlexanderQ689]

MrNateSPF I imagine that would work well for 4-, 6-, & maybe 8-sided dice, but 12 & 20 wouldn't fit all those dots

[@mvmlego1212]

+Alexander -- For that, you could hollow out ditches on each side to form two blocky figure-8s, then fill in the appropriate bars with the different-colored paint to form the numbers, then fill the other ditches in with the other paint. If I worded that confusingly, think about digital clocks.

[@jackywong9504]

You never use dots for 12 and 20 sided dices, do you?

[@ciarfah]

mvmlego1212 what you're looking for is 7-segment numbers :) Hi from the future lol

[@---cr8nw]

A 20 sided die or 30 sided die could be handled with 5 pips and a line on each facet. The pips would be in a row and the line would be under all five pips to orient the binary number.

[@jacobgelven7194]

There can be a fair five sided die. If you take a pentagonal prism and put pentagonal pyramids on the ends, you would have a die that, although it has more than 5 sides, it only has 5 plausible outcomes, because if it was to land on one of the ends with a pyramid, the center of gravity would not allow it to remain in that position.

[@quanjano382]

Niels Kloppenburg you cant have a 5 SIDED die if you have more than 5 sides.

[@paulhauron]

"many of my colleagues don't want to hear about the real world haha" make my day Sr.

[@tomkerruish2982]

I've actually seen a flipped coin (okay, it was a stiff paper disc) with negligible thickness which landed on edge. It came to rest leaning against another object.

[@francogonz]

6:25 - "You are a math god" + (Puts sunglasses) " _Yeah._ " *LUL*

[@danielsmerdel8214]

Math *guy* is what was said, I'm pretty sure

[@kailomonkey]

the end bit of reflection was the best, and the rest was great!

[@uuav]

I wish there was a part 3 with non-flat faced dice.

[@MusiciansReflib]

My kids play Dungeons & Dragons. I saved these videos too show them. Great job!

[@MF-fd2ug]

"well, there isnt because i proofed there isnt"

[@Tattybirch96]

So far, best video uploaded on my birthday.

[@zalikster]

Man, Persi is still one of my favourite people to watch and listen to. Though I don't think I've disliked anyone you've had in your vids...

[@djanmhood]

Thank you very much for all these videos. Thanks to passionate people to let us enter for a few minutes in their universe and thanks to you for bringing it to us. I am pretty sure, some very talented kids will boost their curiosity with your videos, which will change their vision of mathematics.

[@michael1234252]

5:10 try doing that by taking some of the old British 1 pound coins they made 20 years ago and glue a few of them together to see when it'll land on the edge.

[@adamrice7976]

Interesting stuff. I think that you can make a five sided dice... in an uninteresting way, I suppose. So if you made a pencil with 5 sides instead of 6 and sharpened both ends.

[@hps362]

Oh yeah, it would just *kathunk* instead of doing the balancing thing. No you've got an interesting conjecture, and I'm not sure who to believe. I'm gonna play it safe and believe not you.

[@noahkupinsky1418]

The cones created by the sharpened ends of the pencil count as side - that would be 7 sided

[@noahkupinsky1418]

Oh wait you said that two years ago not weeks ago oops

[@alapikomamalolonui6424]

@@noahkupinsky1418 Since it's impossible for the "FairPencilgon"™ to "land" on it's conical ends, assuming a flat landing surface and "gravity", would it be more accurate (on target) to claim the FairPencilgon is a "five landable sided" die? ...and is this yet ANOTHER category of "fair" dice? (( I'm gonna say "Yes" to both those questions. )) Mahalo and aloha! :) 🤙

[@dig8634]

I am fairly certain that this type of die encounters the same problem that the two d4s stuck together into a d6 did. It is technically fair in that the probability of landing on the sides in a perfect math world would be equal, but rolling a d5 pencil would be easier to control than a true fair die would. It wouldn't have as many symmetries as a true fair die does, so controlling it requires less skill and control than with a true fair die.

[@Ronnybanan]

Beautifully said😃

[@mesplin3]

I like him.

[@eyesrajones]

Matt Parker is doing the thick coin problem on his channel right now, January 2018

[@delve_]

+Bobby Jones Yeah, but they didn't solve it. They only created upper and lower bounds, and, even then, the test was flawed. This becomes obvious once you notice the discrepancies among the X and O values at the end. Hopefully there'll be a part 2.

[@cpcraft6984]

First start with a cylinder with the radius larger than the height of the cylinder. Then start to shrink the radius. The volume of a napkin ring around the cylinder will always be the same, and the volume of one of the other two caps goes from larger than the napkin ring to smaller than it. Therefore, there is somewhere between where the volumes are equal.

[@fturla___156]

The probability of landing on a surface of a coin or any type of object is not only dependent on the amount of surface area but also the interaction of the edges of the object to the surfaces it interacts with prior to the 'coin' settling down. Equal areas for each chance potential is not a strict condition to make the probability equal to all others.

[@matthewburton9637]

I learnt basic group theory in January, nice to see someone who enjoys it!

[@maythesciencebewithyou]

I'm sure many people managed to make a coin land on its edge. That possibility alone makes a coin toss probability different than 50/50

[@halulife35]

i just have to bring attention to the quote at 4:11, "how much chocolate you would have to eat in order to make a fair, five-sided die would depend on the dynamics."

[@PhilBagels]

Other than the infinite family of dice based on dipyramids (which would theoretically allow for any even number of faces, the only numbers of faces you can have on a fair die are: 2, 4, 6, 8, 12, 20, 24, 48, 60, and 120. And if you really needed to, you can make a "toblerone" die with 3 sides, or 5 sides or 7 sides, or whatever, with the thing being long enough that it would never land on one of the ends, or simply make the ends rounded do it would fall onto one of the sides. Or make it like a top, which would probably give the fairest distribution, although spinning it would take longer than throwing a die or dice.

[@TSLMachine]

I feel good about having to have watched this yesterday while this video was still unlisted.

[@dlavanty]

what the die is made from also is a huge factor in its fairness. clear dice are the most fair others are not due to the manufacturer usually having imperfections in consistent material in the center

[@LostprophetPL]

Love videos with this man! More interviews with him please!

[@matthewthomas4427]

As I'm watching this I dropped an empty toiler roll and it landed on it's head. Gave me a chuckle.

[@RedsBoneStuff]

You can make a 5-sided dice if you allow curved surfaces. Start with a 5-sided prism and squish its top into a point. As well as the bottom. Then round each of the five remaining surfaces to make each slope as gentle as possible. That way it will never be able to land in a strange position.

[@danieldancey3162]

I was thinking the same thing. Maybe somebody should make one and send it in.

[@RedsBoneStuff]

***** Great minds think alike xD Well, I might make a wooden one, but it would be easier for me to just make a video about it than actually sending it.

[@RedsBoneStuff]

***** Yeah, I want to make some and use them when I have the time :)

[@RicksPoker]

I have a 5 sided die made by Gamescience, and they used that system. They rolled a die tens of thousands of times until the chance of getting an edge was the same as getting one of the flat faces. The die worked on hard surfaces. On a hard surface it would hit, and usually start spinning. Once it was spinning, getting the edge faces was equal to the flat faces. But if you rolled it on a soft surface (say a vinyl gaming map), then the flat faces came up a lot more often. Very interesting talk. Warm regards, Rick.

[@andrewpolk132]

This was really a pleasure to watch. Thank you!

[@jubileeYAVEL]

oh, this guy is the most awesome guy ever

[@ThatBulgarian]

Hey watch it, I'm eating my tobleroni here! :P

[@Greywander87]

Someone else posted this earlier, but this is one way to make a fair die with an odd number of sides: en.wikipedia.org/wiki/Long_dice Basically, take your Toblerone and round off the ends so that the die cannot stand on either end without falling over. Suddenly, you have a three-sided die.

[@tezer2d]

It's possible to buildt a dice with 5 sides if it's allowed to make the sides round. Just make two 5-gon-pyramides, stick them together (like in the last video) and than you round off the corner between one triangle from the lower pyramide and the corresponding triangle from the upper Pyramide so the two triangles become one diamond with a rounded diagonal.

[@gnetkuji]

So assuming the assertion that there is no such thing as a fair die with an odd number of faces is true, the only way to make a fair odd-numbered die would be to double the die to make it even. i.e. the only fair 5-sided die would be a 10-sided die with the numbers 1, 2, 3, 4, and 5 each written on two different faces. Got it.

[@Cythil]

Some prefer the icosahedron but same principle. Take 20 sided die and just put 1-5 on all it faces at equal proportions.

[@nmarbletoe8210]

One can make a fair odd-sided die (practically fair, not symmetrically fair, as the faces are not transitive): a triangular chopstick. oh I just got to the toblerone part! what a great channel!!

[@Pyrotic]

Would you be able to map each and every number to a perfectly symmetrical number on another side if you did that?

[@cOmAtOrAn]

Alternately, you could take a cube and label one of the faces "re-roll." It lacks elegance, but it's effective.

[@Cythil]

cOmAtOrAn I would not say efficient as you risk having to re-roll a few times if you unlucky. But yes, it does the job and is a simple if brutal way to do it.

[@farmonious420]

I love these vids. Math always wins.

[@palmomki]

The point you posed about the distance between perfect maths and the real world was just what I was thinking about through these videos. The main point being that, since the toss of a coin or of a die are deterministic and repeatable (robots that can toss a coin predictably have already been built), the very concept of the "probability" of a face is very blurry, and mathematically this means that the definitions in the model are arbitrary.

[@Kaepsele337]

I think he made a distinction between dice that are "fair by symmetry" which are fair in the real world and dice that are fair because each face is equally likely (but not by symmetry). The later would not be fair in varying circumstances, such as carpet vs table. If you have a die that is fair on a table but not by symmetry, it would not be fair on a carpet. A die that is fair by symmetry is fair regardless of the circumstance. I guess that's where it was unclear why he brought the real world into the discussion.

[@palmomki]

Guest6265+ No, a fair by symmetry die isn't necessarily "fair" regardless of the circumstance. You can always, in theory, control the launch. And the objection "but nobody can actually do that" cannot be a mathematical one, but it's a statistical or at best physical consideration of the behaviour of the real world. But you need a mathematical model that at least approximates the behaviour of a human toss before you can talk rigorously. And maybe they do have such a model, even if they didn't talk about it in these videos, but the point remains that its definition is completely arbitrary and the idea that it validly represents what it's meant to represent comes down to either assumption or empirical evidence - and here we're clearly doing physics rather than mathematics.

[@Waggles1123]

There are a lot of ways to fudge odd-sided dice though. To get a d3, you can label a d6 1,1,2,2,3,3, or you can round 3 edges of a d6 such that you have 3 continuous sides (kind of like stitching together 3 saddle shapes). Sure, rounding edges might take away from the polygonal faces, but it's just as fair as long as you read it consistently.

[@BlameItOnGreg]

You can have a fair odd sided die if you allow for curved faces. If you have any long prism and then bring the vertices on each end to the center points of the end caps, you then have a shape with two vertices and with the edges and faces arching between them. Each curved face is the same size and shape.

[@GregtheMad]

In the array of dice they found, in the top row, the third from the left is made out of what looks like flower pedals. If you color the pedals like flowers you could make a 12 sided flower dice.

[@AlabasterJazz]

In order to get the true probability of something you would have to fully understand and measure all of the forces at play, many of which we don't yet fully understand. However, since the forces of reality seem to be consistent, as you account for more of them, the concept of randomness begins to vanish. The point of rolling a dice, or any other random number generation, is that the combined conditions are so complex that it would be impossible for the human mind to accurately measure the relevant starting conditions and then pre-calculate the outcome prior to it resolving. The concept of Fair is that while the conditions are complex, there is a finite way to resolve them, and each solution is equally likely as the others. In the grand scheme of things, randomness is an illusion and any outcome is entirely pre-determined by all the forces of the universe acting upon each other.

[@neilkadu8745]

0:53 music of marathi song (jhingaat) from the movie Sairat.

[@tomrivlin7278]

Fan theory: Cliff Stoll (Klein bottle guy) and Persi Diaconis (this guy) are mirror universe versions of each other. Cliff is from the universe where everything is Klein bottles.

[@PeterBarnes2]

I can make you a fair 3-sided die. (In abstract, I don't have the tools to do it irl., but I could if I did, and knew how to use them.) Take your Toblerone, get all of the chocolate out, and, perfectly, separate the edges to 3 points in the exact middle (1 point per edge), leaving one connecting, unfilled triangle. Now cut each of the six rectangles so that they form 'bulgy triangles'. They each have 3 corners, but the area is larger than the isosceles triangle contained in them. the two unconnected edges are curves that could be of many different forms, so long as the 'bulgy triangle' is convex, and the limit of the tangent of the curves as they approach the middle triangle is the same as what the tangent was before cutting, and all the curves are the same. You also need to know what curve to cut, beforehand. Now glue the edges together, and it should now have 3 sides. remember to put the glue on the inside, not the outside. You've now made something that is not a 3-sided die, because you made it with scissors and tape, and you aren't a robot. Using a 3d scanner and a 3d printer, make a plastic version of the surface. So long as the CoM is along the line to which the shape has rotational symmetry, it will be a fair die. What? You only wanted polyhedra? But you never even said polyhedra/on! You said 'shapes!' You have at least begun researching these sorts of dice, right?

[@iagocasabiellgonzalez7807]

Excellent video. Thanks

[@jonas2560]

This man is the best.

[@RoaringTRex]

Very cool guy: embracing the practical and the theoretical.

[@frederf3227]

I hypothesize a fair die whose fairness is not derived from symmetry and is unaltered by initial condition (equal partitioning) for all physical response systems (surface, die material, etc.)

[@BurnabyAlex]

I feel that the attitude of knowing that some math isn't always perfect for physics can only help to forward physics, and math.

[@ThAlEdison]

There are fair 5 sided dice. Pentagonal prisms with conical or spherical caps on the end that prevent those sides from being valid (an extension of his Toblerone example). His groups only seemed to list convex polyhedrons, but concave polyhedrons with isohedral envelopes are also fair. Then there is the concept of treating a set of faces as a single unit for transitivity. For example, if you take en.wikipedia.org/wiki/Truncated_triakis_tetrahedron label all of the hexagons with 1-4, and then label the three pentagons who share a corner and label them with the same number as the hexagon opposite their shared corner. You will get fair results from the die. Each set of hexagon+3 pentagons has transitivity to any other set. It would not be a fair 16-sided die, or a fair 8-sided die, but it is fair as a 4-sided die.

[@commandercorner5575]

It's easy to make a fair odd sided die. Similar to the d10, but instead of a hard edge, the faces curve from one point to the other.

[@GrankFarrett]

it's easy, just make cylindrical dice. To make a six sided cylindrical die sharpen both ends of a six sided pencil and write the numbers on each side. Works fine with odd sided pens also.

[@phookadude]

Using curved sides you can make odd sided fair dice. Like a 3 sided toblerone with the ends of the prism nibbled down to opposing points like an American football.

[@diamondflaw]

I know, older video... but I wanted to mention that I have some odd sided die that I believe to be fair. They are made with a polygonal cross section that has a curved taper to each end. Three or five identical faces with the same number of identical edges and two identical nodes - and can't sit on its end.

[@jacobstaten2366]

You could make the tokeroniy thingy a 5 point instead of 3 point profile. You could also make a a 10 sided surface like a common d10 and number both sides 1-5 rather than 1-5 on one half and 6-10 on the other. For a 3 sided die, you could have a 6 sided die that has 2 faces representing 1,2, and 3.

[@euclon]

The roll rate of unique faces of a die come up the same as the time rate of radioactive atoms decaying in a sample of radioactive material. So, for example, if you roll a 20 sided die 20 times then the expectation is that 20(1-e^-(20/20)) or 20(1-e^-1)=12.6 or about 13 unique faces will show (decay) in 20 throws of said die. If you roll that same die 60 times the expectation is that 19 faces will show ie 20(1-e^-3)=19. Then using Excel for example, x,y plot the expected number of unique faces that show against the actual number of unique faces that show for 60 throws. Do this plot roll by roll. Then do a first order (linear) curve fit to the data; set intercept to zero. Let the slope of said curve fit be the judge of the die. I would think that a slope of 0.98 to 1.02 would indicate a fair die.

[@bryankopkin6869]

Technically if you had a long pentagonal prism and rolled it like a ball, not tossing it, it would roll and land on a random side because the sides are all congruent other than the bases, which are taken out of the equation because we are rolling it like a ball

[@SureWouldFriend]

My solution for an odd number die (n) is to take the even sided die (n+1) and change the highest value to "roll again."

[@megadog9305]

I think Matt Parker needs to see this video. He's trying to make a practical version of the three sided coin/dice.

[@Mroziwanman]

I loved these two videos. Fascinating, thanks!

[@TehJumpingJawa]

Such a clear presentation of thought; love listening to Professor Diaconis. Though I too have never heard anyone pronounce Toblerone with an 'E'; '-own', or if you're Italian maybe '-own-eh'. wrt the 'fair' cylinder problem. What about a cylinder whose faces were not flat, but instead the containing sphere's surface reflected in the flat faces of the cylinder? No idea what a cylinder with concave faces would actually be called though?!?

[@JM-us3fr]

I'm one of those people that doesn't care about the reality. Much more beautiful mathematics appears when you abandon application to reality. Also, can you do a video on Sicherman dice?

[@kirby4life123]

Can we just have a channel of the professor talking about fairness

[@mertonhirsch4734]

Note that a toblerone with points instead of flat ends would only require a miniscule curvature.

[@ugluwuglu]

Very informative and amusing videos. I hope there will be more of professor Diaconis in future videos.

[@williamcompitello2302]

Neo would be banned from every casino. "LOADED!" "Sorry, sir, but that's false."

[@empurress77]

A five sided object similar to a standard length pencil would roll with an even chance of landing on any of it's sides. Just as a coin has distinct legal areas of valid chance the "Pencil" die would only count the sides it lands on as valid. The number would be read on it's ends. :) The same could be don just as the previous video showed with any number of sides until you reached a theoretical roundness that would not be any sided at all.

[@hyfy-tr2jy]

You can make a fair die of any number...odd or even... Imagine the full length Toblerone example but instead of 3 sides it has 5 sides, or 6 sides, or "n" sides and then on the flat ends...you can do one of two things...either round it off or have the end have a pyramid that equals the "n" number of sides. This way all the "long sides" have equal surface area and likelihood of coming up, while the ends, should the die land on them, the die will pivot to one of the long rectangular sides thereby never being able to "land" on the ends due to center of gravity no allowing one of the pyramid ends to be touching the table

[@Cythil]

The trick to a prism die fair is just to give it a symmetric unstable bases. Of course then you can make a die with as many sides as you wish. There a bit boring sure. But they work.

[@jeratzel]

Brady, I would like you to ask the professor about his thoughts on pentagonal trapezohedron dice (the common tabletop RPG d10). More specifically: Are they fair? Are they as fair as a pentagonal bipyramid dice? Differences between both of them, and ideas for other 10 sided dice. Thanks, I love your channels, all of them.

[@keiyakins]

Of course, there's a whole field of mathematically unsatisfying but quite practical solutions to the problem, like barrel dice and teetotums.

[@iannoble8626]

There MAY be other "fair" dice as well. For example - somewhere along the continuum cube truncated cube cuboctahedron truncated octahedron octahderon, there must a be a point where, under constant, adequately random launch conditions, the chances of any particular face coming up are identical. Even though the faces are of two distinct shapes (a "fair" d14, in other words). (Call the faces that, at one end, form the cube, C-faces, and those that, at the other, form the octahedron, O-faces. It seems intuitively obvious that the chance of a particular C-face coming up falls smoothly from 1/6 to 0 as the C-faces shrink and the O-faces grow. In parallel those of a particular O-face rise from 0 to 1/8. At some point, therefore - unless something very odd in the way of discontinuity is happening, which seems improbable - there must be a cross-over point where the chance of any individual face - be it C-face or O-face - is 1/14, and the die is "fair".) Certainly, if I were to build a machine to throw such dice in a consistent manner, I ought to be able to choose my shape so as to "tune" my dice to be fair. The question is - is that shape unique, or is it determined by my launch conditions? And if it IS unique, what relationships do the various dimensions have, and are those relationships shared by other, related shapes?

[@crait]

Wow. Love the 5-sided die explanation.

[@darcipeeps]

I love all of the videos from Persi. I wish I could work with him, but alas. I’d love more videos with him though :-)

[@bloergk]

I'll call cameras "automated bradys" from now on

[@Triumvirate888]

I don't know about all the maths behind this, but common sense would dictate that you cannot have an odd-number sided object for dice because whether you are tossing dice or using a coin flip, you need one side to land flat on the table, and one side to land flat facing up. So every side must have an equal and corresponding opposite side.

[@hps362]

tetrahedron. Okay, I guess the vertex, but...

[@PiercingSight]

The sphere around the center of mass should actually work physics-wise. The math would be about the starting orientation and what face it would land on. If the surface area of those sphere parts are equal, that is an indication that if the die started in an orientation where one of those surfaces was at the tippy top, then that would be the side it would land on once gravity takes affect, no matter the roughness or bounciness or whatever of the surface. Such an approach actually works with any number sided die (so long as the sphere is centered on the center of mass). Though this only applies in a situation where perfectly random starting orientations are assumed. So in theory it can be done. Testing it though would require an experiment like that dice roller.

[@B3Band]

A 1-sided die is fair. Theorem ruined.

[@42scientist]

Ehhh do you have a picture ? #sceptism

[@amateur-disco]

It's just a sphere with a number on it. Basically a pool ball.

[@Jimb3rts]

Or a mobius strip, if you'd rather flip it than roll it.

[@MagicSerwyn]

Obviously with round surfaces you can do a fair dice with any number of side, but i guess the theoreme speaks about polyhedrons.

[@BrianBlock]

That does not work because a sphere would not be considered a one-sided die. It would be considered infinitely-sided, because any of the infinite points it could come to rest on need to have a corresponding value on the opposite side of the sphere.

[@johnsavard7583]

In the previous video, you noted that some shapes for dice make it easier to roll a die so that only some of its faces come up. With cubical dice, for example, you could roll it so that it only can come up on four of its faces. And I see you continue on here. I'm not sure yet if you mentioned that in addition to the infinite number of bipyramids and so on, there are even an infinite number of fair dice based on the cube, because you can round off the corners - and the extent to which they are (symmetrically!) rounded off is continuously variable.