I like the sets that have totally different side counts. The idea of pulling them out and telling your friends "no, trust me, this is totally fair" is very funny
The funny part of that to me, in this case, is that while there may not be a demand; this is the exact sorta thing that many board gamers/tabletop gamers are gonna want to buy for the sheer novelty of it Like, there was zero demand for a way to roll a D10000 without it bein absurdly clunky... Then a company just slapped more 0s on their D10s and sold that and id wager most tabletop rpg groups are gonna have at least one person who has that set or wants to get it; just bcuz its novel. If not, someone has a D30 instd; which has equally little demand/use and yet folks buy it cuz its different and new and does this weird obscure thing that we nvr need to do and cud do with literally just 6 D6 Dangle dice that have a niche use in front of our dice obsessed eyes and we will buy those dice just to be able to say we have them heh
@@randomname285 This is also why the cool kids clique is so elitist; they cant have more than 5 ppl in a given clique group of cool kids before they need to start bringin out thousand sided dice and no one has time for that :P
Hearing a MATHEMATICIAN say "I thought those problems were the same!" while I was thinking the exact same thing, when you got to talking about 'is every order equally likely?' was such a huge moment that got glossed over.. Even the "smart" people fall for assumptions and make mistakes. :)
@@FirstLast-gw5mg If ABCD is more likely than BACD, and BADC is more likely than ABDC, everyone can be equally likely to be first, second, third and fourth, but every order is not equally likely.
@@FirstLast-gw5mg Because your position to another player could be skewered. Let's assume for a moment that the four possible outcomes would be Red-Blue-Green-Purple, Blue-Green-Purple-Red, Green-Purple-Red-Blue, Purple-Red-Blue-Green. Now, everyone's equally likely to be 1st, 2nd, 3rd and 4th, but Red has a 75% chance to be before Blue. Also, Blue would always be after Red (though 25% of the time he'd go first, so he'd only go after Red from round two onwards). Sitting after a certain player can be very advantageous or disadvantageous in certain games.
To show that "every order is equally likely" is a different, stronger constraint, you can take it to the extreme: You can use a single 6-sided die to figure out who goes first among 6 players, with the relative order completely fixed. So, like if it's a 1, you get A B C D E F if it's a 2, you get B C D E F A and so on. Everyone has an equal chance of going first, second, third, fourth, fifth, or sixth, but the order is severely restricted!
funnily enough - thats also one of the most common methods to determining player order (if altering the order isnt a major game mechanic in the game itself) just pick the starting player and then go clockwise^^
Seeing James is like bumping into your best friend from high school after years of only speaking to work colleagues and some uni mates, and the conversation is amazing and it's like you last saw each other yesterday.
I just wanted to come in here and say that is it so incredibly wholesome seeing this man still loving his job and teaching what he loves after all these years
The fact that the permutaion-fair problem is different from just "who goes first" problem is evident if you think about just rolling one N-sided die for N people, whoever's number comes up goes first, then clockwise. Everyone is equally likely to get any position, but the whole order, the permutation is as limited as it can be
not about the content but a numberphile video with james grime always takes me back to high school. his videos on prime numbers were the reason i finally got interested in math, something that contributed to my choosing a STEM program to major in at uni. the folks at sixty symbols helped too. i might have to thank brady on the dedication page of my thesis 😆
I think the first set of dice James pulled off for 5 players should be called "Parker dice" because like the Parker square is almost there but not quite. Also that would annoy Matt Parker that has nothing to do with it.
@@edwardpaddock2528 language evolves, and the point of language is to convey meaning; using dice as both singular and plural is a net improvement on meaning conveyance because differentiating from other uses of the word die and making sure your audience will always know what object you're referring to is more useful than differentiating between singular and plural (which will almost always be clear from context anyway).
@@ce7.0 And the meaning is lost, when you use the plural in place of the singular. Dice also has multiple meanings. More of them than die does, in fact.
That's actually a really interesting idea for a Catan variant. If side is half a road, it should play about the same, except that there's one extra half road per tile that allows for settlements not being tightly packed in a new way, and a different scattering of corners are half a road further apart each game.
Hello and Shalom fellow member of the tribe, Mr. James Grime! I saw the Menorah, Torah, and Talit Bag in the background. (I also love happy finds like this. Makes the world feel that much more connected.)
Theoretically constructible from any given set of dice. Simply double the number of sides on one dice, and repeat each value on to two faces. Then triple the number of sides on another dice, and repeat each value on to three faces, and so on.
Yeah, i honestly liked that solution more than the 60 or 120 solution; and it got me wonderin then what the lowest number of total sides wud be needed to achieve a solution, not just for the 5 player scenario, but also the 4 player scenario Can we bring the total sides down to less than 48 for the 4 player scenario? (Tho i will say, the solution at 15:00 is probs my fave of all them)
12:46 is my favorite part of the video and the lighting fast answer to the question "is it needed?" - "no! you can solve it in a simpler ways!" It shows it's all about solving the math problem and not the practical application! Pure mathematician! :)
12:31 That’s the problem, with computer-powered brute force -solutions (besides being potentially less satisfying): They don’t really offer you any significant insight, into the ”Why?” of things.
An existing set of 5 order-preserving dice can be expanded to support a sixth player by adding one six-sided die. One player rolls the d6 to decide their turn place. The remaining 5 roll to decide where around the first player they place. The method for determining which player rolls the d6 is left as an exercise for the reader.
Yes, but this method also lacks the elegance of rolling for initiative and it always working well. Might as well use the card-drawing approach at that point. You could also use the 4-player dice, a D20 modulo 5 and a D6 to get a much smaller 6-player configuration. But that path also leads to unsatisfying mechanisms.
in terms of game design, it would be interesting to have five dice (edit: go-first dice, as shown in the video) where each die has a different number of sides (perhaps even visually distinct shapes) with equal fairness, so each die can act as a unique game piece as well
At some point the dice have so many sides that they won't reliably _stay_ on a side, and you start having trouble seeing quickly which number has been rolled.
The story of the 100-sided dice is pretty much that. Difficult to use because it rolled too long, difficult to read, and didn't really need to exist where people had d100 in the form of two d10s, particularly those with 10s and 1s on different dice. The Zocchihedron is the best known of these. I owned one, it was, at best, a novelty.
I inherited a set of Dice from my dad that included a couple D100. Having them roll across the table is a little bit of a concern. Also figuring out which one that top number is also a little tricky.
I'd just use a D2 (coin), D3, D4 and D5. Say the five players are Red, Green, Blue, Yellow, Orange. Roll (flip) all dice. They go from 1 to X being their D2 determines if Green goes before or after Red D3 determines if Blue goes before, in between or after Red and Green. D4 determines if Yellow goes 1st, 2nd, 3rd or 4th relative to Red, Green & Blue. D5 determines spot of orange. Simplest solution always using a set of dice that have exactly the same set of of options as the possible permutations of 5 players of course. Just need to find some sort of mapping from the dice to the result.
Very cool! In the spirit of the d120 go first die (that just has every order on a side), you can also pick an order of 5 people rolling a d6 and a d20 once. Label the people 0 through 4. The first person can be the d20 roll modulo 5. Relabel the remaining people who haven't been assigned an order yet 0 through 3. Then the d20 modulo 4 gives the next person to go. Relabel the remaining people 0 through 2 and the next person to go is the d6 modulo 3. Relabel the remaining people 0 and 1 the penultimate turn is the d6 modulo 2. The only person not picked so far goes last.
This is great! My four imaginary friends and I will use the five 120-sided dice to decide which one of us gets to use the 20-sided die while the others use the 36-sided dice, when we decide the order of gameplay.
0:30 I'm not sure how many viewers will pick up on it (I can't be the only one) but I really appreciate the imaginary game board being tiled with the newly-proven "Kite Hat" shapes. For those not in the know, a paper recently came out proving that the shape that's tessellating the board there can tile an infinite plane, without ever repeating. Very big deal for geometry nerds lol
"It's there a demand for this?" "No." Y'all haven't met any dice collectors, have you? I play 2 different tabletop games regularly and everyone addy those tables would absolutely love a set of dice like that.
Point of order: as to the question whether there's any demand among gamers for this, I remind Dr. Grime that it was none other than James Ernest that first asked if dice like these were even possible, and indeed the website that sells the dice (Mathsgear) credits him as one of the inventors. It was through him that I learned of their existence. (I ordered a set a year or two ago and they are downright beloved in my gaming circle. Used them just earlier today, in fact.)
As a fellow gamer I love the idea of a set of dice that can determine 'who goes first' but there are phone apps that can do this were each player touches the screen and it decides. That said, a brilliant maths puzzle/exploration. And I'm a big fan of Dr. James and Numberphile. Great work!
This is so interesting! I was puzzled about how they'd "work" or how you'd find those sets, until you put up the breakdown of the set of three, at 7:04.
My first thought was that this feels like it could be used in computer scheduling somewhere down the line to prevent collisions in a fair way. Maybe each device or application could have its own internal "die", and when a new device or application is made, a new die is generated to fit the gaps of the previous ones.
A guy I played a board game with recently had an app where everyone touches their fingers on the screen and it randomly chooses one to go first. Was a kinda cool way to pick. I don't know if it had a function to pick the order too. I honestly don't like huge dice for IRL applications because it gets really hard to tell what number is on top. Like those dice that James has are cool but they're almost round! I would spend more time looking at it trying to work out what number was on top than I would appreciating the mathematicalness of them.
Finally my two favourite hobbies-mathematics and board games-intersect! Though in practical terms, if I wanted to choose a random turn order, I’d just use Chwazi. 😉
That five 360-sided and one 20-sided set can be made all the same size by replacing the 20-sided die with a 360-sided one that has 20 different numbers each appearing 18 times.
I mostly just find it amusing that they're using ~46.8 bits of entropy to select one of 720 equally likely combinations. There are 120 932 352 000 000 different ways that those 5 dice could fall.
I think for a given set of dice. The nicest solution is one where the number of sides in the die with the most number of sides is minimized. So the 36, 36, 36, 36, 20 one is very satisfying to me.
This design will work well for my patented "roll a die" game. An exciting up to 5 player game of deep strategy. Everyone chooses a die, in some agreeable fashion, then everyone rolls a die. High roller wins. No re-rolls necessary with this new dice tech. Quick game, allowing people to quickly get back to staring at their phone.
The final set of 5 dice with the 20 sider and the 4x 36 siders is so close to being workable as actually usable dice. Those 36 sided dice tho are super weird. I wish there was a better shape than that that was more balanced than having the bipyramid shape
I wonder if they went to the trouble of making sure that the average result around each vertex on the catalan solid dice is as close to the same for all of the dice. This helps to make the dice fair, by ensuring that whatever vertex the die is near on stopping has the same average result, which makes it a little less hard to encourage higher rolls by trying to roll the higher numbers on the top of the die. Granted, that's less of a problem with these larger rounder dice like the 120 sider. On that note though, rounder dice are also much more prone to bias in the weight distribution or possible imperfections in the sphere it's inscribed in. Because it's so round, it's more likely to keep rolling until its center of gravity is as low as possible, so if it's not made with enough precision its bias will lead to one side coming up more often. One way to reduce this effect is to keep the edges sharp after you remove it from the mould, and only sand down the point where it was cut away from the mould. The fairest dice have the most uniform sides, and the sharpest edges so that they stop quickly. More rolling might feel more random, but it's actually less.
I'm curious about the details of the math behind what makes the sets permutation fair. Just a cursory look at the three set example given, the A die has both the highest high number and the highest low number, while the C die has the lowest of both, which doesn't intuitively seem like it would be fair. I'm sure the numbers in between somehow make up for it in probability, though, so I'd like to see just how they do. Although even if provably statistically fair, if I rolled the highest number on my die but someone else still beat me, it sure wouldn't feel fair.
Someone must have the highest number in the set, and it’s obviously going to be higher than anyone else’s highest number. The balance is that they don’t get as many high numbers.
@@ragnkja But it doesn't just have the highest number, it's also got the highest of the lowest numbers, (in the particular example three set given, it has both 18 and 3). And it doesn't have fewer high numbers; each set has two. But the fact that set B in this case (I messed up in my original post. It's B, not A.) has both 18 and 3 makes it sound like it has both a slightly higher chance to get first place (since it has a number no one else can beat) and a slightly lower chance to get last (since its lowest number can still win vs both other sets), so by intuition it seems like it would average out to a higher position. My intuition would be that the die with the highest number would also have the lowest one to balance that out: that if it has the ability to win vs all other dice, it should also have the ability to lose vs all of them. I'm sure that intuition is incorrect, of course, as is often the case with statistics, which is why I'm curious to see how it actually works out.
As a stats teacher - honestly, I'd just have a computer brute-force computing the probabilities to verify. 5 120-sided dice is only some 25 billion combinations.
my guess based on the video would be: they work from a set distribution that is known or trivial to be permutation fair and then perform operations to extend that set that maintain that property at some point this likely started with a 2 player version and some proof of which operations will maintain the property sidenote: so it feels more fair when you both rolled the highest number on the die and then you have to reroll until one of you rolls higher than the other? the entire point is to determine the order... if you want the property of "no draws" then you cant have duplicated numbers... so one die will have to have the highest number in the set the specific distribution of numbers isnt quite as relevant as how many of the sides of each die beat how many sides of each of the other dice - so in your example... the die with the highest high and highest low number... could for example have a load of other really low numbers...and it would lose to all those other dice, except in the rare cases where it rolls its best, while the other dice average much higher (which isnt how they are arranged of course, but im just trying to illustrate how looking at those two things in isolation is fairly pointless)
Of course if you can have a different number of sides per dice (and allow a tweak in the algorithm), there is a very simple extensible method: the N-th player gets an N-sided die. Instead of looking at all the results at once, we maintain a list and go from player 1 to player N. The number that player rolled is the place that player gets inserted into the list (roll a 1 you get inserted at the front, roll an N, get inserted last). For example, with 4 players: 1st player, Alice: doesn't have to roll anything (equivalent to rolling a d1). The list is [Alice] 2nd player, Bob: rolls a d2, gets a 2. The list is now [Alice, Bob] 3rd player, Charlie: rolls a d3, gets a 1. [Charlie, Alice, Bob] 4th player, Denise: rolls a d4, gets 3. [Charlie, Alice, Denise, Bob]
Oh wow I was just thinking of the single 120 sided die option just before the talk of "demand" was brought up... There goes any sense I had of being clever!
This reminds me of the video about flipping a coin that's equally likely to land on the edge as on one of the faces. Did anyone ever solve that ratio question?
I've watched your videos off and on for a few years and enjoy the enthusiasm you bring to bear. I'm retired and tutor math at the local library (with a math/physics/engineering/teaching) background. I'm interested in whether you have topics that would be of interest to an average-to-bright 12-year-old. So far I've been doing a lot with prime numbers and how to use them with fractions.
As an avid dice collector I find this extremely interesting. Are there any potential plans in the works for the possibility of people being able to buy these?
I understood parts of explanation. Pass on the complements to James grime. So UA-cam needs to provide ways in which one can contact not only the video maker but also people in the video.
Interesting that none of the dice are allowed duplicate numbers. Obviously you cannot have repeats on multiple dice, other wise you could get a collision, but if there is a die that has consecutive numbers, those could both be the same number. I wonder what the lowest value you could have, starting at 1, with consecutive numbers on a single die being allowed to be duplicated.
I like that idea. For instance, that would reduce the amount of numbers in the configuration at 7:07 from 1-18 to 1-15, with sides like this: A- 2, 4, 6, 10, 12, 14 B- 3, 3, 7, 9, 11, 15 C- 1, 5, 8, 8, 13, 13
Any place where duplicating the number will solve the problem, you can replace those duplicates with their sequential values, making sure that every side that is higher is also bumped. You can do the same in reverse with the existing dice to 'compress' the number of numbers used, if any single die has sequential values.
Using factorial base, you can roll a single die (or combo). For 5 players, d12 and d10. Suppose you get 103. That's 4*24 + 1*6 + 0*2 + 1*1 => 4101 fac. The player four seats ahead of the roller goes first, skip one among the remaining players and the next one goes second, skip zero and the next goes third, skip one and the next goes fourth. (A result of 120 is just 0, so just clockwise order starting with the roller.)
I've always preferred his original series. In particular, it is a video that contains content about the spiral law of prime numbers. It is because of the genuine inquiring mind. This video is closer to the kind of relaxation that viewers should do before engaging in new research. I thought this was just right for brain exercise rather than content.
Truly mathematician way to avoid "oh we rolled the same, we who rolled the same have to roll again to decide which one of us goes first". "That's not fun!"
Why isn't anyone acknowledging how amazing it is that you can even choose one die and it'll still, with equal probability, determine that you go first.
In a game you could also use those whenever there is a decision to be made between 2 or more players, not just for determining order once. Whenever two or more players "battle" for something in a game, everyone involved could throw his die and it would never be a draw and is always fair. And it doesn't matter which or how many players battle against each other, there would always be 1 fair winner.
I actually searched for a set of 5 dice. I thought it would be very elegant if we could use the 5 Platonic solids to build the set. But I couldn’t find a solution as go first dice.
That can't work for a permutation fair set of dice. If you select the d4, the d8 and the d20, it's impossible to have the permutations for these three players be equally likely, since there is 4×8×20 possible outcomes, but 3! permutations, and 3! does not divide 4×8×20.
@@QuantumHistorian Unfortunately, that doesn't work. For all permutations of five players to be equally likely, then all sub-permutations of three players must also be equally likely. For a smaller example, if with two players, the order AB was more likely than BA, then increasing to three players would mean that the union of ABC, ACB and CAB must collectively be more likely than the union of BAC, BCA and CBA, and so there's no way to get those six permutations equally likely.
@@xfgjnsfgj I think you're confusing two different things. "For all permutations of five players to be equally likely, then all sub-permutations of three players must also be equally likely" is true, if you roll all the dice and then take the marginal sum by taking the union of orders where one of the players is ignored. That does not mean that removing any dice must result in a fair game with one fewer players. The two are different operations. The former is true, I see no reason to believe the latter must also be.
I sent you a solution of mine on twitter. It is a comment under one of the video posted there. 5 (10 sided) dice values 1-51. High low pairs equal 51. 5 sets created that rotate them so high lowest is rotated across all dice. I did a colored spread sheet to make it simple to understand. Using that method you should be able to generate any number of go first dice easily.
i know that's not fair because there are 100000 possibilities for the roll and 120 possible permutations. if they were equal then there would be 833.3333 ways to roll each permutation, which is clearly ridiculous
That last 5-dice solution seems like it's easily the best one to me. Less sides means they're smaller, which is far more practical in every way if you're actually planning to use these for their intended purpose. I'd much rather have each person throw a few ordinary 6-sided dice (perhaps of different colours) than have to deal with those 120-sided things.
Yeah, using such huge dice is kind of ridiculous. There are only 120 permutations of the numbers 1 through 5 without repeats... you could literally just roll _one_ d120 and then refer to a table to tell you what the player order should be.
For Bram's method there and the 6 player solution... you could just take that 20 sided die and split each side 18 ways. That turns it into another 360 sided die to match the other 5. Presumably, you could then use Bram's method again to add another 7th die. Each would have thousands of sides, but it should work.
If there is a solution for 6 dice with 1 20-sided die, and 5 360-sided dice, there is also a solution with 6 360-sided dice, by just dividing each of the sides of the smaller die into 18 sides with consecutive numbers.
People are tribalistic. Even though it's mathmatically proven that the dice are fair, the people using them will swear that the odd-one-out D20 is blessed or cursed. Probably the latter.
I'm lost at the idea that any position in the order could be equally likely but the permutations of the dice could be unequally likely. If the red die has an even 1:5 probability of being 1st, 2nd, 3rd, 4th, or 5th when you roll all 5 dice, and the same is true for the other 4 dice in the set, then how could the permutations end up skewed in one particular direction?
Suppose there are three players and the probabilities are P(ABC) = P(BCA) = P(CAB) = 1/3 P(ACB) = P(CBA) = P(BAC) = 0 Then each player has a 1/3 chance of being 1st, 2nd, or 3rd but half the arrangements are impossible.
@@FirstLast-gw5mg My point is that, in principle, "place-fairness" does not imply "order-fairness". For a much more complex example that does come about through dice rolls you can work out the probabilities for one of the sets of dice in the video.
@@martinepstein9826 Well, it's not _just_ place fairness, it's also the fact that they're independent dice. If they're not independent dice it's easy to see that place fairness does not guarantee order fairness. The sets of dice in the video are rather complex, is it possible to use a simpler scenario that demonstrates this discrepancy? E.g. some other commenter suggested using a d1, d2, d4, etc., but when I looked at the probabilities for the 3-player "go first," this set of 3 dice (d1 = 4, d2 = 2/6, d4 = 1/3/5/7) didn't even guarantee place fairness, much less order fairness... there are 6 order permutations and 8 dice permutations, so just at a glance this couldn't give order fairness.
James Grime, are you reading the comments? I remember you like cryptography. Shuffling card and a single die with all the orders are centralized. The cards also not parallel, and everybody could look their position before the process is completed. But you can give these dies to different parties, they all roll independently, you can verify that the number they gave you is valid for the dice you gave them, and they are not sure of their own position in the order. We could further obscure the number behind a hash, and we have a secure distributed way to pick the order.
“Imagine 4 friends”
This is gonna be tough but I’ll play along James!
sad but true
Reminds me of the first video, friendly numbers. "Aww, friends."
Well, they're complex friends... they're partly imaginary.
3 friends for some. “Harvey the Rabbit” never leaves their side.
@@FirstLast-gw5mg depends on the mood as to whether they appear of not IRL.
I like the sets that have totally different side counts. The idea of pulling them out and telling your friends "no, trust me, this is totally fair" is very funny
Let them choose which one they want, so they'll start panicking thinking you're trying to trick them somehow :D
@@jonasba2764 I would always call dibs on the one with the most faces.
@@jonasba2764 they'll be like... Wait a minute, didn't you play that game with me where no matter which one I pick, you'll win most rolls?
@@jonasba2764 Well, I could clearly not choose the dice in front of me!
My friends already don’t believe me that the 12 sided dice for 4 players are fair
I just love how James has been on the channel since the beginning and he's still as exuberant as ever.
I was just thinking this is basically the same James from 11.11.11!
15 seconds in and I'm already hyped with how enthusiastic he is!
If anything I think he's gotten more exuberant as time goes on.
James probably drinks a lot of coffee
How to spot a pure mathematician: "...is there a demand?" "No, no, you can solve this problem in simpler ways."
"You could do it in about 2 seconds, but that's not as FUN as spending years solving a harder problem!"
@@IceMetalPunk Gotta look smart!
yeah sure you could do it the loser normie way with cards but all the cool kids are using five 120 sided dice
The funny part of that to me, in this case, is that while there may not be a demand; this is the exact sorta thing that many board gamers/tabletop gamers are gonna want to buy for the sheer novelty of it
Like, there was zero demand for a way to roll a D10000 without it bein absurdly clunky... Then a company just slapped more 0s on their D10s and sold that and id wager most tabletop rpg groups are gonna have at least one person who has that set or wants to get it; just bcuz its novel. If not, someone has a D30 instd; which has equally little demand/use and yet folks buy it cuz its different and new and does this weird obscure thing that we nvr need to do and cud do with literally just 6 D6
Dangle dice that have a niche use in front of our dice obsessed eyes and we will buy those dice just to be able to say we have them heh
@@randomname285 This is also why the cool kids clique is so elitist; they cant have more than 5 ppl in a given clique group of cool kids before they need to start bringin out thousand sided dice and no one has time for that :P
Hearing a MATHEMATICIAN say "I thought those problems were the same!" while I was thinking the exact same thing, when you got to talking about 'is every order equally likely?' was such a huge moment that got glossed over.. Even the "smart" people fall for assumptions and make mistakes. :)
I wish it wasn't glossed over, because I'm still not sold on why they're not the same.
@@FirstLast-gw5mg
If ABCD is more likely than BACD, and BADC is more likely than ABDC, everyone can be equally likely to be first, second, third and fourth, but every order is not equally likely.
@@FirstLast-gw5mg Because your position to another player could be skewered. Let's assume for a moment that the four possible outcomes would be Red-Blue-Green-Purple, Blue-Green-Purple-Red, Green-Purple-Red-Blue, Purple-Red-Blue-Green. Now, everyone's equally likely to be 1st, 2nd, 3rd and 4th, but Red has a 75% chance to be before Blue. Also, Blue would always be after Red (though 25% of the time he'd go first, so he'd only go after Red from round two onwards). Sitting after a certain player can be very advantageous or disadvantageous in certain games.
Meant to say "directly after Red in turn order".
@@ThelsdeKwant Thank you
0:25 I love that the game board uses the newly found aperiodic monotile hat/shirt-shapes! :D
Now I want to make a game like that
And the new hat tile is a polykite too ! Wearing hats and flying kites are mutually exclusive due to the wind requirements. 😆
I saw that too! Came to the comments to see if anyone else noticed!
Shoot, I just typed my own comment about it!
Settlers of aperiodic Catan
To show that "every order is equally likely" is a different, stronger constraint, you can take it to the extreme:
You can use a single 6-sided die to figure out who goes first among 6 players, with the relative order completely fixed. So, like
if it's a 1, you get A B C D E F
if it's a 2, you get B C D E F A
and so on.
Everyone has an equal chance of going first, second, third, fourth, fifth, or sixth, but the order is severely restricted!
Thank you so much! This should be pinned.
They mention this in the video.
@@jayjasespud they briefly mention that they're different constraints; they don't give a clear concise illustration of the difference like this
excellent explanation!
funnily enough - thats also one of the most common methods to determining player order (if altering the order isnt a major game mechanic in the game itself)
just pick the starting player and then go clockwise^^
Seeing James is like bumping into your best friend from high school after years of only speaking to work colleagues and some uni mates, and the conversation is amazing and it's like you last saw each other yesterday.
I just wanted to come in here and say that is it so incredibly wholesome seeing this man still loving his job and teaching what he loves after all these years
The fact that the permutaion-fair problem is different from just "who goes first" problem is evident if you think about just rolling one N-sided die for N people, whoever's number comes up goes first, then clockwise. Everyone is equally likely to get any position, but the whole order, the permutation is as limited as it can be
not about the content but a numberphile video with james grime always takes me back to high school. his videos on prime numbers were the reason i finally got interested in math, something that contributed to my choosing a STEM program to major in at uni. the folks at sixty symbols helped too. i might have to thank brady on the dedication page of my thesis 😆
I think the first set of dice James pulled off for 5 players should be called "Parker dice" because like the Parker square is almost there but not quite. Also that would annoy Matt Parker that has nothing to do with it.
I love the bit about annoying Matt 😂😂
Dice is plural. Die is singular. "Dices" is what what one does to cabbage to make coleslaw.
@@edwardpaddock2528 language evolves, and the point of language is to convey meaning; using dice as both singular and plural is a net improvement on meaning conveyance because differentiating from other uses of the word die and making sure your audience will always know what object you're referring to is more useful than differentiating between singular and plural (which will almost always be clear from context anyway).
@@ce7.0 I'm the guy who said "dies" to refer to multiple dice
@@ce7.0 And the meaning is lost, when you use the plural in place of the singular.
Dice also has multiple meanings. More of them than die does, in fact.
Hey, the board is the aperiodic tiling recently discovered
The hat!
Is that meant to be catan?
A keen eye!
Yeah nice that it was included :)
@borandiUK looks like it
That's actually a really interesting idea for a Catan variant. If side is half a road, it should play about the same, except that there's one extra half road per tile that allows for settlements not being tightly packed in a new way, and a different scattering of corners are half a road further apart each game.
Hello and Shalom fellow member of the tribe, Mr. James Grime! I saw the Menorah, Torah, and Talit Bag in the background. (I also love happy finds like this. Makes the world feel that much more connected.)
Gotta love a Dr. Grime video! You can tell he's super passionate about his field!
I've known Eric Harshbarger for years but haven't talked with him in some time. Glad to know he is still working on something so cool!
He's currently my precalc professor!
I'd love to have a set of dice where each one is different sided but can all be rolled against eachother fairly
Theoretically constructible from any given set of dice. Simply double the number of sides on one dice, and repeat each value on to two faces. Then triple the number of sides on another dice, and repeat each value on to three faces, and so on.
Yeah, i honestly liked that solution more than the 60 or 120 solution; and it got me wonderin then what the lowest number of total sides wud be needed to achieve a solution, not just for the 5 player scenario, but also the 4 player scenario
Can we bring the total sides down to less than 48 for the 4 player scenario?
(Tho i will say, the solution at 15:00 is probs my fave of all them)
@@SylviaRustyFae from the wiki: total of 30 sides is optimal (d4+d6+d8+d12) for 4 players.
@@tomsirgedas4221 I love that theyre 4 different sized dice too :3
My thought as well.
12:46 is my favorite part of the video and the lighting fast answer to the question "is it needed?" - "no! you can solve it in a simpler ways!"
It shows it's all about solving the math problem and not the practical application! Pure mathematician! :)
12:31 That’s the problem, with computer-powered brute force -solutions (besides being potentially less satisfying): They don’t really offer you any significant insight, into the ”Why?” of things.
0:27 I like that that board is rocking a bit of tessellation of hats.
As a serious board gamer, I particularly like this episode. It was especially good to see meeples. Thanks Brady.
An existing set of 5 order-preserving dice can be expanded to support a sixth player by adding one six-sided die. One player rolls the d6 to decide their turn place. The remaining 5 roll to decide where around the first player they place. The method for determining which player rolls the d6 is left as an exercise for the reader.
Yes, but this method also lacks the elegance of rolling for initiative and it always working well. Might as well use the card-drawing approach at that point.
You could also use the 4-player dice, a D20 modulo 5 and a D6 to get a much smaller 6-player configuration. But that path also leads to unsatisfying mechanisms.
in terms of game design, it would be interesting to have five dice (edit: go-first dice, as shown in the video) where each die has a different number of sides (perhaps even visually distinct shapes) with equal fairness, so each die can act as a unique game piece as well
That'd be cool, and I think it's not that hard to do (unless you're talking in this case about no repeat numbers)
Its been done. Any D20 system essentially does this
@@GodwynDi Well that's not true, is it? Dice in DnD can roll the same number.
@@jayjasespud That was not part of OPs requirements.
@@GodwynDi but not with these kinds of go-first dice. i didn't specify because i figured it was implied.
James is SO excited and usually does SUCH a great job explaining his concepts. Maybe I’m just extra dense today but this all flew right over my head 😂
James is the God of maths, simply because of his sheer enthusiasm. Stay with us, James, we need you.
13:30 The fun part of Mathematics is just doing the Maths, solving the problem. Demand or utility is secondary, if not entirely trivial.
At some point the dice have so many sides that they won't reliably _stay_ on a side, and you start having trouble seeing quickly which number has been rolled.
There's an old video, might be Matt Parker, where they explain that 120 is really as high as you can go for a regular polygon.
The story of the 100-sided dice is pretty much that. Difficult to use because it rolled too long, difficult to read, and didn't really need to exist where people had d100 in the form of two d10s, particularly those with 10s and 1s on different dice. The Zocchihedron is the best known of these. I owned one, it was, at best, a novelty.
Perhaps use more dimensions for the dice and the rolling "surface" ??
@@MarkTillotson We already lose enough dice in our paltry three dimensions!
I inherited a set of Dice from my dad that included a couple D100. Having them roll across the table is a little bit of a concern. Also figuring out which one that top number is also a little tricky.
I'd just use a D2 (coin), D3, D4 and D5.
Say the five players are Red, Green, Blue, Yellow, Orange.
Roll (flip) all dice. They go from 1 to X being their
D2 determines if Green goes before or after Red
D3 determines if Blue goes before, in between or after Red and Green.
D4 determines if Yellow goes 1st, 2nd, 3rd or 4th relative to Red, Green & Blue.
D5 determines spot of orange.
Simplest solution always using a set of dice that have exactly the same set of of options as the possible permutations of 5 players of course. Just need to find some sort of mapping from the dice to the result.
Very cool! In the spirit of the d120 go first die (that just has every order on a side), you can also pick an order of 5 people rolling a d6 and a d20 once. Label the people 0 through 4. The first person can be the d20 roll modulo 5. Relabel the remaining people who haven't been assigned an order yet 0 through 3. Then the d20 modulo 4 gives the next person to go. Relabel the remaining people 0 through 2 and the next person to go is the d6 modulo 3. Relabel the remaining people 0 and 1 the penultimate turn is the d6 modulo 2. The only person not picked so far goes last.
OH HECK YEAH
I've been following this topic since it first popped up with James Ernest and Harshbarger. This is a great result!
This is great! My four imaginary friends and I will use the five 120-sided dice to decide which one of us gets to use the 20-sided die while the others use the 36-sided dice, when we decide the order of gameplay.
You don't need permutation fairness, just position fairness. So you can use the initial set of five 60-sided dice!
are the sides of a die always an even amount?
I love dice. I've been working on dice probabilities for different systems of rolling dice for over 3 decades.
Grime is the OG numberphile presenter. I always click when I see him on the thumbnail
0:30 I'm not sure how many viewers will pick up on it (I can't be the only one) but I really appreciate the imaginary game board being tiled with the newly-proven "Kite Hat" shapes.
For those not in the know, a paper recently came out proving that the shape that's tessellating the board there can tile an infinite plane, without ever repeating. Very big deal for geometry nerds lol
I saw that too!
This makes me so happy! I want some "Go First Dice for 5" now!
What a great tour James brought us on made all the more enjoyable by the fantastic graphics..
I absolutely love when Numberphile posts!
"Trembling with anticipation"! Always for numberphile videos!
Bram Cohen is also a frequent contributor to Oskar van Deventer's work!
"It's there a demand for this?"
"No."
Y'all haven't met any dice collectors, have you? I play 2 different tabletop games regularly and everyone addy those tables would absolutely love a set of dice like that.
0:25 Ah yes, Saddlers of Cat-Town, my favourite!
Sneaky monotile there too...
0:27 I love the Einstein tiling they're using here.
Point of order: as to the question whether there's any demand among gamers for this, I remind Dr. Grime that it was none other than James Ernest that first asked if dice like these were even possible, and indeed the website that sells the dice (Mathsgear) credits him as one of the inventors. It was through him that I learned of their existence. (I ordered a set a year or two ago and they are downright beloved in my gaming circle. Used them just earlier today, in fact.)
As a fellow gamer I love the idea of a set of dice that can determine 'who goes first' but there are phone apps that can do this were each player touches the screen and it decides. That said, a brilliant maths puzzle/exploration. And I'm a big fan of Dr. James and Numberphile. Great work!
This is so interesting! I was puzzled about how they'd "work" or how you'd find those sets, until you put up the breakdown of the set of three, at 7:04.
There was one time with my brothers that we all rolled dice to start. The 3 of us had to roll 3 times each because we got 7 consecutive 6's.
Welp I just bought some Go first dice for a friend who enjoys math and a gamer. Thanks for the gift idea
My first thought was that this feels like it could be used in computer scheduling somewhere down the line to prevent collisions in a fair way.
Maybe each device or application could have its own internal "die", and when a new device or application is made, a new die is generated to fit the gaps of the previous ones.
"Forget easy, is it elegant?" I was never even close this guy's league, but I've fallen down that rabbit hole more than a few times.
A guy I played a board game with recently had an app where everyone touches their fingers on the screen and it randomly chooses one to go first. Was a kinda cool way to pick. I don't know if it had a function to pick the order too.
I honestly don't like huge dice for IRL applications because it gets really hard to tell what number is on top. Like those dice that James has are cool but they're almost round! I would spend more time looking at it trying to work out what number was on top than I would appreciating the mathematicalness of them.
D120s are quite round. I own one. Need a very flat surface. More of a mathematical curiosity, than a practical die.
chwazi is one such app
Finally my two favourite hobbies-mathematics and board games-intersect! Though in practical terms, if I wanted to choose a random turn order, I’d just use Chwazi. 😉
That five 360-sided and one 20-sided set can be made all the same size by replacing the 20-sided die with a 360-sided one that has 20 different numbers each appearing 18 times.
I mostly just find it amusing that they're using ~46.8 bits of entropy to select one of 720 equally likely combinations.
There are 120 932 352 000 000 different ways that those 5 dice could fall.
It sounds like they don't like to repeat the same number, but a run of 18 consecutive numbers also works.
I think for a given set of dice. The nicest solution is one where the number of sides in the die with the most number of sides is minimized. So the 36, 36, 36, 36, 20 one is very satisfying to me.
0:24 Nice cameo appearance by the newly-discovered 13-sided shape that can tile the plan aperiodically. I hope you'll be doing a video on this soon!
This design will work well for my patented "roll a die" game. An exciting up to 5 player game of deep strategy. Everyone chooses a die, in some agreeable fashion, then everyone rolls a die. High roller wins. No re-rolls necessary with this new dice tech. Quick game, allowing people to quickly get back to staring at their phone.
Nice Settlers of Hatan board
The final set of 5 dice with the 20 sider and the 4x 36 siders is so close to being workable as actually usable dice. Those 36 sided dice tho are super weird. I wish there was a better shape than that that was more balanced than having the bipyramid shape
It's always fun when Grime talks about dice.
I wonder if they went to the trouble of making sure that the average result around each vertex on the catalan solid dice is as close to the same for all of the dice. This helps to make the dice fair, by ensuring that whatever vertex the die is near on stopping has the same average result, which makes it a little less hard to encourage higher rolls by trying to roll the higher numbers on the top of the die. Granted, that's less of a problem with these larger rounder dice like the 120 sider.
On that note though, rounder dice are also much more prone to bias in the weight distribution or possible imperfections in the sphere it's inscribed in. Because it's so round, it's more likely to keep rolling until its center of gravity is as low as possible, so if it's not made with enough precision its bias will lead to one side coming up more often. One way to reduce this effect is to keep the edges sharp after you remove it from the mould, and only sand down the point where it was cut away from the mould. The fairest dice have the most uniform sides, and the sharpest edges so that they stop quickly. More rolling might feel more random, but it's actually less.
I'm curious about the details of the math behind what makes the sets permutation fair. Just a cursory look at the three set example given, the A die has both the highest high number and the highest low number, while the C die has the lowest of both, which doesn't intuitively seem like it would be fair. I'm sure the numbers in between somehow make up for it in probability, though, so I'd like to see just how they do.
Although even if provably statistically fair, if I rolled the highest number on my die but someone else still beat me, it sure wouldn't feel fair.
Someone must have the highest number in the set, and it’s obviously going to be higher than anyone else’s highest number. The balance is that they don’t get as many high numbers.
It's probably statistically fair, but not game theory fair.
@@ragnkja But it doesn't just have the highest number, it's also got the highest of the lowest numbers, (in the particular example three set given, it has both 18 and 3). And it doesn't have fewer high numbers; each set has two. But the fact that set B in this case (I messed up in my original post. It's B, not A.) has both 18 and 3 makes it sound like it has both a slightly higher chance to get first place (since it has a number no one else can beat) and a slightly lower chance to get last (since its lowest number can still win vs both other sets), so by intuition it seems like it would average out to a higher position. My intuition would be that the die with the highest number would also have the lowest one to balance that out: that if it has the ability to win vs all other dice, it should also have the ability to lose vs all of them.
I'm sure that intuition is incorrect, of course, as is often the case with statistics, which is why I'm curious to see how it actually works out.
As a stats teacher - honestly, I'd just have a computer brute-force computing the probabilities to verify. 5 120-sided dice is only some 25 billion combinations.
my guess based on the video would be: they work from a set distribution that is known or trivial to be permutation fair and then perform operations to extend that set that maintain that property
at some point this likely started with a 2 player version and some proof of which operations will maintain the property
sidenote: so it feels more fair when you both rolled the highest number on the die and then you have to reroll until one of you rolls higher than the other?
the entire point is to determine the order... if you want the property of "no draws" then you cant have duplicated numbers... so one die will have to have the highest number in the set
the specific distribution of numbers isnt quite as relevant as how many of the sides of each die beat how many sides of each of the other dice - so in your example... the die with the highest high and highest low number... could for example have a load of other really low numbers...and it would lose to all those other dice, except in the rare cases where it rolls its best, while the other dice average much higher (which isnt how they are arranged of course, but im just trying to illustrate how looking at those two things in isolation is fairly pointless)
Of course if you can have a different number of sides per dice (and allow a tweak in the algorithm), there is a very simple extensible method: the N-th player gets an N-sided die. Instead of looking at all the results at once, we maintain a list and go from player 1 to player N. The number that player rolled is the place that player gets inserted into the list (roll a 1 you get inserted at the front, roll an N, get inserted last). For example, with 4 players:
1st player, Alice: doesn't have to roll anything (equivalent to rolling a d1). The list is [Alice]
2nd player, Bob: rolls a d2, gets a 2. The list is now [Alice, Bob]
3rd player, Charlie: rolls a d3, gets a 1. [Charlie, Alice, Bob]
4th player, Denise: rolls a d4, gets 3. [Charlie, Alice, Denise, Bob]
James never disappoints!
Oh wow I was just thinking of the single 120 sided die option just before the talk of "demand" was brought up... There goes any sense I had of being clever!
So would a set of go first dice that are positionally fair but not permutationally fair be "Parker go first dice"?
Love the Catan board with the new einstein tiling! Fun little easter egg :D
This reminds me of the video about flipping a coin that's equally likely to land on the edge as on one of the faces. Did anyone ever solve that ratio question?
I've watched your videos off and on for a few years and enjoy the enthusiasm you bring to bear. I'm retired and tutor math at the local library (with a math/physics/engineering/teaching) background. I'm interested in whether you have topics that would be of interest to an average-to-bright 12-year-old. So far I've been doing a lot with prime numbers and how to use them with fractions.
As an avid dice collector I find this extremely interesting. Are there any potential plans in the works for the possibility of people being able to buy these?
I did search for and find the ones for up to 4 players available to buy, presumably much simpler to make than the bigger ones.
no
I understood parts of explanation. Pass on the complements to James grime. So UA-cam needs to provide ways in which one can contact not only the video maker but also people in the video.
Interesting that none of the dice are allowed duplicate numbers. Obviously you cannot have repeats on multiple dice, other wise you could get a collision, but if there is a die that has consecutive numbers, those could both be the same number. I wonder what the lowest value you could have, starting at 1, with consecutive numbers on a single die being allowed to be duplicated.
I like that idea. For instance, that would reduce the amount of numbers in the configuration at 7:07 from 1-18 to 1-15, with sides like this:
A- 2, 4, 6, 10, 12, 14
B- 3, 3, 7, 9, 11, 15
C- 1, 5, 8, 8, 13, 13
Any place where duplicating the number will solve the problem, you can replace those duplicates with their sequential values, making sure that every side that is higher is also bumped.
You can do the same in reverse with the existing dice to 'compress' the number of numbers used, if any single die has sequential values.
I love how James always looks happy
"team of dice experts" sounds badass
Never thought i'd find someone who could get more excited about dice than me.
Using factorial base, you can roll a single die (or combo). For 5 players, d12 and d10. Suppose you get 103. That's 4*24 + 1*6 + 0*2 + 1*1 => 4101 fac. The player four seats ahead of the roller goes first, skip one among the remaining players and the next one goes second, skip zero and the next goes third, skip one and the next goes fourth. (A result of 120 is just 0, so just clockwise order starting with the roller.)
I've always preferred his original series.
In particular, it is a video that contains content about the spiral law of prime numbers.
It is because of the genuine inquiring mind.
This video is closer to the kind of relaxation that viewers should do before engaging in new research.
I thought this was just right for brain exercise rather than content.
Truly mathematician way to avoid "oh we rolled the same, we who rolled the same have to roll again to decide which one of us goes first".
"That's not fun!"
The point was to avoid a never-ending process. Theoretically you could keep tying and never begin your game.
Why isn't anyone acknowledging how amazing it is that you can even choose one die and it'll still, with equal probability, determine that you go first.
question: can you show a certain set of dice has the lowest number of total sides and be permutation fair?
I've got a comment somewhere here where I solved that assuming that each dice has the same number of sides.
Go First Dice are awesome and I've been using them for years. At least for games with 4 or fewer players.
Is it just me, or does James have a Klein bottle bong?
In a game you could also use those whenever there is a decision to be made between 2 or more players, not just for determining order once. Whenever two or more players "battle" for something in a game, everyone involved could throw his die and it would never be a draw and is always fair. And it doesn't matter which or how many players battle against each other, there would always be 1 fair winner.
I actually searched for a set of 5 dice. I thought it would be very elegant if we could use the 5 Platonic solids to build the set. But I couldn’t find a solution as go first dice.
That can't work for a permutation fair set of dice. If you select the d4, the d8 and the d20, it's impossible to have the permutations for these three players be equally likely, since there is 4×8×20 possible outcomes, but 3! permutations, and 3! does not divide 4×8×20.
@@xfgjnsfgj nice observation. Since there’s no factor of 3 in 4,8,20, they can’t even give the “first player” equally likely with probability 1/3.
@@xfgjnsfgj You can solve that ny picking a dodecahedron dice though.
@@QuantumHistorian Unfortunately, that doesn't work. For all permutations of five players to be equally likely, then all sub-permutations of three players must also be equally likely. For a smaller example, if with two players, the order AB was more likely than BA, then increasing to three players would mean that the union of ABC, ACB and CAB must collectively be more likely than the union of BAC, BCA and CBA, and so there's no way to get those six permutations equally likely.
@@xfgjnsfgj I think you're confusing two different things. "For all permutations of five players to be equally likely, then all sub-permutations of three players must also be equally likely" is true, if you roll all the dice and then take the marginal sum by taking the union of orders where one of the players is ignored. That does not mean that removing any dice must result in a fair game with one fewer players. The two are different operations. The former is true, I see no reason to believe the latter must also be.
It's been a while. Time to catchup on some numberphile
I like how can you visually see him ageing but he still goes on about maths with the excitement of a little kid
I sent you a solution of mine on twitter. It is a comment under one of the video posted there. 5 (10 sided) dice values 1-51. High low pairs equal 51. 5 sets created that rotate them so high lowest is rotated across all dice. I did a colored spread sheet to make it simple to understand. Using that method you should be able to generate any number of go first dice easily.
i know that's not fair because there are 100000 possibilities for the roll and 120 possible permutations. if they were equal then there would be 833.3333 ways to roll each permutation, which is clearly ridiculous
Now you need to make a set of dice to decide who gets what shaped dice.
It's Go First Dice all the way down!
Great, constructive, useful. This sort of probabilitys will follow a multiplication similar as factorials 😂 just awesome
That last 5-dice solution seems like it's easily the best one to me. Less sides means they're smaller, which is far more practical in every way if you're actually planning to use these for their intended purpose. I'd much rather have each person throw a few ordinary 6-sided dice (perhaps of different colours) than have to deal with those 120-sided things.
Yeah, using such huge dice is kind of ridiculous. There are only 120 permutations of the numbers 1 through 5 without repeats... you could literally just roll _one_ d120 and then refer to a table to tell you what the player order should be.
@@FirstLast-gw5mg that's the solution mentioned at 13:27 in the video :p
i love that the catan game was tiled with the shirt tile!
Isn't putting 1-N in a bag and pulling turn orders always fair? Or is there some quirky statistical element I'm missing?
Why use five 1-sided dice that are pulled from a bag at random when you could have a d120 though.
@@FirstLast-gw5mg true, or even better why did I post a comment before finishing the video when they address my comment at the end lol 😅
For Bram's method there and the 6 player solution... you could just take that 20 sided die and split each side 18 ways. That turns it into another 360 sided die to match the other 5.
Presumably, you could then use Bram's method again to add another 7th die. Each would have thousands of sides, but it should work.
James ❤
If there is a solution for 6 dice with 1 20-sided die, and 5 360-sided dice, there is also a solution with 6 360-sided dice, by just dividing each of the sides of the smaller die into 18 sides with consecutive numbers.
People are tribalistic. Even though it's mathmatically proven that the dice are fair, the people using them will swear that the odd-one-out D20 is blessed or cursed. Probably the latter.
If only because it’s significantly easier to read and less likely to be bumped.
The ideal dice might be fair, but the physical dice probably have manufacturing defects.
oh you're killing me with the hat tiles in the video, I can't wait for your explanation of it:)
A twenty sided DIE. Four ten sided DICE.
The amount of permutations are mind boggling. Still a 1 in 5 chance of picking the dice with the highest number to begin with.
I'm lost at the idea that any position in the order could be equally likely but the permutations of the dice could be unequally likely. If the red die has an even 1:5 probability of being 1st, 2nd, 3rd, 4th, or 5th when you roll all 5 dice, and the same is true for the other 4 dice in the set, then how could the permutations end up skewed in one particular direction?
If ABCDE is more likely than ABCED and CBAED is equally more likely than CBADE, that’s one example of such a bias.
Suppose there are three players and the probabilities are
P(ABC) = P(BCA) = P(CAB) = 1/3
P(ACB) = P(CBA) = P(BAC) = 0
Then each player has a 1/3 chance of being 1st, 2nd, or 3rd but half the arrangements are impossible.
@@martinepstein9826 With fair dice, a scenario such as the one you described is impossible.
@@FirstLast-gw5mg My point is that, in principle, "place-fairness" does not imply "order-fairness". For a much more complex example that does come about through dice rolls you can work out the probabilities for one of the sets of dice in the video.
@@martinepstein9826 Well, it's not _just_ place fairness, it's also the fact that they're independent dice. If they're not independent dice it's easy to see that place fairness does not guarantee order fairness.
The sets of dice in the video are rather complex, is it possible to use a simpler scenario that demonstrates this discrepancy?
E.g. some other commenter suggested using a d1, d2, d4, etc., but when I looked at the probabilities for the 3-player "go first," this set of 3 dice (d1 = 4, d2 = 2/6, d4 = 1/3/5/7) didn't even guarantee place fairness, much less order fairness... there are 6 order permutations and 8 dice permutations, so just at a glance this couldn't give order fairness.
James Grime, are you reading the comments? I remember you like cryptography. Shuffling card and a single die with all the orders are centralized. The cards also not parallel, and everybody could look their position before the process is completed. But you can give these dies to different parties, they all roll independently, you can verify that the number they gave you is valid for the dice you gave them, and they are not sure of their own position in the order. We could further obscure the number behind a hash, and we have a secure distributed way to pick the order.
Too bad that it scales so badly. Decades of research just to add one more dice.
@@Pystro At least for the application I propose, keeping the dice small is not a strong requirement.
Looks like James discovered a Parker solution to the 5 die problem.
Such a wholesome video. Thanks Brady and James! :)