This video has been years in the making actually. I'm quite excited about this video, it's a problem
that I've actually got a part in as well. I've been given permission to give you an exclusive look at
this new solution - it's a secret! It's not a secret, I'm going to tell you all. So in this box we have
our new solution to this old problem; and building up to this I think I should start with what the
problem is of course. Imagine four friends and they're going to play a game and they want to
decide who goes first, second, third and fourth - fine. So to do that they decide to roll dice. They roll
one each and whoever has the highest number goes first, second highest number goes second; you get
the idea. The only problem with that is two players might roll the same number - oh no, a draw. So what
are they going to do? They could roll again. Well potentially that could go on forever. Is there a
set of four dice, so each player can have one, roll them together; there are no drawers and all equally
likely to be first, second, third, and fourth. That's the question. Well that does exist.
They're called Go First dice; they were invented by Eric Harshbarger and Robert Ford about 10 years ago.
I've got these actually, so here they are. So these are four 12-sided dice. They use the numbers from 1
to 48 so there's no repeats. So you're not going to get a draw, that's easily solved. And they've
been designed so that you're equally likely to be first, second, third, and fourth - very nice.
As a bonus though, it also works for subsets. So if you had three friends they could pick
any three of these dice and that would do the same thing; they would be equally likely
to be first, second, and third; or if you have two friends they could pick any two of these dice
and you'd be equally likely to be first and second - really nice. So the question then was, what's it
gonna look like for five players? Can we make a set of five dice: no draws and you're equally
likely to be first, second, third, fourth and fifth? That's the question we've got a new solution for
but before I show you that new solution I want to show you the best answers we've had so far - just to
build up to this. The first one I want to show you is one of my own. Okay, I get to do this, it's my video, I
can do that, right yeah okay. I don't know what you imagine it's going to look like but this is it.
So this is a set of 5 60 sided dice. They use the numbers 1 to 300 and, just like
they're supposed to, you roll them you're equally likely to be first, second, third, fourth and fifth.
Works for the subsets as well. So that's kind of nice so there you go. - (Brady: I can just see the numbers)
You see, I should have painted in the numbers - couldn't be bothered, couldn't be bothered. So
this is my solution. I came up with this uh years ago, not to be confused by the way with things like
non-transitive dice or other things you may have seen before. You know, it sounds similar but it is
a different problem, this is a different solution to that. So this is kind of nice. I might tell
you a little bit about how we came across it. So I did this with an internet friend of mine, Brian
Pollock, and well we did a computer search but just- to do a computer search would take a long
time. If you search through every possible set of five dice it's just a huge number, we're talking
like many many many Googols of possibilities, right. Too many to search through. So we did a
clever search. So what we did is we looked for dice with a particular nice mathematical
structure and that allowed us to reject the wrong answers really quickly. So we sort of narrowed our
search down to these particularly nice kind of dice. So you could reject the wrong ones easily,
then the ones you can't reject we checked. That's what we did, and that's what we found: we found the
60-sided dice. We knew that the answer- they're all the same size like these are - we knew that the dice
were going to be 30-sided or a multiple of 30. We didn't find 30-sided dice but we did find these
60-sided dice. So that's nice, I was proud of this! This was my solution to it; we can do better though.
Because this solution, you're equally likely to be first, second, third, fourth and fifth; what about if
we make every order of the players equally likely? So a, b, c, d, e is just as likely as e, d, c, b, a.
That might sound like the same thing - that's what I thought. When I said, is every order
equally likely or every place equally likely? - (As in who gets the highest, second highest-)
Yeah I thought that- isn't that the same? So no. Having every order is actually
a stronger sense of fairness. This set that I discovered, every place is equally likely,
you're equally likely to be first, second, third, fourth and fifth. Every order is not
equally likely with this. In fact they're a, b, c, d, e is less likely than e, d, c, b, a with this set.
(Quick question. Is every- what about with these ones?) - Every order is equally likely. It's
something we appreciated slightly after the fact. Oh, but should we make it every
order equally likely? Yeah so that's a downfall of those. I'm- like I said I, you know, we came up
with this idea, I'm quite proud of it, but can we make an even better set? Every order equally
likely, that would be nice. Now the original inventors of the Go First dice - that's Eric and
Robert - they came up with a solution. Put my set to one side, I'll show you Eric's solution.
Got it- d'you want me- you see I've got it in this box. - (Ok, it's a special box. )
Yeah, special box for it. Well, they look like this. - (Ooh) - Yeah, that's the right reaction
(You answered my next question actually, do the dice all have the same number of sides?) - So if you
relax that condition here is a set with all weird sides. So they're not all the same number of sides
but this is every order equally likely. We've got a 20-sided dice here; we've got a 36 sided dice;
here we've got two 48s and we've got a 54-sided dice. So this is what we've got. They're all kind
of weird values but every order equally likely. And true for the subsets! I might be worth telling
you how these were invented. The idea was you start with a set that you already know is permutation fair.
That's what we call it when all the orders are equally likely. So let's start with a set of
three dice. They're going to be 6-sided dice and they're going to have these numbers on them:
so let's say A here is going to be 2, 5, 7, 12, 14 and 17. B, I'll do B, is 3, 4, 8, 11, 13, 18.
And we'll do C which is 1, 6, 9, 10, 15 and 16. There we go, 6-sided
dice; they're using numbers 1 to 18, every order is equally likely. What we can do though
is turn that set of 3 into a set of 4. So this is this is how you do it. What we're going to
do first, we're going to double the size of these dice, right. So we're going to repeat it: 2, 5,
7, 12, 14, 17 - I'm trying to fit them into my piece of paper - do that again and for
C, okay 1, 6, 9, 10, 15 and 16. And so that we don't repeat the numbers, so we can tell them apart,
I'm going to add 100 to the first copy and 200 to the second copy so that now they're different
numbers. Look I've got gaps now; there are numbers less than 100 that I could use. I've got a gap
here between 118 and 201 - I could use that gap - and I've got a gap here larger than 218. So we're going
to create a new one called D. We're going to put values in those gaps and we can calculate how many
values to put in those gaps and keep the set of dice still permutation fair. In this example if you
put one number in the first gap; could be anything I want, call it 50. I could put 50 here, I could
put four numbers in that second gap - let's call it 150 and then 151, 152, 153, there you go. And then
one number in that last gap - 250? That will do it. That's now a set of four dice which will still be
permutation fair. We can shrink those numbers down as well so that it's all consecutive. But now the
new one I've introduced is now 6-sided; there's no guarantee that that new die will have the same
number of sides as the other dice that you started with and that's why they start to have weird sizes.
There are things you can do; like you can make the dice like have the lowest common multiple of that-
the dice start to get quite large though, the size of these dice do get quite large. It doesn't have to be two
copies either; sometimes you get a nicer set of dice if you do like three copies and then fill
those gaps. Or like five copies or ten copies - sometimes you just get a nicer result by doing
that instead. Now those two sets of dice I've just shown you were the best answers we had up to about
two months ago. And, Brady, I was going to make this video about two months ago and that was going to
be the end of the video. Just before I came and did that video uh we were contacted; our little team of
dice inventors were contacted by a mathematician called Michael Purcell and he said I've got a new
solution for you. Now I'm going to show you that, that's it, that's what I've been building up to!
(Oh, that's the box!) - That's the box over here. So what is it going to be? This is Michael's solution - you ready?
(I'm trembling with anticipation) - Right, here we go - this is it. This is a set of five
120-sided dice; all the same number of sides, which is nice, and every order is equally likely as well.
So I think this is one of my favorite sets so far. His ideas were completely different to the
ideas that we were using. He started with a set of five 12-sided dice, but they weren't completely
permutation fair. Any subset of four dice did work, it was only partially there. What he did next is he
made 10 copies of that, kind of like this method here. So he made those 12-sided dice into
120-sided dice. That doesn't completely solve it, they're still not permutation fair. What he
did next is in the copies he started to swap the rows over. So in the second copy he swapped
the first row and the third row; in the third copy he swapped the first row and the fifth row. So he
started doing some permutations as we call them on the rows within those copies that he made. By
doing that it kind of balanced out all the sort of biases, all the little problems, to sort of cancel
themselves out. And the result was, this set of five 120-sided dice where every order is equally likely.
The important thing about that is we kind of knew you could do something like that; there was a team
- actually they've got a YouTube channel as well so I'm going to give them a little bit of a shout out.
The polylog YouTube channel; they realised that if you did every permutation on the rows that would
balance out all the biases. They worked that out. But if you do every permutation these are going
to be huge dice. What Michael has done is he found that you could do it with just 10 permutations, not
just using every permutation. And there are some open questions, because he did it with a computer.
Is there some mathematical reason or mathematical way to pick the permutations that balance out all
those biases? We're not sure about that, we're still thinking about that. You know,
is is there a way to pick the initial seed set of dice - that initial set of dice before
we made copies? Is there a good way to pick that? Is there a good way to pick the permutations?
They're still problems we're working on but yes at least with a computer search we can
use those ideas and we can get this set of five 120-sided dice. (Two questions) - Yeah I'm ready
(All right. First question is, I'm not questioning your motivation for doing this, I think it's)
(really cool. But are Go First dice like necessary? Is there a demand in the gaming community for Go)
(First dice?) - No, no. So you can solve this problem in simpler ways. So if I wanted to arrange five
people in some order I would just have five cards with 1, 2, 3, 4, 5 written on
it and then you shuffle that up and they can pick a card. I mean that's that's that's not the point.
There are easier solutions to this. Another easy solution by the way to this is you could just have
one die with 120 sides on it. Each side would have an order of the five people and then roll that. You
could do it with one die; that's not the point either, that's not fun. The fun is five players
have one die each, and they all roll them, they all have something to do, and then it orders them.
That's the fun part of it. - (And the other obvious question that springs to mind is what about 6 dice?)
Now 6, yes. So Eric uh who originally invented this and is really like our boss in all of this;
he has a website where he's collating all like the best results. So for 6 shall I just- shall I just
look at what the answer for 6 is? Ohh, no I'll save that thought actually. Can I go back a step? Dun dun duh - cool.
There is a second punch line to this video because Michael contacted us with this new
result and we were so excited by it. Our little team of dice experts got working on it; well
one in particular Bram Cohen, and he started to mix and match our methods. So he started to
use a bit of Michael's method, a bit of Eric's method, started to match them together, just
to see if he could find a better solution - and this is his solution. So I've got another set!
This is a set of four 36-sided dice and one 20-sided dice. So it's almost like they're all
the same size, they're very similar sizes like oh, you're almost all the same size. Also the dice are just
so much smaller than than any of these solutions we've got so far. So if you were just looking at
adding up all the sides and seeing that- you know, trying to minimise that number, then in
that case by that measure uh this is the best set we've found. - (So that's more efficient but that one)
(there's more perfect because they're all the same) - Yeah. Some of us like to minimise the total number
of sides and some of us like to have them all the same - because it makes it more like a toy I
feel, but also minimise that number as well. So for 6 players the best results we've got now
um using actually Bram's method, the best result we've got a set which is a 20-sided die
and then five 360-sided dice. - (That's quite asymmetrical) - Yeah yeah yeah. Well it's-
and that's because of how Bram's method worked. So he actually used Michael's method to make these
set of dice, and then by induction he created one extra dice. Which is why this one has a different
size to all the other dice. So that's why this is the 20-sided dice there because
it's fitting in the gaps. So this is why it's a mix and match of of these two methods here.
Can we do better? Can we find a smaller set of dice? Personally I want to find a set of 5
30-sided dice. Theoretically we can make a set of five 30-sided dice but they might not exist. If they
don't exist can we make a set of five 60-sided dice? Or maybe this set of dice here, maybe this is
kind of minimum in its way - or is it? Again these are questions we still don't know the answers to.
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