**After years of work, I am excited to give you an exclusive look at our new solution to an old problem. Four friends want to decide who goes first, second, third and fourth, so they decide to roll dice. The problem is that two players might roll the same number, resulting in a draw. To solve this, there is a set of four 12-sided dice, called Go First Dice, invented by Eric Casparger and Robert Ford about 10 years ago. These dice use the numbers from 1 to 48, so there are no repeats and all are equally likely to be first, second, third and fourth. As a bonus, it also works for subsets.
The question then was, what would it look like for five players? Could we make a set of five dice, with no draws, and be equally likely to be first, second, third, fourth and fifth? We have a new solution for this, but before I show you, I want to show you the best answers so far.
My own solution is a set of five 60-sided dice, using the numbers 1 to 300. Just like they are supposed to, you roll them and you are equally likely to be first, second, third, fourth and fifth, and it works for the subsets as well.
To come up with this solution, I did this with an internet friend of mine, Brian Pollock. To do a computer search would take a long time, as it would search through many, many Googles of possibilities. To narrow down the search, we looked for dice with a particular mathematical structure, that allowed us to reject the wrong answers quickly. The ones we couldn’t reject, we checked and that’s what we found - the 60-sided dice. We knew that the answer was 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 is my solution to it. We can do better though, because this solution makes it so that you’re equally likely to be first, second, third, fourth, or 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 asked, “Is it every order equally likely or every place equally likely, as in who gets the highest, second highest, number right?” Yeah, I thought that isn’t that the same? 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, or fifth. Every order is not equally likely with this - in fact, A-B-C-D-E is less likely than E-D-C-B-A with this set.
Quick question - is every order equally likely with these ones? That’s something we appreciated slightly after the fact. Okay, but should we make it every order equally likely? Yeah, so that’s a downfall of those. I’m like I said, I came up with this idea, and 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. Yeah, special box for it? Yeah, well they look like this - oh, yeah, that’s the right reaction. You answered my next question actually, because 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 is 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 six-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 the 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 - six-sided dice, but his in numbers 1 to 18 - every order is equally likely.
What we can do though is turn that set of three into a set of four. This is how you do it - what we’re going to do first is double the size of these dice, right? So, we’re going to repeat it - 2, 5, 7, 12, 14, and 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, it could be anything I want - I call it 50. We were contacted by a mathematician called Michael Purcell who introduced us to a new solution: a set of five 120-sided dice, all the same number of sides, and every order is equally likely. This is one of our favorite sets so far, as it was completely different to the ideas we were using. Michael started with a set of five 12-sided dice that weren’t completely permutation fair, but any subset of four dice did work. He then made 10 copies of that set, resulting in a set of five 120-sided dice. bit more and some of us like to have all the same size dice and this is a nice balance between the two
A team of dice experts, led by Michael and Eric, found a way to create a set of five 120-sided dice where every order is equally likely. This was done by making copies of the dice, then swapping the rows in each copy. Brown Cohen then mixed and matched their methods to find an even better solution - a set of four 36-sided dice and one 20-sided dice. This set is more efficient, as it minimizes the total number of sides, while also having all the same size dice.