New type of dice guarantees no tie when deciding who goes first

It's not stated plainly in the article what the problem is, so here:

Each participant rolls a die. For there to be no possibility of a tie, no die can share a face number with another die—every face across all dice must be unique. For it to be fair, the distribution of numbers across all faces must be such that no die has an advantage over another die—the odds of rolling the highest number must be exactly the same for each die. The problem is in finding the combination of faces across five dice that satisfies these constraints. One difficulty of this is that each added player changes the whole equation—the odds get recalculated and new faces must be chosen. The secondary goal is to minimize the number of faces on the die.

This Wikipedia article goes over things:

* https://en.wikipedia.org/wiki/Go_First_Dice

As well as the pages of the project:

* http://gofirstdice.ericharshbarger.org/

A physical example of dice (USD 35):

* https://www.mathartfun.com/thedicelab.com/GFD5.html

* https://www.youtube.com/shorts/yMtTqiAhol8

* UK store: https://mathsgear.co.uk/collections/dice/products/go-first-d...

In addition to the above 5-player go first, they also have 4- and 3-player go first:

* https://www.mathartfun.com/dSpecial.html

Roll dice as if you're rolling a fractional base six number of indefinite precision, stopping when one player wins.

A roll of 5, 2, and 4 is treated like 5.24

So if Alex and Bob both roll a '3', they just keep extending the precision until one is higher.

You may ask what this gives you over just re-rolling ties. Well, this preserves order. For example, if several people are rolling initiative, and there's a few rerolls for ties, you may end up with this initiative sequence:

    John:  5
    Betsy: 4
    Alex:  3.16
    Bob:   3.15
    Phil:  2

If Alex and Bob had to reroll, the order can get confusing. It also gives you a magnitude: Betsy rolled 100% better than Phil, but Alex only came in 0.3% better than Bob.

Interesting. For 3 players this set of 3 6-sided dice would work:

  #1: 1 2 3 4 17 18
  #2: 5 6 7 14 15 16
  #3: 8 9 10 11 12 13

But if you had those 3 dice but only 2 players you could not just have each player grab one of them and roll. If one of them happened to grab #1 they would only win 1/3 of the time instead of the desired 1/2.

With 2 players they would have to use just #2 and #3.

That's because the way I came up with those numbers is as follows.

1. Number the players 1, 2, and 3. We want #1 to win exactly 1/3 of the time. We could do that by given them a 3-sided die 1 1 H1, where all the numbers on the other dice are lower than H1 and higher than 1.

2. In the cases where #1 rolls 1, we want #2 to win half the time. Give them a 2-sided die 2 H2 where the remaining die has all numbers between 2 and H2.

3. Assuming the remain die is also 2-sided we will need a total of 6 different numbers. Using 1-6 our set of dice is (1 1 6), (2 5), (3 4).

4. Most people would probably prefer that they all have the same number of sides instead of 3, 2, 2. LCM of those is 6, so double the 3-sided and triple the two 2-sided: (1 1 1 1 6 6), (2 2 2 5 5 5), (3 3 3 4 4 4).

5. People might object to having the same number more than once on a die. We have 18 total sides so lets renumber from 1-18. Our 4 1s become 1-4, our 3 2s become 5-7, and so on, given the set of 3 6-sided dice at the start.

It seems pretty clear that this generalizes to more than 3 players, with the more players the more sides the dice will have. But all of those suffer from that annoyance of needed to exclude specific dice when you are trying to decide the starting order for less than the maximum number of players.

Do the dice in the article avoid that annoyance? I have no idea how I would go about making something like that.

Also note that my dice only determine who goes first. It would be really nice if they could be used to determine complete order. Mine fail for that because #1 is always either the highest or the lowest.

It would be possible to use #1s number on a losing roll to give their place: 1 2 means they go second and 3 4 they go third. You could even print something on the dice saying that, but I think most people would find it more elegant if it was a simple highest goes first, second highest second, and so on.

Do the dice in the article do that, or are they also just solving the who goes first problem?

The big open question is whether a set of 5 (permutation fair) 30-sided dice exists.

I’ve been working on that on and off since 2012. I picked it up again about a month ago and have made dramatic speed improvements to my search, but exhausting the whole space I’m searching will still take my computer an estimated 70 years.

I think it's more about guaranteeing the whole sequence, rather than who goes first?

At least for two players, if you use a two sided die (a coin), have player one win ties on ones, player two win ties of twos - and otherwise highest wins - then that is trivially done?

I would have to do a little more math to see if it generalizes by induction... I'm not sure you would get a guaranteed sequence - but I think at least guaranteed fair winner works by just increasing the die (7, 9 and 11 would be tricky because if physics again... I suppose. Unless you just ignore highest tie for missing player (reroll on extremely rare 9 9s on a d10 for nine players)?

Ed: I suppose we break smaller ties, by letting closest and highest win (for ten players, 4, 6 and 7 roll 5 - 6 is closest and over/highest of the close players to 5, then come 7?)

Ed2: nevermind we end up biased towards "high" players that often win on "high" ties, like 5 or 6.

So it was more of a physical problem rather than a mathematical one [1]:

> Harshbarger says he and his colleagues always knew the dice were mathematically possible.

> The mystery was whether that mathematical solution could be translated into the physical geometry of a die — something that could actually be manufactured and rolled.

> “I knew there was a solution with something crazy like 1,440 sides for each die,” he said. “That's not makeable.”

[1] https://www.cbc.ca/radio/asithappens/dice-mystery-board-game...

I might be missing something, but a single 6 sided die can have all permutations for 3 players; a 24 sided die for 4 players; a 120 sided die has all permutations necessary for 5 players. Each one of these would even firmly support fewer players.

By the time you got to 120 sides you probably couldn’t label the sides with the order and would need a lookup table or something. That’s a disadvantage, I suppose.

For those as confused as me, I'm pretty sure this isn't "new". Numberphile talked about doing even better than this several years ago: https://www.youtube.com/watch?v=5q32heFz1bs

Can somebody explain why not just make a die with 5! sides, and roll it once to decide the order? With each side having a unique order printed e.g. 12345 -> 12354 -> ...

Especially, that 120-sided dice are already invented and commercially available.

If you want to buy these, they are commercially available https://mathartfun.com/dSpecial.html

(no affiliation)

I think there have been discussions about some of these sets here as well.

It didn’t mention the underlying theory? Is it just that these large dimensions reduce the collision probability?

I think I'm having trouble understanding why this is so complicated/requires so many sides.

If I imagine a 3-sided die, for simplicity, you should be able to have this result if the sets are [1,5,9],[2,6,7],[3,4,8]. And so on for larger numbers of players. Why doesn't this work?

Normal url: https://www.cbc.ca/radio/asithappens/dice-mystery-board-game...

Oh, Eric is a friend of mine. I've got a set of five non-uniform go first dice that he 3D printed for me.

Dude runs a mean D&D campaign too. And his Lego mural and sculpture portfolio are something to behold: http://www.ericharshbarger.org/lego/portfolio.html

Better source: http://www.ericharshbarger.org/dice/go_first_dice.html

TFA claims it's "new" in 2026, but the current state of the art seems to still be that of 2022.

I bought actual dice like these in 2024 from https://mathsgear.co.uk/collections/dice/products/go-first-d...

So well, is TFA just a big pile of slop?

You can just pull tokens out of a bag to generate a random sequence of virtually any length. Am I missing something? Why make it so complicated?

If anyone wants a practical way to decide who goes first, in a single roll, with no possibility of a tie, this works:

For 2 players, the dealer rolls a standard 6-sided die, and the result determines who goes first:

  1, 2, 3 => Player A
  4, 5, 6 => Player B

For 3 players, the dealer rolls a standard 6-side die:

  1, 2 => Player A
  3, 4 => Player B
  5, 6 => Player C

For 4 players, the dealer rolls a standard d20 die:

  1, 2, 3, 4, 5 => Player A
  6, 7, 8, 9, 10 => Player B
  11, 12, 13, 14, 15 => Player C
  16, 17, 18, 19, 20 => Player D

For 5 players, the dealer rolls a standard d20 die:

  1, 2, 3, 4 => Player A
  5, 6, 7, 8 => Player B
  9, 10, 11, 12 => Player C
  13, 14, 15, 16 => Player D
  17, 18, 19, 20 => Player E

If the dealer has a tetrahedral (d4) die, they can use that instead for the 4-player problem.

This is really all you need to decide who goes first. It's perfectly fair. It uses standard dice that most people already have in their game drawer.

The article is actually about a different, much harder problem about the full playing order. You don't actually need that extra complexity to decide who goes first.