New type of dice guarantees no tie when deciding who goes first (cbc.ca)

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.

* 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

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

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

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?

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.

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.

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.

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

> 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...

(no affiliation)

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

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?

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

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?