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
1, 2 => Player A
3, 4 => Player B
5, 6 => Player C
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
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
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.
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.
There are possibly other applications in navigation, radio communication, or other domains that might make this a more interesting problem than just figuring out game ordering.
Personally, I feel there is an easier approach: Take a regular 12-sided die, assign numbers modulo N (N being the amount of players). Now you assign the first turn fairly for 2, 3, 4, 6 or 12 players. Add a d20 and that covers 5 or 10 players, too.
That’s the pragmatic solution. The article is about an interesting mathematical puzzle, not actually solving a real problem.
Also, an n-sided die is fair for 2 to n players if you discard rolls above the number of players.
It used to be something people were corrected on, but these days dice is accepted as the singular too and die is starting to be seen as a little antiquated.
* 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:
#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.
If it were just a matter of a few thousand dollars of computer time (say, less than $5000) the money would already be spent and I’d have an answer.
We’ll see, I may build the tooling to distribute the search and enlist help from others interested.
It’s only been about 2 weeks since I was able to drop the runtime from “age of the universe” levels to just decades.
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. All players flips a coin.
2. Players that got heads go before players that got tails, forming (up to) two groups.
3. If a group has more than one player go back to #1 to determine the order within that group.
(no affiliation)
I think there have been discussions about some of these sets here as well.
I’m loving the non-transitive dice. My son’s birthday is in (checks calendar) ten months, but he’s getting these dice now anyway. So cool.
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.
What do you mean? The question is who goes first.
As a matter of practice, what happens in a board game is that everyone takes a position around the board before choosing who goes first. If turns proceed in a fixed sequence, that position will determine the sequence. If the order of turns is specified by the game (for example, many feature a turn order track), then that order will be used. You never need to decide on a sequence longer than one person.
But even if that wasn't the case, the article couldn't be more explicit:
> Eric Harshbarger was asked by a board game designer if he could come up with dice that would determine who goes first — without the possibility of a tie.
> The idea was simple: settle the first turn quickly and get on with the game.
This from the article appears somewhat questionable:
>> “It was a question that did not have an obvious answer and that's something that a mathematician will often jump at.”
The problem they're bragging about solving is using dice to quickly and unambiguously select one of five options with equal probability.
The obvious answer should be that you roll a single 10- or 20-sided die, divide by 2 or 4, round up, and there you have it.
> 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...
PopSci journalists must get so excited every time they get to write about a math result with a combinatorial flavor, letting them use their favorite "wow factor" phrase.
But for a mechanism to ship in a physical game? Big dice are difficult and expensive. Printing a deck of card that has the number 1 (or 0) to N on it will always be cheaper and has solved the problem for at least N up to 52.
Especially, that 120-sided dice are already invented and commercially available.
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.
The only problem with that is that you need to pick an order to roll the dice in order to pick the order to rank the players.
But happily, we have a set of dice that pick the order to roll the dice. Now we only need to pick an order to roll the dice to pick the order to roll the dice to pick the order to play.
But ...
ETA: Oh I just looked at the linked article again and it was just published. Weird.
They're not actually interested in determining who goes first or anything practical like that. The article even implicitly admits as much. After all, you could just use normal dice and roll again on ties or, like you suggested, come up with an extra protocol built on top of them.
"We want dice that determine who goes first without a tie" is just a catchier way of saying "We want N dice that all have the same probability of showing the higher number but zero probability of a tie," which is an interesting mathematical problem.
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?
You also can't generally "and so on" constrained combinatorial arrangements like this.
> You also can't generally "and so on" constrained combinatorial arrangements like this.
I know you can't generally but in the specific case I proposed you can (3 sets of 3, 4 sets of 4, 5 sets of 5, each dice taking one ordinal of each set).
You might want to check that your proposed generalization criticism applies to the generalization at all before suggesting the arrangement doesn't work :)
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?
It is a pile of slop and I am surprised that your comment is downvoted.
I don't know why would HN crown prefer this pile of journo fluff to the article by, you know, the guy who actually made the thing.
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
The “Go First” name is catchy for laypeople, but permutation fairness is the strongest and most interesting property.
There are sets we call “all subset place fair” which means any subset of the dice can be used and can fairly choose 1st, 2nd, and so forth, but this property is slightly weaker and doesn’t always make every ordering equally likely for every subset.
Wikipedia distinguishes between "go first fair" and "permutation fair". I believe Harshbarger wanted to find permutation fair dice.
Sure you can fly a helicopter to the top of Everest, but it’s not the same as climbing Everest even if the outcome is the same.
The original challenge was to design dice (I assume a single die would satisfy the requirement since the number of dice wasn't the point) that would quickly determine which of five players would go first. Nothing about the challenge required that there be five dice. A five-sided die would certainly be the simplest way, and would be marginally faster than having five players each roll a separate die and then compare the numbers.
Hey, that’s a good idea—now we just need to decide who rolls the die. …Well, if we had a set of die that each player could roll one of, and the highest number rolled gets to roll the 120-sided die. Of course, we’d have to ensure that there’s no chance of a tie.
Hmm… this idea has legs…
And for the pedants, you arent rolling every microsecond for millennia to make it plausibly likely that you have that many ties. You are rolling a handful of times before a game.
Its an interesting math problem to solve, but its way more effort to solve than the real world solutions.
I don't get the gotcha you're trying to do. Yes, the simple pattern you posted can be extended, but it has very little to do with the problem. If you had suggested [1,2,3] [4,5,6] [7,8,9] worked I would still have made my comment; I think the fact that you found a pattern that could be extended easily should have itself been a counter-indicator. Compare with something like the Fano plane[2].
I would want you to take away from this that posting the thing you came up with in "20 seconds" isn't as neutral a contribution to conversation as you thought. Things are often complicated and worth engaging with. It would have been cool if you kept going far enough to see there's a nice argument that it's actually impossible to solve it with n-sided dice for n people if n>2.
[1] https://xkcd.com/793/ [2] https://en.wikipedia.org/wiki/Combinatorial_design
Gee, you need a way to rank the players to decide.
> The golf-ball-sized dice are now available to buy ...
I agree the math problem does not map well to a real problem players are having.
You don't have to buy them to use them. For example I own a baggie of D(4-20) dice but I don't and have never played tabletop RPGs.
This set may end up too expensive for my "i know i'll never use it" budget, but other than that I'd get one.
I think it's an extraordinarily hard problem computationally, even with exceptional theoretical backing.
Where people may be able to beat me is if no solution exists-- there may be some highly nontrivial, but findable unsatisfiability argument.
I by no means was suggesting that the trivial solution I, a non-mathematician, thought up in 20 seconds was somehow out of reach to a math professor who spent years on the problem. I knew I was wrong.
I didn't see why until I actually went through the solutions by hand.
Flip one coin. If heads, player A is the winner. If tails, player B is the winner.
But apparently the constraints are: every player rolls one die once.
That doesn't eliminate the natural scheme, it just adds some useless die rolls. You can have each of two players flip a coin and choose player A when player A flips heads or player B when player A flips tails.