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.
JackFr 20 minutes ago [-]
Article was really poor in stating this clearlly, to the extent it made no sense to me. Thank you.
kbelder 24 minutes ago [-]
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.
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?
bmenrigh 2 hours ago [-]
A set of permutation fair dice work for any subset of dice and players. So a 4-dice perm-fair set allows any three dice to be used and is guaranteed to be perm-fair for 3 as well.
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.
trhway 1 hours ago [-]
there seems to be no need to match number of dice with number of players - for example in 2 players the second player throwing second die (2,3) is really a noop as outcome is decided by the first player's die (1,4)
bmenrigh 2 hours ago [-]
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.
Petersipoi 2 hours ago [-]
Only 70 years? If that's the case, why isn't it solved yet? Just rent 4,000 computers and have it done in less than a week.
bmenrigh 2 hours ago [-]
Well I’m first exhausting some promising regions of the search space which should complete in less than 2 weeks. After that I strongly suspect a solution doesn’t exist (for the column grouped space I’m searching).
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.
bawolff 27 minutes ago [-]
If you're being serious, i assume its because its primarily a novelty problem, and the economic investment of 4000 computers isn't worth it just for the bragging rights of solving the problem.
jonhohle 37 minutes ago [-]
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.
eru 2 minutes ago [-]
I think the (implied) constraint is that each person gets to roll one die, and the highest die roll wins.
e12e 3 hours ago [-]
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.
thaumasiotes 3 hours ago [-]
> I think it's more about guaranteeing the whole sequence, rather than who goes first?
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.
mrgoldenbrown 2 hours ago [-]
From a site selling the dice: "With these dice all players will have an equal chance of ranking first, second, third and fourth. And only one roll is required."
Wikipedia distinguishes between "go first fair" and "permutation fair". I believe Harshbarger wanted to find permutation fair dice.
abrookewood 2 hours ago [-]
Pretty sure he means this:
- assume heads beats tails
- if both players get opposite results (HT, TH) then the winner is obvious
- if both players get heads, player A is the winner
- if both players get tails, player B is the winner.
physicalecon 2 hours ago [-]
[dead]
archargelod 2 hours ago [-]
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.
fnordpiglet 29 minutes ago [-]
Because the question isn’t can you make a rice with 5! Sides but can you make it so 5 players can roll simultaneously individual dice with a minimal number of sides per dice and never tie and always be fair.
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.
kmoser 5 minutes ago [-]
From the article: "During a dinner conversation at a gaming convention in 2012, 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."
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.
scosman 19 minutes ago [-]
I wanted to say "You actually can't fly a helicopter to the top of Everest, the air is too thin", but apparently it's been done exactly twice. Stripped down specially chopper and ideal weather conditions.
pvillano 2 hours ago [-]
120 sided dice roll for a long time and don't have much room for printing
2 hours ago [-]
mcphage 2 hours ago [-]
> Can somebody explain why not just make a die with 5! sides, and roll it once to decide the order?
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…
Zebfross 49 minutes ago [-]
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
mckn1ght 31 minutes ago [-]
According to the wikipedia article linked at https://news.ycombinator.com/item?id=49559416 this discovery is from around the same time as that video came out. This submission should be tagged (2023).
ETA: Oh I just looked at the linked article again and it was just published. Weird.
27 minutes ago [-]
selcuka 3 hours ago [-]
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.”
I think it is not a matter of whether it is more of a "physical problem" or a "mathematical one". They knew a solution existed but it was physically unfeasible, which prompted them to mathematically optimize the solution by using fewer sides. They found a solution using only 120 sides, and a lower bound of 30 is known but brute-forcing it is still computationally expensive.
bombcar 3 hours ago [-]
Reword it as “the smallest possible set where X is still true” and you combine both.
I think there have been discussions about some of these sets here as well.
ninjalanternshk 58 minutes ago [-]
What a cool company. They’ve got the coolest nerdiest things.
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.
margalabargala 3 hours ago [-]
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?
arjvik 3 hours ago [-]
This is not exactly perfectly fair - appears that the first player wins 11/27 times and the other two win 8/27 each.
margalabargala 3 hours ago [-]
Ah. Now that I go through it it makes sense. You're right.
pfedak 3 hours ago [-]
You might want to check that your proposed solution works at all before suggesting the problem is trivial. What's the probability that the first die goes first with your numbers?
You also can't generally "and so on" constrained combinatorial arrangements like this.
margalabargala 2 hours ago [-]
I certainly would have, had I suggested the problem trivial! I'll keep that in mind for if I suggest things are trivial rather than suggesting I lack understanding.
> 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 :)
whycome 2 hours ago [-]
Didn’t they do the exact opposite of suggesting the problem was trivial?
cwillu 3 hours ago [-]
Saying “I'm having trouble understanding” is not the same thing as “this is trivial”.
Dylan16807 2 hours ago [-]
"I'm having trouble understanding why this is so complicated" could be interpreted in a non-dismissive way but it defaults to dismissive. And when it comes with a supposed solution attached, and that solution is really simple, that reinforces it sounding dismissive.
margalabargala 2 hours ago [-]
That wasn't my intention.
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.
Petersipoi 2 hours ago [-]
Only the people who can't make eye contact with others interpreted your comment as hostile. Unfortunately, that describes a high number of people on this site.
40 minutes ago [-]
webstrand 2 hours ago [-]
I didn't read it as dismissive. I think the "Why doesn't this work?" at the end is key, implying they know they're probably wrong, but don't know why. It's pretty common to form and present hypothesis like this, hoping that anyone who knows the actual theory can easily provide a counterexample showing why it's wrong. It's not at all intended to be dismissive.
whycome 2 hours ago [-]
Why does that default to dismissive for you? It absolutely did not come off that way for me and I’m wondering if it’s because of experience or culture or something else.
2 hours ago [-]
leoqa 3 hours ago [-]
It didn’t mention the underlying theory? Is it just that these large dimensions reduce the collision probability?
trhaynes 3 hours ago [-]
My naive first idea was that (for two people) one set would have even numbers and the other odds. But then the even number person is more likely to win. So it's something around which numbers are on which dice.
ixwt 3 hours ago [-]
If you had two dice, one with odds 1->11, and the other 2->12, and you made 1 beat 12, they would have even chances of winning then, right?
bombcar 3 hours ago [-]
I was wondering that too - it seems you start with a linked list of sorts, and then evenly distribute the links to the dice. But I’ve obviously not thought it out.
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.
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:
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.* 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
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 “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.
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.
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.
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.
Wikipedia distinguishes between "go first fair" and "permutation fair". I believe Harshbarger wanted to find permutation fair dice.
Especially, that 120-sided dice are already invented and commercially available.
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…
ETA: Oh I just looked at the linked article again and it was just published. Weird.
> 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.
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.
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 :)
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.
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?