Mathematicians invent five-player dice that guarantee a winner — no ties
New type of dice guarantees no tie when deciding who goes first
A 15-year quest to create dice that decide who goes first in a board game without ties has finally succeeded for five players. Mathematician Eric Harshbarger and a team of researchers, including Canadian software engineer Paul Meyer, designed a set of five 60-sided dice that are statistically fair and always produce a unique winner from a single roll. The challenge was translating a mathematical solution into physical dice that could be manufactured. The discovery came after Meyer used a brute-force computer search to find a practical design, reducing the number of sides from a theoretical 1,440 to 60.
“Do you understand that we've been looking for this for a long time?”
- madibo3156
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.
- throw0101a
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:
- tzs
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 […]
- bmenrigh
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.
- kbelder
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.
- nkmnz
Reminds me of how my wife, at the start of the pandemic, beat me TWELVE TIMES IN A ROW in rock-paper-scissors for going first playing Azul. That's like... anyways, she lost all of the Azul games, so I guess we're even.
- toast0
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.
- orlp
If all you have a coin there is a simple algorithm that's equivalent to sorting by random real numbers in [0, 1].
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.
It's not a finite process though - it could go on forever if really unlucky. But this is unavoidable, since the number of permutations on n players with n > 2 has factors not divisible by 2 there is no finite series of n coin tosses that could without any bias create a permutation, as the number of outcomes is 2^n.