Puzzler develops game-changing Go First dice
A set of four dice that always roll a winner is a landmark advance
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Roll play … the smartest dice in the room. Photograph: Eric Harshbarger
Not since the 1971 Luke Rhinehart novel The Dice Man has there been someone so enthusiastic about the possibilities of dice.
Eric Harshbarger, a Lego sculptor from Auburn, Alabama, manufactures and sells dozens of types of dice, such as one that displays six different varieties of beverage, aimed at indecisive coffee shop customers.
Recently, however, one issue about dice really got him thinking.
When two or more people roll a die each in order to see who scores highest – what you do, for example, when deciding who goes first in a board game – there is always the chance of a tie.
In the event of a tie, of course, you roll again.
But then there is still the chance of a tie. And this can go on ad infinitum.
In other words, the process is not as efficient as it could be. Eric wondered if he could come up with a set of fair dice such that one roll of each die is enough to establish an absolute winner.
In devising a solution – and thus saving the board game players of the world hours and hours of lost time – Eric and a friend have made the greatest innovation in dice technology in recent years.
Their set of four "Go First" dice (pictured above) are such that when two, three or all four of the dice are rolled together:
1) no ties are possible.
2) each die has an equal chance of displaying the highest number.
Eric's friend is Robert Ford, a mathematics professor at Dalton State College, Georgia. Initially they were considering a set of eight cubic dice, but Robert worked out that it was impossible to have a set of cubic dice that satisfied the two conditions.
He then looked at a set of four dodecahedral dice – the 12-sided dice that are used in Dungeons & Dragons – and after a week found a solution, which include all the numbers from 1 to 48 with no repeats:
Die 1: 1, 8, 11, 14, 19, 22, 27, 30, 35, 38, 41, 48
Die 2: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47
Die 3: 3, 6, 12, 13, 17, 24, 25, 32, 36, 37, 43, 46
Die 4: 4, 5, 9, 16, 20, 21, 28, 29, 33, 40, 44, 45
These dice satisfy Eric's requirements: if you roll any subset of them, each die has an equal chance of winning.
They also have the nice property that you can arrange each dice such that the sum of opposite sides is always 49, in the same way that in cubic dice the sum of opposite sides is always 7.
Yet the Go First dice have an even more mathematically pleasing – and useful – property.
When any three of the dice are rolled together, each die has an equal chance of being first, second or third, and when all four dice are rolled together each die has an equal chance of being first, second, third or fourth.
If you want to check, Eric has listed all possible outcomes of all possible combinations here.
This is neat. Now before playing a board game with four players, a single throw of each dice will give you a fair order of play.
In fact, in the many scenarios in life when you want to order four things randomly but fairly – such as organising a rota – the dice allow you to do so.
You can buy a set of Go First dice here.
The solution above is not unique. Eric has subsequently discovered another 7 sets of four dice that satisfy his initial conditions (and also the stricter conditions of an equal chance of coming second, third or fourth).
The total number of solutions, however, is still an open problem.
Eric Harshbarger, a Lego sculptor from Auburn, Alabama, manufactures and sells dozens of types of dice, such as one that displays six different varieties of beverage, aimed at indecisive coffee shop customers.
Recently, however, one issue about dice really got him thinking.
When two or more people roll a die each in order to see who scores highest – what you do, for example, when deciding who goes first in a board game – there is always the chance of a tie.
In the event of a tie, of course, you roll again.
But then there is still the chance of a tie. And this can go on ad infinitum.
In other words, the process is not as efficient as it could be. Eric wondered if he could come up with a set of fair dice such that one roll of each die is enough to establish an absolute winner.
In devising a solution – and thus saving the board game players of the world hours and hours of lost time – Eric and a friend have made the greatest innovation in dice technology in recent years.
Their set of four "Go First" dice (pictured above) are such that when two, three or all four of the dice are rolled together:
1) no ties are possible.
2) each die has an equal chance of displaying the highest number.
Eric's friend is Robert Ford, a mathematics professor at Dalton State College, Georgia. Initially they were considering a set of eight cubic dice, but Robert worked out that it was impossible to have a set of cubic dice that satisfied the two conditions.
He then looked at a set of four dodecahedral dice – the 12-sided dice that are used in Dungeons & Dragons – and after a week found a solution, which include all the numbers from 1 to 48 with no repeats:
Die 1: 1, 8, 11, 14, 19, 22, 27, 30, 35, 38, 41, 48
Die 2: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47
Die 3: 3, 6, 12, 13, 17, 24, 25, 32, 36, 37, 43, 46
Die 4: 4, 5, 9, 16, 20, 21, 28, 29, 33, 40, 44, 45
These dice satisfy Eric's requirements: if you roll any subset of them, each die has an equal chance of winning.
They also have the nice property that you can arrange each dice such that the sum of opposite sides is always 49, in the same way that in cubic dice the sum of opposite sides is always 7.
Yet the Go First dice have an even more mathematically pleasing – and useful – property.
When any three of the dice are rolled together, each die has an equal chance of being first, second or third, and when all four dice are rolled together each die has an equal chance of being first, second, third or fourth.
If you want to check, Eric has listed all possible outcomes of all possible combinations here.
This is neat. Now before playing a board game with four players, a single throw of each dice will give you a fair order of play.
In fact, in the many scenarios in life when you want to order four things randomly but fairly – such as organising a rota – the dice allow you to do so.
You can buy a set of Go First dice here.
The solution above is not unique. Eric has subsequently discovered another 7 sets of four dice that satisfy his initial conditions (and also the stricter conditions of an equal chance of coming second, third or fourth).
The total number of solutions, however, is still an open problem.
