Go First Dice Wiki

This wiki serves as a repository the active research about “Go First Dice”. As it will likely be authored by more individuals than Eric Harshbarger alone, the information should be more current, accurate, and inclusive than that which is available on Eric's original Go First Dice Webpage.

“Go First Dice” are a set of dice that allows some number of players to each roll a different single die (picked arbitrarily from the set) and the following conditions hold:

1. There will never be ties.
2. Each possible ordering of the players (determined by highest result, next highest result, etc) has a mathematically equal chance of occurring.
3. The above conditions hold for every subset of the whole set.

This wiki details the history of such sets of dice, some of the mathematics behind them, current sets of dice that satisfy the above conditions, what advances may still be made in the research, and even where some such sets may be purchased.

The concept of Go First Dice was first proposed to Eric Harshbarger during the summer of 2010 by James Ernest (founder of Cheapass Games). At the time James asked if Eric might be able to develop a set of dice (presumedly 6-sided) that would have “go first” properties. Both understood that this was really a “solution looking for a problem” (there are plenty of perfectly acceptable ways to quickly determine who might go first in a game), but mathematically, wouldn't it be great if a set of dice could be created such that players could each take any one of the dice and roll against one another such that:

1. There would not be a tie,
2. Each player has an equal chance of rolling highest (note this is a weaker condition than stated above in the Introduction – see the “Fairness” entry below),
3. And, we would like the above conditions to hold even if any subset of eight players roll against one another (for example, if only five players are participating, each would take a die, and roll and each would then have exacly a 1/5 chance of rolling the high number).

The first condition is easy to satisfy: simply use different numbers on each die. For example, if eight 6-sided dice were being used, just figure out how to distribute the numbers 1 through 48 across the forty-eight faces among all the dice. There could never be a tie then. The real work, of course, comes in satisfying the second and third conditions. And, it turns out that the original proposal (using eight 6-sided dice) is quickly proven to be mathematically impossible…

Below you will find more information about where all of these ideas have led.

• Explanations
  • Terminology
  • The Different Types of Fairness
  • Fast Perm-Checking code
  • “Weaving” Method for searching
  • Using a High Performance Computing Cluster
• Results
  • Significant Solutions
  • Homogeneous Solution Counts
  • Inhomogeneous Solution Counts
  • Induction Chart
  • Records
• More Information/External Links
• Wiki Participants